Non–commutative geometry, dynamics, and ∞–adic Arakelov geometry

Caterina Consani^{∗}and Matilde Marcolli ^{†}

We dedicate this work to Yuri Manin, with admiration and gratitude

Abstract

In Arakelov theory a completion of an arithmetic surface is achieved by enlarging the group of divisors by formal linear combinations of the “closed fibers at infinity”. Manin described the dual graph of any such closed fiber in terms of an infinite tangle of bounded geodesics in a hyperbolic handlebody endowed with a Schottky uniformization. In this paper we consider arithmetic surfaces over the ring of integers in a number field, with fibers of genus g ≥ 2. We use Connes’ theory of spectral triples to relate the hyperbolic geometry of the handlebody to Deninger’s Archimedean cohomology and the cohomology of the cone of the local monodromy N at arithmetic infinity as introduced by the first author of this paper. First, we construct a spectral triple, where the non–commutative space is given by the reduced C^{∗}–algebra of the Schottky group acting on the cohomology of the cone via a representation induced by the presence of a polarized Lefschetz module structure. In this setting we recover the alternating product of the Archimedean factors from a zeta function of the spectral triple. Then, we introduce a second spectral triple, which is related to Manin’s description of the dual graph of the fiber at infinity. Here the non–commutative space is a C^{∗}–algebra representing the “reduction mod infinity” and acting on a “dynamical homology and cohomology” pair, defined in terms of the bounded geodesics in the handlebody and of a dynamical system T. Morita equivalence plays an important role in the definition of the action. The operator Φ, that represents the “logarithm of a Frobenius–type operator” on the Archimedean cohomology, gives the Dirac operator on these spectral triples. The Archimedean cohomology embeds in the dynamical cohomology, compatibly with the action of a real Frobenius F̄_{∞}, so that the duality isomorphism on the cohomology of the cone of N corresponds to the pairing of dynamical homology and cohomology. This suggests the existence of a duality between the monodromy N and the dynamical map 1 − T.

∗Partially supported by NSERC grant 72016789 †Partially supported by Humboldt Foundation Sofja Kovalevskaja Award

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Contents

1 Introduction. 3 1.1 Preliminary notions and notation . . . . . . . . . . . . . . . . . . . . . 6

2 Cohomological Constructions. 9 2.1 A bigraded complex with monodromy and Lefschetz operators . . . . . 10 2.2 A polarized Hodge–Lefschetz structure . . . . . . . . . . . . . . . . . . 11 2.3 Cohomology groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14 2.4 Relations with Deligne’s cohomology . . . . . . . . . . . . . . . . . . . 17 2.5 Archimedean Frobenius and regularized determinants . . . . . . . . . . 22

3 Arithmetic spectral triple. 27 3.1 Spectral triples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 3.2 Hodge-Lefschetz modules and simultaneous uniformization . . . . . . . 32 3.3 Some zeta functions and determinants . . . . . . . . . . . . . . . . . . 39 3.4 Zeta function of the special fiber and Reidemeister torsion . . . . . . . 42

4 Shift operator and dynamics. 45 4.1 The limit set and the shift operator . . . . . . . . . . . . . . . . . . . . 46 4.2 Coding of geodesics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50 4.3 Cohomology and homology of S_{T} . . . . . . . . . . . . . . . . . . . . . 52 4.4 Cuntz-Krieger algebra . . . . . . . . . . . . . . . . . . . . . . . . . . . 56

5 Dynamical spectral triple. 60 5.1 Dynamical cohomology of the fiber at infinity . . . . . . . . . . . . . . 61 5.2 Dynamical homology and duality isomorphisms . . . . . . . . . . . . . 65 5.3 Hilbert completions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67 5.4 Representations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68 5.5 Reduction mod ∞ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72 5.6 The Dirac operator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73 5.7 Morita equivalence and spectral triples . . . . . . . . . . . . . . . . . . 75

6 Further structure at arithmetic infinity 76

1 Introduction.

The aim of this paper is to show how non–commutative geometry provides a connection between two constructions in Arakelov theory concerning the Archimedean fibers of a one-dimensional arithmetic fibration. On the one hand, we consider the cohomological construction introduced by the first author in [14], that was inspired by the theory of limiting mixed Hodge structures on the limit fiber of a geometric degeneration over a disc and it is related to Deninger’s archimedean cohomology and regularized determinants (cf. [18]). On the other hand, we re–interpret Manin’s geometric realization of the dual graph of the fiber at infinity as an infinite tangle of bounded geodesics inside a real 3–dimensional hyperbolic handlebody (cf. [24]) in the context of dynamical systems. The problem of relating the results of [24] with the cohomological constructions of Deninger was already addressed by Manin in [24], but to our knowledge no further progress in this direction was made since then.

Let 𝕂 be a number field and let O_{𝕂} be the ring of integers. The choice of a model X_{O_{𝕂}} of a smooth, algebraic curve X over 𝕂 defines an arithmetic surface over Spec(O_{𝕂}). A closed vertical fiber of X_{O_{𝕂}} over a prime ℘ in O_{𝕂} is given by X_{℘}: the reduction mod ℘ of the model. It is well known that a completion of the fibered surface X_{O_{𝕂}} is achieved by adding to Spec(O_{𝕂}) the Archimedean places represented by the set of all embeddings α : 𝕂 ↪ ℂ. The Arakelov divisors on the completion \overline{X_{O_{𝕂}}} are defined by the divisors on X_{O_{𝕂}} and by formal real combinations of the closed vertical fibers at infinity. Arakelov’s geometry does not provide an explicit description of these fibers and it prescribes instead a Hermitian metric on each Riemann surface X_{/ℂ}, for each archimedean prime α. It is quite remarkable that the Hermitian geometry on each X_{/ℂ} is sufficient to develop an intersection theory on the completed model, without an explicit knowledge of the closed fibers at infinity. For instance, Arakelov showed that intersection indices of divisors on the fibers at infinity are obtained via Green functions on the Riemann surfaces X_{/ℂ}.

Inspired by Mumford’s p-adic uniformization of algebraic curves [33], Manin realized that one could enrich Arakelov’s metric structure by a choice of a Schottky uniformization. In this way, the Riemann surface X_{/ℂ} is the boundary at infinity of a 3-dimensional hyperbolic handlebody 𝔛_{Γ}, described as the quotient of the real hyperbolic 3–space ℍ^{3} by the action of the Schottky group Γ. The handlebody contains in its interior an infinite link of bounded geodesics, which are interpreted as the dual graph of the closed fiber at infinity, thus providing a first geometric realization of that space.

A consequence of this innovative approach is a more concrete intuition of the idea that, in Arakelov geometry, the “reduction modulo infinity” of an arithmetic variety should be thought of as “maximally degenerate” (or totally split: all components are of genus zero). This is, in fact, the reduction type of the special fiber admitting a Schottky uniformization (cf. [33]).

In this paper we consider the case of an arithmetic surface over Spec(O_{𝕂}) where the fibers are of genus g ≥ 2. The paper is divided into two parts.

The first part consists of Sections 2 and 3. Here we consider the formal construction of a cohomological theory for the “maximally degenerate” fiber at arithmetic infinity, developed in [14]. Namely, the Riemann surface X_{/ℂ} supports a double complex (K^{·,·}, d′, d′′) endowed with an endomorphism N. This complex is made by real differential forms with certain “weight-type” conditions on the indices and was constructed as an Archimedean analogue of the one defined by Steenbrink on the semi-stable fiber of a degeneration over a disc [49]. The hyper-cohomologies of (K^{·}, d = d′ + d′′) and (Cone(N)^{·}, d) are infinite dimensional, graded real vector spaces. We show that their summands are isomorphic twisted copies of a same real de-Rham cohomology group of X. The arithmetic meaning of K^{·} arises from the fact that the cohomology of (Coker(N)^{·}, d) computes the real Deligne cohomology of X_{/ℂ}, and the regularized determinant of an operator Φ on the subspace ℍ^{·}(K^{·}, d)^{N=0} of the hyper-cohomology of the complex (K^{·}, d) recovers the Archimedean factors of [18]. The complex K^{·} carries an important structure of bigraded polarized Lefschetz module à la Deligne and Saito (cf. [43]). In particular one obtains an induced inner product on the hyper-cohomology and a representation of SL(2, ℝ) × SL(2, ℝ).

The first part of this paper concentrates on the cohomology H^{·}(X^{∗}) of Cone(N). In the classical case of a semi–stable degeneration over a disc, the cohomology H^{·}(|G|) of the dual graph of the special fiber can be described in terms of graded pieces, under the monodromy filtration on the cohomology of the geometric generic fiber, so that H^{·}(|G|) provides at least a partial information on the mixed Hodge structure on H^{·}(X^{∗}) (here X^{∗} denotes the complement of the special fiber in the model and the cohomology of this space has a second possible description as hypercohomology of the complex Cone(N)^{·}). In the arithmetic case of a degeneration on the ring of integers of a local field, the cohomology group H^{·}(X^{∗}) is still endowed with a graded structure, which is fundamental in arithmetic for determining the behavior of the local Euler factors at integer points on the left of the critical strip on the real line. In fact, the cohomology H^{·}(X^{∗}) carries more arithmetical information than just the cohomology of the dual graph of the special fiber H^{·}(|G|): think for instance of the relation between the torsion subgroup of gr_{2}^{w}H^{2}(X^{∗})(1) and the Tamagawa number.

Using non–commutative geometry, we link the cohomology H^{·}(X^{∗}) at arithmetic infinity as defined in [14] to the hyperbolic handlebody considered by Manin.

More precisely, we prove that the bigraded polarized Lefschetz module structure on the complex (K^{·}, d = d′ + d′′), together with the data of the Schottky uniformization of the Riemann surface X(ℂ) and of its hyperbolic filling 𝔛_{Γ}, induces a representation of the real (reduced) C^{∗}–algebra C^{∗}(Γ) of the Schottky group on the Hilbert space completion of H^{·}(X^{∗}) with respect to the inner product defined by the polarization on K^{·}. This data, together with the operator Φ that determines the Archimedean factors of [18], define the structure of a spectral triple (C^{∗}(Γ), H^{∗}(X^{∗}), Φ) à la Connes. In this theory, Φ has the role of a Dirac operator, hence it defines a metric structure over the non-commutative space C^{∗}(Γ).

The non-commutative space C^{∗}(Γ) has the same non-commutative topology (Ktheory) of the handlebody 𝔛_{Γ}. This can be seen through strong Morita equivalence and a result of Kasparov [22] relating C^{∗}(Γ) to the ring of functions on the handlebody 𝔛_{Γ}. The hyperbolic geometry of the handlebody is encoded in the Beltrami differentials of Bers’ simultaneous uniformization, which we use in order to construct a representation of C^{∗}(Γ) on H^{·}(X^{∗}) via the representation of SL(2, ℝ) × SL(2, ℝ) determined by the Lefschetz structure on K^{·}.

This result allows us to reinterpret Deninger’s regularized determinants describing the Archimedean factors in terms of an integration theory on the “non-commutative manifold” (C^{∗}(Γ), H^{∗}(X^{∗}), Φ). Theorem 3.17 shows that the alternating product of the Γ-factors L_{ℂ}(H^{q}(X_{/ℂ}, ℂ), s) is recovered from a particular zeta function of the spectral triple. In §3.4 we interpret this alternating product as a Reidemeister torsion associated to the fiber at arithmetic infinity.

The second part of the paper (Section 4 and 5) concentrates on Manin’s description of the dual graph G of the “fiber at infinity” of an arithmetic surface in terms of the infinite tangle of bounded geodesics in the hyperbolic handlebody 𝔛_{Γ}.

More precisely, the suspension flow S_{T} of a dynamical system T provides our model of the dual graph G of the fiber at infinity, which maps surjectively over the tangle of bounded geodesics considered in [24]. The map T is a subshift of finite type which partially captures the dynamical properties of the action of the Schottky group on its limit set Λ_{Γ}.

The first cohomology group of S_{T} is the ordered cohomology of the dynamical system T, in the sense of [8] [34] and it provides a model of the first cohomology of the dual graph of the fiber at infinity. The group H^{1}(S_{T}) carries a natural filtration, which is related to the periodic orbits of the subshift of finite type. We give an explicit combinatorial description of homology and cohomology of S_{T} and of their pairing.

We define a dynamical cohomology H_{dyn}^{1} of the fiber at infinity as the graded space associated to the filtration of H^{1}(S_{T}). Similarly, we introduce a dynamical homology H_{1}^{dyn} as the sum of the spaces in the filtration of H_{1}(S_{T}). These two graded spaces have an involution which plays a role analogous to the real Frobenius F̄_{∞} on the cohomological theories of Section 2.

Theorem 5.5 relates the dynamical cohomology to the Archimedean cohomology by showing that the Archimedean cohomology sits as a particular subspace of the dynamical cohomology in a way that is compatible with the grading and with the action of the real Frobenius. The map that realizes this identification is obtained using the description of holomorphic differentials on the Riemann surface as Poincaré series over the Schottky group, which also plays a fundamental role in the description of the Green function in terms of geodesics in [24]. Similarly, in Theorem 5.10 we identify a subspace of the dynamical homology that is isomorphic to the image of the Archimedean cohomology under the duality isomorphism acting on ℍ^{·}(Cone^{·}(N)). This way we reinterpret this arithmetic duality as induced by the pairing of dynamical homology and cohomology.

In Proposition 4.19 we show that the Cuntz–Krieger algebra O_{A} associated to the subshift of finite type T (cf. [15] [16]) acts on a completion of the cohomology H^{1}(S_{T}). This algebra carries a refined information on the action of the Schottky group on its limit set, and it induces an action by bounded operators on a Hilbert space completion H_{dyn}^{1} of the dynamical cohomology. In Proposition 5.13 we show that this representation extends to a representation of the C^{∗}-algebra A = (C(Λ_{Γ}) ⊗ C_{0}(ℍ^{3})) ⋊ Γ. As a non-commutative space, this algebra provides an analog at arithmetic infinity of the p-adic reduction map considered in [24] and [33], which is realized in terms of certain finite graphs in a quotient of the Bruhat-Tits building.

Similarly, in Proposition 5.16 we construct a representation, on a Hilbert space completion H_{1}^{dyn} of the dynamical homology, of a C^{∗}-algebra strongly Morita equivalent to A, which is the algebra of sections of a flat C(Λ_{Γ})-bundle over the handlebody 𝔛_{Γ}.

We introduce a Dirac operator D on the Hilbert space H_{dyn}^{1}⊕H_{1}^{dyn}, whose restriction to the subspaces isomorphic to the Archimedean cohomology and its dual, recovers the Frobenius-type operator Φ of Section 2.

Using Connes’ technique of transferring spectral geometry between Morita equivalent non-commutative spaces, we obtain a second spectral triple, which we call dynamical spectral triple, over the non-commutative space A acting on the Hilbert space H_{dyn}^{1} ⊕ M ⊗ H_{1}^{dyn}, where M is the bimodule implementing the Morita equivalence.

In the last section of the paper, we outline some possible further questions and directions for future investigations.

Since this paper draws from the language and techniques of different fields (arithmetic geometry, non-commutative geometry, dynamical systems), we thought it necessary to include enough background material to make the paper sufficiently self contained and addressed to readers with different research interests.

Acknowledgments. In the course of this project we learned many things from different people to whom we are very grateful: to Paolo Aluffi for many useful suggestions for §4.3, to Alain Connes for beautiful lectures on spectral triples and noncommutative geometry, to Curt McMullen for very enlightening conversations on the dynamics of the shift T and on Schottky groups, to Victor Nistor for various remarks on cross product algebras, to Mika Seppälä for useful information on real algebraic curves, and of course to Yuri Manin for sharing his vision and enlightenment on many aspects of this project.

1.1 Preliminary notions and notation

The three-dimensional real hyperbolic space ℍ^{3} is the quotient

ℍ^{3} = PGL(2, ℂ)/SU(2).  (1.1)

It can also be described as the upper half space ℍ^{3} ≃ ℂ × ℝ^{+} endowed with the hyperbolic metric.

The group PSL(2, ℂ) is the group of orientation preserving isometries of ℍ^{3}. The action is given by

γ : (z, y) ↦ (\frac{(az + b)\overline{(cz + d)} + ac̄y^{2}}{|cz + d|^{2} + |c|^{2}y^{2}} , \frac{y |ad − bc|}{|cz + d|^{2} + |c|^{2}y^{2}} ) ,  (1.2)

for (z, y) ∈ ℂ × ℝ^{+} and

γ = ( \begin{matrix}a & b \\ c & d \end{matrix} ) ∈ SL(2, ℂ).

The complex projective line ℙ^{1}(ℂ) can be identified with the conformal boundary at infinity of ℍ^{3}. The action (1.2) extends to an action on \overline{ℍ}^{3} := ℍ^{3} ∪ ℙ^{1}(ℂ), where PSL(2, ℂ) acts on ℙ^{1}(ℂ) by fractional linear transformations

γ : z ↦ \frac{(az + b)}{(cz + d)}.

We begin by recalling some classical facts about Kleinian and Fuchsian groups (cf.

[4] [7] [31]).

A Fuchsian group G is a discrete subgroup of PSL(2, ℝ), the group of orientation preserving isometries of the hyperbolic plane ℍ^{2}. A Kleinian group is a discrete subgroup of PSL(2, ℂ), the group of orientation preserving isometries of three-dimensional real hyperbolic space ℍ^{3}.

For g ≥ 1, a Schottky group of rank g is a discrete subgroup Γ ⊂ PSL(2, ℂ), which is purely loxodromic and isomorphic to a free group of rank g. Schottky groups are particular examples of Kleinian groups.

A Schottky group that is specified by real parameters so that it lies in PSL(2, ℝ) is called a Fuchsian Schottky group. Viewed as a group of isometries of the hyperbolic plane ℍ^{2}, or equivalently of the Poincaré disk, a Fuchsian Schottky group G produces a quotient G\ℍ^{2} which is topologically a Riemann surface with boundary.

In the case g = 1, the choice of a Schottky group Γ ⊂ PSL(2, ℂ) amounts to the choice of an element q ∈ ℂ^{∗}, |q| < 1. This acts on ℍ^{3} by

( \begin{matrix}q^{1/2} & 0 \\ 0 & q^{−1/2} \end{matrix} ) (z, y) = (qz, |q|y).

One sees that 𝔛 = ℍ^{3}/(q^{ℤ}) is a solid torus with the elliptic curve X_{/ℂ} = ℂ^{∗}/(q^{ℤ}) as its boundary at infinity. This space is known in the theory of quantum gravity as Euclidean BTZ black hole [28].

In general, for g ≥ 1, the quotient space

𝔛_{Γ} := Γ\ℍ^{3}  (1.3)

is topologically a handlebody of genus g. These also form an interesting class of Euclidean black holes (cf. [28]).

We denote by Λ_{Γ}, the limit set of the action of Γ. This is the smallest non–empty closed Γ–invariant subset of ℍ^{3}∪ℙ^{1}(ℂ). Since Γ acts freely and properly discontinuously on ℍ^{3}, the set Λ_{Γ} is contained in the sphere at infinity ℙ^{1}(ℂ). This set can also be described as the closure of the set of the attractive and repelling fixed points z^{±}(g) of the loxodromic elements g ∈ Γ. In the case g = 1 the limit set consists of two points, but for g ≥ 2 the limit set is usually a fractal of some Hausdorff dimension 0 ≤ δ = dim_{H}(Λ_{Γ}) < 2.

We denote by Ω_{Γ} the domain of discontinuity of Γ, that is, the complement of Λ_{Γ} in ℙ^{1}(ℂ). The quotient

X_{/ℂ} = Γ\Ω_{Γ}  (1.4)

is a Riemann surface of genus g and the covering Ω_{Γ} → X_{/ℂ} is called a Schottky uniformization of X_{/ℂ}. Every complex Riemann surface X_{/ℂ} admits a Schottky uniformization.

The handlebody (1.3) can be compactified by adding the conformal boundary at infinity X_{/ℂ} to obtain

\overline{𝔛}_{Γ} := 𝔛_{Γ} ∪ X_{/ℂ} = Γ\(ℍ^{3} ∪ ℙ^{1}(ℂ)).  (1.5)

Let {g_{i}}_{i=1}^{g} be a set of generators of the Schottky group Γ. We write g_{i+g} = g_{i}^{−1}. There are 2g Jordan curves γ_{k} on the sphere at infinity ℙ^{1}(ℂ), with pairwise disjoint interiors D_{k}, such that the elements g_{k} are given by fractional linear transformations that map the interior of γ_{k} to the exterior of γ_{j} with |k − j| = g. The curves γ_{k} give a marking of the Schottky group.

The choice of a Schottky uniformization for the Riemann surface X_{/ℂ} provides a choice of a set of generators a_{i}, i = 1, … g, for Ker(I_{∗}), where I_{∗} : H_{1}(X_{/ℂ}, ℤ) → H_{1}(𝔛_{Γ}, ℤ) is the map induce by the inclusion of X_{/ℂ} in \overline{𝔛}_{Γ} as the conformal boundary at infinity. The a_{i} are the images under the quotient map Ω_{Γ} → X_{/ℂ} of the Jordan curves γ_{i}.

Recall that, if 𝕂 is a number field with n = [𝕂 : ℚ], there are n archimedean primes which correspond to the embeddings α : 𝕂 ↪ ℂ. Among these n archimedean primes, there are r embeddings into ℝ, and s pairs of conjugate embeddings in ℂ not contained in ℝ, so that n = r + 2s.

If X is an arithmetic surface over Spec(O_{𝕂}), then at each archimedean prime we obtain a Riemann surface X_{/ℂ}. If the archimedean prime corresponds to a real embedding, the corresponding Riemann surface has a real structure, namely it is a smooth real algebraic curve X_{/ℝ}.

A smooth real algebraic curve X_{/ℝ} is a Riemann surface X_{/ℝ} together with an involution ι : X_{/ℝ} → X_{/ℝ} induced by complex conjugation z ↦ z̄. The fixed point set X_{ι} of the involution is the set of real points X_{ι} = X_{/ℝ}(ℝ) of X_{/ℝ}. If X_{ι} ≠ ∅, the components of X_{ι} are simple closed geodesics on X_{/ℝ}. A smooth real algebraic curve is called orthosymmetric if X_{ι} ≠ ∅ and the complement X_{/ℝ}\X_{ι} consists of two connected components. If X_{ι} ≠ ∅, then X_{/ℝ} can always be reduced to the orthosymmetric case upon passing to a double cover.

Even when not explicitly stated, all Hilbert spaces and algebras of operators we consider will be separable, i.e. they admit a dense (in the norm topology) countable subset.

An involutive algebra is an algebra over ℂ with a conjugate linear involution ∗ (the adjoint) which is an anti-isomorphism. A C^{∗}-algebra is an involutive normed algebra, which is complete in the norm, and satisfies ∥ab∥ ≤ ∥a∥ · ∥b∥ and ∥a^{∗}a∥ = ∥a∥^{2}. The analogous notions can be defined for algebras over ℝ.

The condition of strong Morita equivalence between two C^{∗}-algebras A_{1} and A_{2} means that there exists a bimodule M, which is a right Hilbert module over A_{1} with an A_{1}-valued inner product ⟨·, ·⟩_{A_{1}}. and a left Hilbert module over A_{2} with an A_{2}- valued inner product ⟨·, ·⟩_{A_{2}}, such that the following properties hold:

1. The closure of the linear span of

{⟨ξ_{1}, ξ_{2}⟩_{A_{i}} : ξ_{1}, ξ_{2} ∈ M}

gives all of A_{i}, for both i = 1, 2.

2. The two inner products satisfy

⟨ξ_{1}, ξ_{2}⟩_{A_{1}}ξ_{3} = ξ_{1}⟨ξ_{2}, ξ_{3}⟩_{A_{2}}.

for all ξ_{1}, ξ_{2}, ξ_{3} ∈ M.

3. The algebras A_{1} and A_{2} act on M by bounded operators, namely

⟨a_{2}ξ, a_{2}ξ⟩_{A_{1}} ≤ ∥a_{2}∥^{2}⟨ξ, ξ⟩_{A_{1}} ⟨a_{1}ξ, a_{1}ξ⟩_{A_{2}} ≤ ∥a_{1}∥^{2}⟨ξ, ξ⟩_{A_{2}}

for all a_{1} ∈ A_{1}, a_{2} ∈ A_{2}, ξ ∈ M.

A Hilbert module M for a C^{∗}-algebra A is finite projective if there exists an idempotent P in the matrix algebra M_{N}(A), P^{2} = P, such that M = P A^{N}.

2 Cohomological Constructions.

In this chapter we give an explicit description of a cohomological theory for the archimedean fiber of an Arakelov surface. The general theory, valid for any arithmetic variety, was defined in [14]. This construction provides an alternative definition and a refinement for the archimedean cohomology H_{ar}^{∗} introduced by Deninger in [18]. The spaces H^{·}(X̃^{∗}) (cf. Definition 2.8) are infinite dimensional real vector spaces endowed with a monodromy operator N and an endomorphism Φ (cf. Section 2.5). The groups H_{ar}^{∗} can be identified with the subspace of the N-invariants (i.e. Ker(N)) over which (the restriction of) Φ acts in the following way. The monodromy operator determines an integer, even graduation on H^{·}(X̃^{∗}) = ⊕_{p∈ℤ}gr_{2p}^{w}H^{·}(X̃^{∗}) where each graded piece is still infinite dimensional. We will refer to it as to the weight graduation. This graduation induces a corresponding one on the subspace H^{·}(X̃^{∗})^{N=0} := ⊕_{·≥2p}gr_{2p}^{w}H^{·}(X^{∗}). The summands gr_{2p}^{w}H^{·}(X^{∗}) are finite dimensional real vector spaces on which Φ acts as a multiplication by the weight p.

When X_{/κ} is a non-singular, projective curve defined over κ = ℂ or ℝ, the description of gr_{2p}^{w}H^{·}(X^{∗}) (· ≥ 2p) is particularly easy. Proposition 2.23 shows that H^{·}(X̃^{∗})^{N=0} is isomorphic to an infinite direct sum of Hodge-Tate twisted copies of the same finite-dimensional vector space. For κ = ℂ, this space coincides with the de-Rham cohomology H_{DR}^{∗}(X_{/ℂ}, ℝ) of the Riemann surface X_{/ℂ}.

For the reader acquainted with the classical theory of mixed Hodge structures for an algebraic degeneration over a disk (and its arithmetical counterpart theory of Frobenius weights), it will be immediately evident that the construction of the arithmetical cohomology defined in this chapter runs in parallel with the classical one defined by Steenbrink in [49] and refined by M. Saito in [43]. The notation: H^{·}(X̃^{∗}), H^{·}(X^{∗}), H^{·}(Y) followed in this section is purely formal. Namely, X̃^{∗}, X^{∗} and Y are only symbols although this choice is motivated by the analogy with Steenbrink’s construction in which X̃^{∗}, X^{∗} and Y describe resp. the geometric generic fiber and the complement of the special fiber Y in the model. The space H^{·}(X̃^{∗}) is the hypercohomology group of a double complex K^{·,·} of real, differential twisted forms (cf. Section 2.1: (2.1)) on which one defines a structure of polarized Lefschetz module that descends to its hypercohomology (cf. Theorem 2.6 and Corollary 2.7).

The whole theory is inspired by the expectation that the fibers at infinity of an arithmetic variety should be thought to be semi-stable and more specifically to be ‘maximally degenerate or totally split’. We like to think that the construction of the complex K^{·,·} on the Riemann surface X_{/κ}, whose structure and behavior gives the arithmetical information related to the ‘mysterious’ fibers at infinity of an arithmetic surface, fits in with Arakelov’s intuition that Hermitian geometry on X_{/κ} is enough to recover the intersection geometry on the fibers at infinity.

2.1 A bigraded complex with monodromy and Lefschetz operators

Let X_{/κ} be a smooth, projective curve defined over κ = ℂ or ℝ. For a, b ∈ ℕ, we shall denote by (A^{a,b} ⊕ A^{b,a})_{ℝ} the abelian group of real differential forms (analytic or C^{∞}) on X_{/κ} of type (a, b) + (b, a).

For p ∈ ℤ, the expression (A^{a,b} ⊕ A^{b,a})_{ℝ}(p) means the p-th Hodge-Tate twist of (A^{a,b} ⊕ A^{b,a})_{ℝ}, i.e.

(A^{a,b} ⊕ A^{b,a})_{ℝ}(p) := (2π\sqrt{−1})^{p}(A^{a,b} ⊕ A^{b,a})_{ℝ}.

Let i, j, k ∈ ℤ. We consider the following complex (cf. [14], §4 for the general construction)

K^{i,j,k} = \begin{cases}⨁_{\begin{matrix}a+b=j+1 \\ |a−b|≤2k−i \end{matrix}}(A^{a,b} ⊕ A^{b,a})_{ℝ}(\frac{1 + j − i}{2}) & if 1 + j − i ≡ 0(2), k ≥ max(0, i) \\ 0 & otherwise. \end{cases}

(2.1)

On the complex K^{i,j,k} one defines the following differentials

d′ : K^{i,j,k} → K^{i+1,j+1,k+1};  d′′ : K^{i,j,k} → K^{i+1,j+1,k}
d′ = ∂ + \overline{∂}  d′′ = \sqrt{−1}(\overline{∂} − ∂).

These maps satisfy the property that d′^{2} = 0 = d′′^{2} (cf. [14] Lemma 4.2). Since X_{/κ} is a projective variety (hence Kähler) one uses the existence of the fundamental real (closed) (1, 1)-form ω to define the following Lefschetz map l. The operator N that is described in the next formula plays the role of the logarithm of the local monodromy at infinity

N : K^{i,j,k} → K^{i+2,j,k+1}, N(f) = (2π\sqrt{−1})^{−1}f  (2.2)

l : K^{i,j,k} → K^{i,j+2,k}, l(f) = (2π\sqrt{−1})f ∧ ω  (2.3)

These endomorphisms are known to commute with d′ and d′′ and satisfy [l, N] = 0 (cf. op.cit. Lemma 4.2). One sets K^{i,j} = ⊕_{k}K^{i,j,k} and writes K^{∗} = ⊕_{i+j=∗}K^{i,j} to denote the simple complex endowed with the total differential d = d′ + d′′ and with the action of the operators N and l.

Remark 2.1 In the complex (2.1) the second index j is subject to the constraint a + b = j + 1 (where a + b is the total degree of the differential forms). This implies that j assumes only a finite number of values: −1 ≤ j ≤ 1, in fact 0 ≤ a + b ≤ 2 (X is a Riemann surface).

2.2 A polarized Hodge–Lefschetz structure

In this paragraph we will review the theory of polarized bigraded Hodge-Lefschetz modules due to Deligne and Saito. The main result is Theorem 2.6 which states that the complex K^{·,·} defined in (2.1) together with the maps N and l as in (2.2) and (2.3) determine a Lefschetz module. A detailed description of the structure of polarized Hodge-Lefschetz modules is contained in [43]; for a short and quite pleasant exposition we refer to [21].

Definition 2.2 A bigraded Lefschetz module (K^{·,·}, L_{1}, L_{2}) is a bigraded real vector space K = ⊕_{i,j}K^{i,j} with endomorphisms

L_{1} : K^{i,j} → K^{i+2,j}  L_{2} : K^{i,j} → K^{i,j+2}  (2.4)

satisfying [L_{1}, L_{2}] = 0. Furthermore, the operators L_{i} are required to satisfy the following conditions

1. L_{1}^{i} : K^{−i,j} → K^{i,j} is an isomorphism for i > 0

2. L_{2}^{j} : K^{i,−j} → K^{i,j} is an isomorphism for j > 0.

Bigraded Lefschetz modules correspond to representations of the Lie group SL(2, ℝ)× SL(2, ℝ) (cf. [21] or [43]). Therefore, given a bigraded Lefschetz module (K^{·,·}, L_{1}, L_{2}) this corresponds to the representation

σ : SL(2, ℝ) × SL(2, ℝ) → Aut(K^{·,·})

satisfying

σ \{( \begin{matrix}a & 0 \\ 0 & a^{−1} \end{matrix} ) , ( \begin{matrix}b & 0 \\ 0 & b^{−1} \end{matrix} )\} (x) = a^{i}b^{j}x for x ∈ K^{i,j}  (2.5)

dσ \{( \begin{matrix}0 & 1 \\ 0 & 0 \end{matrix} ) , 0\} = L_{1}  (2.6)

dσ \{0, ( \begin{matrix}0 & 1 \\ 0 & 0 \end{matrix} )\} = L_{2}.  (2.7)

The Weyl reflection

w = ( \begin{matrix}0 & 1 \\ −1 & 0 \end{matrix} ) ∈ SL(2, ℝ)

defines the elements w̃ = {w, w}, w_{1} = {w, 1}, w_{2} = {1, w} ∈ SL(2, ℝ) × SL(2, ℝ). They determine isomorphisms σ(w̃) : K^{i,j} → K^{−i,−j}, σ(w_{1}) : K^{i,j} → K^{−i,j} and σ(w_{2}) : K^{i,j} → K^{i,−j}.

Definition 2.3 A bigraded Lefschetz module is a Hodge–Lefschetz module if each K^{i,j} carries a pure real Hodge structure and the L_{i} ( as in Definition 2.2) are morphisms of real Hodge structures.

For convenience, we recall the definition of a pure Hodge structure over κ = ℂ or ℝ. For a summary of mixed Hodge theory we refer to [50].

Definition 2.4 A pure Hodge structure H over κ is a finite dimensional ℂ-vector space H = ⊕_{p,q}H^{p,q}, together with a conjugate linear involution c and in case κ = ℝ a ℂ-linear involution F_{∞} such that

1. c(H^{p,q}) = H^{q,p}

2. the inclusion of H_{ℝ} := H^{c=id} into H induces an isomorphism H = H_{ℝ} ⊗_{ℝ} ℂ

3. in case κ = ℝ, F_{∞} commutes with c and verifies F_{∞}(H^{p,q}) = H^{q,p}. The action of F_{∞} on the space H^{p,p} decomposes it as H^{p,p} = H^{p,+} ⊕ H^{p,−}. We denote the dimensions of the eigenspaces by h^{p,±} := dim_{ℂ} H^{p,±(−1)^{p}}. In the case κ = ℝ, H is called a real, pure Hodge structure.

Example 2.5 An example of pure Hodge structure is given by the singular (Betti) cohomology H_{B}^{∗}(X_{/ℂ}, ℂ) on the Riemann surface X_{/ℂ}. The ℂ-linear involution F_{∞} is induced by the complex conjugation on the Riemann surface.

On a bigraded Lefschetz module (K^{·,·}, L_{1}, L_{2}) we consider the additional data of a differential d and a pairing ψ:

d : K^{i,j} → K^{i+1,j+1}, ψ : K^{−i,−j} ⊗ K^{i,j} → ℝ(n), n ∈ ℕ

satisfying the following properties:

1. d^{2} = 0 = [L_{i}, d]

2. ψ(x, y) = (−1)^{n}ψ(y, x)

3. ψ(dx, y) = ψ(x, dy)

4. ψ(L_{i}x, y) + ψ(x, L_{i}y) = 0

5. ψ(·, L_{1}^{i}L_{2}^{j}·) is symmetric and positive definite on K^{−i,−j} ∩ Ker(L_{1}^{i+1}) ∩ Ker(L_{2}^{j+1}).

If (K, L_{1}, L_{2}, ψ) is a polarized bigraded Lefschetz module (i.e. (K, L_{1}, L_{2}) is a bigraded Lefschetz module satisfying the properties 1.-5.), then the bilinear form

⟨·, ·⟩ : K ⊗ K → ℝ(n), ⟨x, y⟩ := ψ (x, σ(w̃)y)  (2.8)

is symmetric and positive definite.

Theorem 2.6 The differential complex K^{·,·} defined in (2.1) endowed with the operators L_{1} = N (cf. (2.2)) and L_{2} = l (cf. (2.3)) is a polarized bigraded Lefschetz module. The polarization is given by

ψ : K^{−i,−j,k} ⊗ K^{i,j,k+i} → ℝ(1)

ψ(x, y) := (\frac{1}{2π\sqrt{−1}}) ϵ(n − j)(−1)^{k} ∫_{X(ℂ)} x ∧ Cy.

Here, for m ∈ ℤ: ϵ(m) := (−1)^{\frac{m(m+1)}{2}} and C(x) := (\sqrt{−1})^{a−b}x is the Weil operator, for x a differential form of type (a, b) (cf. [53] §V.1).

Proof. We refer to [14] Lemmas 4.2, 4.5, 4.6 and Proposition 4.7). ⋄

Such elaborate construction on the complex K^{·,·} allows one to set up a harmonic theory as in [14] pp. 350-1, so that the polarized bigraded Lefschetz module structure passes to the hypercohomology ℍ^{∗}(K^{·}, d). More precisely, one defines a Laplace operator on K^{·,·} as

□ := d(^{t}d) + (^{t}d)d

where ^{t}d is the transpose of d relative to the bilinear form ⟨·, ·⟩ defined in (2.8). Then, □ commutes with the action of SL(2, ℝ) × SL(2, ℝ) (cf. [53] Lemma at p. 153). Using the properties of the bilinear form ⟨·, ·⟩ one gets

ℍ^{∗}(K^{·}, d) = Ker(d) ∩ Ker(^{t}d) = Ker(□)

and □ is invariant for the action of SL(2, ℝ) × SL(2, ℝ). The following result holds Corollary 2.7 The data (ℍ^{∗}(K^{·}, d), N, l, ψ) define a polarized, bigraded Hodge-Lefschetz module. Proof. The statement follows from the isomorphism of complexes

K^{·} ≃ Ker(□) ⊕ Image(□)

and from the facts that d = 0 on Ker(□) and that the complex Image(□) is d-acyclic. These three statements taken together imply the existence of an induced action of SL(2, ℝ) × SL(2, ℝ) on the hypercohomology of K^{·} (cf. [21] for details). ⋄

2.3 Cohomology groups

It follows from the definition of the double complex (K^{·,·}, d′, d′′) in (2.1) that the total differential d = d′+d′′ satisfies d^{2} = 0 and commutes with the operator N. In particular, d induces a differential on the graded groups

Ker(N)^{·,·} = ker(N : K^{·,·} → K^{·+2,·}), Coker(N)^{·,·} = coker(N : K^{·,·} → K^{·+2,·})

as well as on the mapping cone of N

Cone(N)^{·,·} = Cone(N : K^{·,·} → K^{·+2,·}) := K^{·,·}[1] ⊕ K^{·+2,·}, D(a, b) = (−d(a), N(a) + d(b))

Definition 2.8 For any non-negative integer q and p ∈ ℤ, define

gr_{2p}^{w}H^{q}(X̃^{∗}) = \frac{Ker(d : K^{q−2p,q−1} → K^{q−2p+1,q})}{Im(d : K^{q−2p−1,q−2} → K^{q−2p,q−1})},  (2.9)
gr_{2p}^{w}H^{q}(Y) = \frac{Ker(d : Ker(N)^{q−2p,q−1} → Ker(N)^{q−2p+1,q})}{Im(d : Ker(N)^{q−2p−1,q−2} → Ker(N)^{q−2p,q−1})},  (2.10)
gr_{2p}^{w}H_{Y}^{q} (X) = \frac{Ker(d : Coker(N)^{q−2p,q−3} → Coker(N)^{q−2p+1,q−2})}{Im(d : Coker(N)^{q−2p−1,q−4} → Coker(N)^{q−2p,q−3})},  (2.11)
gr_{2p}^{w}H^{q}(X^{∗}) = \frac{Ker(d : Cone(N)^{q−2p+1,q−2} → Cone(N)^{q−2p+2,q−1})}{Im(d : Cone(N)^{q−2p,q−3} → Cone(N)^{q−2p+1,q−2})}.  (2.12)

We define: H^{q}(X̃^{∗}) := ℍ^{q}(K^{·}), H^{q}(Y) := ℍ^{q}(Ker(N)^{·}), H^{q}(X^{∗}) := ℍ^{q}(Cone(N)^{·}) and H_{Y}^{q} (X) := ℍ^{q}(Coker(N)^{·}). These groups are identified with

\begin{matrix}H^{q}(X̃^{∗}) = ⨁_{p∈ℤ} gr_{2p}^{w}H^{q}(X̃^{∗}), & H^{q}(Y) = ⨁_{p∈ℤ} gr_{2p}^{w}H^{q}(Y), \\ H^{q}(X^{∗}) = ⨁_{p∈ℤ} gr_{2p}^{w}H^{q}(X^{∗}), & H_{Y}^{q} (X) = ⨁_{p∈ℤ} gr_{2p}^{w}H_{Y}^{q} (X). \end{matrix}

Remark 2.9 Note that the even graduation is a consequence of the parity condition q + r ≡ 0 (2) imposed on the indices of the complex (2.1).

Because dim X_{/κ} = 1, it is easy to verify from the definition of K^{·,·} that H^{q}(X̃^{∗}) and H^{q}(Y) are ≠ 0 only for q = 0, 1, 2. Furthermore, one easily finds that H^{q}(X^{∗}) ≠ 0 for q = 0, 1, 2, 3 and H_{Y}^{q} (X) ≠ 0 only when q = 2, 3, 4.

The definition of these groups is inspired by the theory of degenerations of Hodge structures (cf. [49]) where the symbols X̃^{∗}, Y and X^{∗} have a precise geometric meaning: namely, they denote resp. the smooth fiber, the special fiber and the punctured space 𝔛 − Y, where 𝔛 is the chosen model for a degeneration over a disk. In our setup instead, X̃^{∗}, Y, and X^{∗} are only symbols but the general formalism associated to the hypercohomology of a double complex endowed with an operator commuting with the total differential can still be pursued and in fact it gives interesting arithmetical information. In the following we will show that the groups that we have just introduced enjoy similar properties as the graded quotients of the weight-filtration on the corresponding cohomology groups of op.cit. .

It is important to remark that the hypercohomology of the complex Cone(N)^{·} contains both the information coming from the cohomologies of Ker(N)^{·} and Coker(N)^{·}, as the following proposition shows

Proposition 2.10 The following equality holds:

H^{q}(X^{∗}) = ⨁_{p∈ℤ}gr_{2p}^{w}H^{q}(X^{∗}) =

= ⨁_{2p≤q−1} gr_{2p}^{w}H^{q}(Y) ⊕ gr_{q}^{w}H^{q}(X^{∗}) ⊕ gr_{q+1}^{w}H^{q}(X^{∗}) ⊕ ⨁_{2p>q+1} gr_{2p}^{w}H_{Y}^{q+1}(X).

The proof of proposition 2.10 as well as an explicit description of each addendum in the sum is a consequence of the following lemmas

Lemma 2.11 For all p ∈ ℤ and for q ∈ ℕ there are exact sequences

⋯ → gr_{2p}^{w}H^{q}(X^{∗}) → gr_{2p}^{w}H^{q}(X̃^{∗}) →^{N} gr_{2p−2}^{w}H^{q}(X̃^{∗}) → gr_{2p}^{w}H^{q+1}(X^{∗}) → ⋯ (2.13)

⋯ → gr_{2p}^{w}H^{q}(Y) → gr_{2p}^{w}H^{q}(X^{∗}) → gr_{2p}^{w}H_{Y}^{q+1}(X) → gr_{2p}^{w}H^{q+1}(Y) → ⋯ (2.14)

Furthermore, the maps gr_{2p}^{w}H_{Y}^{q} (X) → gr_{2p}^{w}H^{q}(Y) in (2.14) are zero unless q = 2p, in which case they coincide with the morphism

\frac{(A^{p−1,p−1})_{ℝ}(p − 1)}{Im(d′′)} ⟶^{2π\sqrt{−1}d′d′′} Ker(d′ : (A^{p,p})_{ℝ}(p) → ( ⨁_{\begin{matrix}a+b=2p+1 \\ |a−b|≤1 \end{matrix}} A^{a,b})_{ℝ}(p)).

Proof. We refer to [6]: Lemmas 3 and Lemma 4 and to [14]: Lemma 4.3. ⋄

Lemma 2.12 For all p ∈ ℤ and for q ∈ ℕ:

1. The group gr_{2p}^{w}H^{q}(Y) is zero unless 2p ≤ q, in which case

gr_{2p}^{w}H^{q}(Y) = \begin{cases}Ker(d′ : (A^{p,p})_{ℝ}(p) → ⨁_{\begin{matrix}a+b=2p+1 \\ |a−b|≤1 \end{matrix}}(A^{a,b})_{ℝ}(p)) & if q = 2p \\ \frac{Ker(d′ : ( ⨁_{\begin{matrix}a+b=q \\ |a−b|≤q−2p \end{matrix}} A^{a,b})_{ℝ}(p) → ( ⨁_{\begin{matrix}a+b=q+1 \\ |a−b|≤q−2p+1 \end{matrix}} A^{a,b})_{ℝ}(p))}{Im(d′)} & if q ≥ 2p + 1. \end{cases}

2. The group gr_{2p}^{w}H_{Y}^{q} (X) is zero unless 2p ≥ q, in which case

gr_{2p}^{w}H_{Y}^{q} (X) = \begin{cases}Coker(d′′ : ( ⨁_{\begin{matrix}a+b=2p−3 \\ |a−b|≤1 \end{matrix}} A^{a,b})_{ℝ}(p − 1) → (A^{p−1,p−1})_{ℝ}(p − 1)) & if q = 2p \\ \frac{Ker(d′′ : ( ⨁_{\begin{matrix}a+b=q−2 \\ |a−b|≤2p−q \end{matrix}} A^{a,b})_{ℝ}(p − 1) → ( ⨁_{\begin{matrix}a+b=q−1 \\ |a−b|≤2p−q−1 \end{matrix}} A^{a,b})_{ℝ}(p − 1))}{Im(d′′)} & if q ≤ 2p − 1. \end{cases}

Proof. We refer to [14] Lemmas 4.3. ⋄

Proof.(of Prop. 2.10) It is a straightforward consequence of Lemma 2.11. ⋄

Corollary 2.13 The monodromy map

N : gr_{2p}^{w}H^{q}(X̃^{∗}) → gr_{2(p−1)}^{w}H^{q}(X̃^{∗}) is \{ \begin{matrix}injective & if q < 2p − 1 \\ bijective & if q = 2p − 1 \\ surjective & if q ≥ 2p. \end{matrix}

Therefore, the following sequences are exact

q ≥ 2p : 0 → gr_{2p}^{w}H^{q}(X^{∗}) → gr_{2p}^{w}H^{q}(X̃^{∗}) →^{N} gr_{2(p−1)}^{w}H^{q}(X̃^{∗}) → 0,  (2.15)
q ≤ 2(p − 1) : 0 → gr_{2p}^{w}H^{q}(X̃^{∗}) →^{N} gr_{2(p−1)}^{w}H^{q}(X̃^{∗}) → gr_{2p}^{w}H^{q+1}(X^{∗}) → 0. (2.16)

In particular, one obtains

Ker(N) = \{ \begin{matrix}gr_{2p}^{w}H^{q}(X^{∗}) & if q ≥ 2p \\ 0 & if q ≤ 2p − 1. \end{matrix}

Proof. This follows from Lemma 2.11: (2.13) and from Lemma 2.12.⋄

Remark 2.14 For future use, we explicitly remark that when q ≥ 2p one has the following decomposition:

gr_{2p}^{w}H^{q}(X̃^{∗}) = ⊕_{k≥q−2p} \frac{Ker(d : K^{q−2p,q−1,k} → ⋯)}{Im(d)} =  (2.17)

= gr_{2p}^{w}H^{q}(X^{∗}) ⊕_{k≥q−2p+1} \frac{Ker(d : K^{q−2p,q−1,k} → ⋯)}{Im(d)}.

Hence, when q = 0, 1, 2, the group gr_{2p}^{w}H^{q}(X^{∗}) coincides with the homology of the complex (⊕_{k≥q−2p}K^{q−2p,q−1,k}, d) at k = q − 2p ≥ 0.

It is important to recall that the presence of a structure of polarized Hodge-Lefschetz module on the hypercohomology ℍ^{∗}(K^{·}, d) = H^{∗}(X̃^{∗}) allows one to state the following results

Proposition 2.15 For q, p ∈ ℤ satisfying the conditions q − 2p > 0, q ≥ 0, the operator N induces isomorphisms

N^{q−2p} : gr_{2(q−p)}^{w}H^{q}(X̃^{∗}) →^{≃} gr_{2p}^{w}H^{q}(X̃^{∗}).  (2.18)

Furthermore, for q ≥ 2p the isomorphisms (2.18) induce corresponding isomorphisms

(gr_{2p}^{w}H^{q}(X̃^{∗})^{N=0} ≃) gr_{2p}^{w}H^{q}(X^{∗}) →_{≃}^{N^{2p−q}} gr_{2(q−p+1)}^{w}H^{q+1}(X^{∗}).  (2.19)

Proof. For the proof of (2.18) we refer to [14]: Proposition 4.8. For a proof of the isomorphisms (2.19) we refer to Corollary 2.13 and either the proof of Proposition 2.21 or to op.cit. .⋄

2.4 Relations with Deligne’s cohomology

The main feature of the complex (2.1) is its relation with the real Deligne cohomology of X_{/κ}. This cohomology (cf. Definition 2.16) measures how the natural real structure on the singular cohomology of a smooth projective variety is behaved with respect to the de-Rham filtration. One of the most interesting properties of Deligne’s cohomology is its connection with arithmetics. Proposition 2.18 describes a precise relation between the ranks of some real Deligne’s cohomology groups and the orders of pole, at nonpositive integers, of the Γ-factors attached to a (real) Hodge structure H = ⊕_{p,q}H^{p,q} over ℂ (or ℝ). We recall that these factors are defined as (cf. [47])

L_{ℂ}(H, s) = ∏_{p,q} Γ_{ℂ}(s − min(p, q))^{h^{p,q}}  (2.20)
L_{ℝ}(H, s) = ∏_{p<q} Γ_{ℂ}(s − p)^{h^{p,q}} ∏_{p} Γ_{ℝ}(s − p)^{h^{p,+}}Γ_{ℝ}(s − p + 1)^{h^{p,−}}

where s ∈ ℂ, h^{p,q} = dim_{ℂ} H^{p,q} and h^{p,±} is the dimension of the ±(−1)^{p}-eigenspace of the ℂ-linear involution F_{∞} on H (cf. Definition 2.4). One sets

Γ_{ℂ}(s) = (2π)^{−s}Γ(s), Γ_{ℝ}(s) = 2^{− \frac{1}{2}} π^{− \frac{s}{2}} Γ(\frac{s}{2}) where Γ(s) = ∫_{0}^{∞} e^{−t}t^{s} \frac{dt}{t}. (2.21)

The following relation is known to hold (cf. e.g. [18]: Section 2)

Γ_{ℂ}(s) = Γ_{ℝ}(s)Γ_{ℝ}(s + 1).  (2.22)

The Γ-function Γ(s) is analytic in the whole complex plane except for simple poles at the non-positive integer points on the real axis.

Definition 2.16 The real Deligne cohomology H_{D}^{∗}(X_{/ℂ}, ℝ(p)) of a projective, smooth variety X_{/ℂ} is the cohomology of the complex

ℝ(p)_{D} : ℝ(p) → O_{X(ℂ)} → Ω^{1} → ⋯ → Ω^{p−1} → 0.

The first arrow is the inclusion of the constant sheaf of Hodge-Tate twisted constants into the structural sheaf of the manifold: ℝ(p) := (2π\sqrt{−1})^{p}ℝ ⊆ ℂ ⊆ O_{X(ℂ)}. Ω^{i} denotes the sheaf of holomorphic i-th differential forms on the manifold.

The real Deligne cohomology of X_{/ℝ} is defined as the subspace of elements invariant with respect to the de-Rham conjugation

H_{D}^{∗}(X_{/ℝ}, ℝ(p)) := H_{D}^{∗}(X_{/ℂ}, ℝ(p))F̄_{∞}.

Precisely: F̄_{∞} corresponds to the de-Rham conjugation under the canonical identification between de-Rham and Betti cohomology H_{DR}^{∗}(X_{/ℂ}, ℂ) = H_{B}^{∗}(X_{/ℂ}, ℂ) on the manifold X_{/ℂ}.

Remark 2.17 To understand correctly the meaning of F̄_{∞} in the definition, it is worth to recall that both de-Rham and singular cohomology of the manifold X_{/ℂ} have real structures. The real structure on the singular cohomology H_{B}^{∗}(X_{/ℂ}, ℝ) ⊗_{ℝ} ℂ = H_{B}^{∗}(X_{/ℂ}, ℂ) is given by the ℝ-linear involution: ¯ (on the right hand side) which is induced by the complex conjugation on the coefficients. On the other hand, by GAGA the algebraic de-Rham cohomology of X_{/ℝ} defines a real structure H_{DR}^{∗}(X_{/ℝ}, ℝ)⊗_{ℝ}ℂ = H_{DR}^{∗}(X_{/ℂ}, ℂ) on the analytic de-Rham cohomology. The complex conjugation on the pair (X_{/ℂ}, Ω^{·}) induces a ℝ-linear involution on the right hand side called de-Rham conjugation. Such de-Rham conjugation corresponds to the operator F̄_{∞} on the Betti cohomology (cf. example 2.5) through the identification H_{DR}^{∗}(X_{/ℂ}, ℂ) = H_{B}^{∗}(X_{/ℂ}, ℂ) on the manifold X_{/ℂ}.

For example, let assume that X_{/ℝ} is a smooth, real algebraic curve, that is, a symmetric Riemann surface with an involution ι : X_{/ℝ} → X_{/ℝ} induced by the complex conjugation. Then, ι determines a corresponding involution ι^{∗} in cohomology. On H^{0}(X_{/ℝ}, ℝ) this is the identity, whereas on H^{2}(X_{/ℝ}, ℝ) it reverses the orientation, hence is identified with −id. On H^{1}(X_{/ℝ}, ℝ) ι^{∗} acts as a non-trivial involution that exchanges H_{ℝ}^{1,0} and H_{ℝ}^{0,1}, hence it separates the cohomology group into two eigenspaces corresponding to the eigenvalues of ±1: H^{1}(X_{/ℝ}, ℝ) = E_{1} ⊕ E_{−1}, with dim E_{1} = dim E_{−1} = g = genus of X. On twisted cohomology groups such as H^{∗}(X_{/ℝ}, ℝ(p)), F̄_{∞} acts as the composition of ι^{∗} with the involution that acts on the real Hodge structure ℝ(p) = (2πi)^{p}ℝ ⊂ ℂ as (−1)^{p}.

From the short exact sequence of complexes

0 → Ω_{<p}^{·}[−1] → ℝ(p)_{D} → ℝ(p) → 0

one gets the following long exact cohomology sequence that explains the statement we made at the beginning of this chapter (F^{p}H_{DR}^{∗}(X_{/ℂ}, ℂ) = the Hodge filtration on de-Rham cohomology of the complex manifold)

⋯ → H^{∗}(X_{/ℂ}, ℝ(p)) → H_{DR}^{∗}(X_{/ℂ}, ℂ)/F^{p} → H_{D}^{∗+1}(X_{/ℂ}, ℝ(p)) → H^{∗+1}(X_{/ℂ}, ℝ(p)) → ⋯

Proposition 2.18 Let X_{/κ} be a smooth, projective variety over κ = ℂ or ℝ. Let i, m be two integers satisfying the conditions: i ≥ 0, 2m ≤ i. Set n = i + 1 − m. Then

− dim_{ℝ} H_{D}^{i+1}(X_{/κ}, ℝ(n)) = ord_{s=m} L_{κ}(H_{B}^{i}(X_{/ℂ}, ℂ), s)

Here H_{B}^{i}(X_{/ℂ}, ℂ) denotes Betti’s cohomology of the manifold X_{/ℂ} endowed with its pure Hodge structure over κ.

Proof. We refer to [44] Section 2.⋄

The definition of the double complex K^{·,·} in (2.1) was motivated by the expectation that the geometry on X_{/κ} supports interesting arithmetical information on the archimedean fiber(s) of an arithmetic variety. Proposition 2.18 shows that the real Deligne cohomology of X_{/κ} carries such information. The main goal was to construct a complex and an operator N acting on it which carries interesting arithmetic information so that the hypercohomology of Cone(N)^{·} becomes isomorphic to the real Deligne cohomology of X_{/κ}. Proposition 4.1 of [14] shows that the complex K^{·,·} has such property. When X_{/κ} is a projective, non singular curve defined over κ = ℂ or ℝ, this can be stated as follows

Proposition 2.19 Let p, q be two non-negative integer. Then, for each fixed valued of p, the complex Cone(N : K^{q−2p,q−1} → K^{q−2p+2,q−1}) is quasi-isomorphic to the complex

… →^{d′′} \underbrace{(A^{p−2,p−1}(p − 1) ⊕ A^{p−1,p−2}(p − 1))_{ℝ}}_{q=2p−2} →^{d′′} \underbrace{A_{ℝ}^{p−1,p−1}(p − 1)}_{q=2p−1} ⟶^{2π\sqrt{−1}d′d′′} \underbrace{A_{ℝ}^{p,p}(p)}_{q=2p} →^{d′}

→^{d′} \underbrace{(A^{p+1,p}(p) ⊕ A^{p,p+1}(p))_{ℝ}}_{q=2p+1} →^{d′} … (2.23)

whose homology, in each degree q, is isomorphic to the real Deligne cohomology of X_{/ℂ}.

Furthermore, taking the F̄_{∞}-invariants of such homology yields a description of H_{D}^{q}(X_{/ℝ}, ℝ(p)).

Proof. We refer to [14] Section 4, Proposition 4.1.⋄

Remark 2.20 Notice that the left-side of (2.23), with respect to the central map 2π\sqrt{−1}d′d′′ is quasi-isomorphic to the complex obtained by exchanging the map d′ with d′′.

Proposition 2.21 For q ≥ 2p ≥ 0, the following isomorphisms hold

H_{D}^{q}(X_{/ℂ}, ℝ(p)) ≃ H_{D}^{q+1}(X_{/ℂ}, ℝ(q + 1 − p))  (2.24)
H_{D}^{q}(X_{/ℝ}, ℝ(p)) ≃ H_{D}^{q+1}(X_{/ℂ}, ℝ(q + 1 − p))^{(−1)^{q}F̄_{∞}=id}  (2.25)

Proof. We consider in detail the case q = 2p. From Proposition 2.15, the following composite map (N^{2}) is an isomorphism

gr_{2(p+1)}^{w}H^{2p}(X̃^{∗}) ↪^{N_{2(p+1)}} gr_{2p}^{w}H^{2p}(X̃^{∗}) ↠^{N_{2p}} gr_{2(p−1)}^{w}H^{2p}(X̃^{∗}).

This implies, using the results of Corollary 2.13, that gr_{2p}^{w}H^{2p}(X^{∗}) = KerN_{2p} is mapped isomorphically to the group gr_{2(p+1)}^{w}H^{2p+1}(X^{∗}) = Coker(N_{2(p+1)}): this isomorphism is induced by the sequence of maps (2.13) in Lemma 2.11 (case q = 2p). It follows from Proposition 2.19 that gr_{2p}^{w}H^{2p}(X^{∗}) ≃ H_{D}^{2p}(X_{/ℂ}, ℝ(p)) and that gr_{2p}^{w}H^{2p}(X^{∗})^{F̄_{∞}=id} ≃ H_{D}^{2p}(X_{/ℝ}, ℝ(p)). Similarly, one gets from the same proposition that gr_{2(p+1)}^{w}H^{2p+1}(X^{∗}) ≃ H_{D}^{2p+1}(X_{/ℂ}, ℝ(p + 1)) and therefore we obtain (2.24). Taking the invariants for the action of (−1)^{q}F̄_{∞} yields (2.25). The proof in the case q ≥ 2p + 1 is a generalization of the one just finished. For details on this part we refer to [14]: pp 352-3. ⋄

It is well known that the algebraic de-Rham cohomology H_{DR}^{∗}(X_{/ℂ}, ℝ(p)), (p ∈ ℤ) is the homology of the complex

0 → ℝ(p) → A_{ℝ}^{0,0}(p) →^{d′} (A^{1,0} ⊕ A^{0,1})_{ℝ}(p) →^{d′} A_{ℝ}^{1,1}(p) → 0; d′ = ∂ + ¯∂.

Using Lemma 2.12 together with Proposition 2.19 and Remark 2.20 we obtain the following description

Proposition 2.22 Let X = X_{/κ} be a smooth, projective curve over κ = ℂ or ℝ. For κ = ℂ, the following descriptions hold

\begin{aligned}H^{0}(Y) & = ⨁_{p≤0} gr^{w}H^{0}(Y) = ⨁_{p≤0} H^{0}(X_{/ℂ}, ℝ(p)) \\ H^{1}(Y) & = ⨁_{p≤0} gr_{2p}^{w}H^{1}(Y) = ⨁_{p≤0} H^{1}(X_{/ℂ}, ℝ(p)) \\ H^{2}(Y) & = gr_{2}^{w}H^{2}(Y) ⊕ ⨁_{p≤0} gr_{2p}^{w}H^{2}(Y) = A_{ℝ}^{1,1}(1) ⊕ ⨁_{p≤0} H^{2}(X_{/ℂ}, ℝ(p)). \end{aligned}  (2.26)

\begin{matrix}H_{Y}^{2} (X) = & gr_{2}^{w}H_{Y}^{2} (X) ⊕ ⨁_{p≥2} gr_{2p}^{w}H_{Y}^{2} (X) ≃ A_{ℝ}^{0,0} ⊕ ⨁_{p≥2} H_{D}^{1}(X_{/ℂ}, ℝ(p)) \\ H_{Y}^{3} (X) = & ⨁_{p≥2} gr_{2p}^{w}H_{Y}^{3} (X) ≃ ⨁_{p≥2} H_{D}^{2}(X_{/ℂ}, ℝ(p)) \\ H_{Y}^{4} (X) = & ⨁_{p≥2} gr_{2p}^{w}H_{Y}^{4} (X) ≃ ⨁_{p≥2} H_{D}^{3}(X_{/ℂ}, ℝ(p)). \end{matrix}  (2.27)

For X_{/ℝ} similar results hold by taking F̄_{∞}-invariants on both sides of the equalities.

Using Proposition 2.22, the description of H^{q}(X^{∗}) given in Proposition 2.10 can be made more explicit.

Proposition 2.23 Let X_{/κ} be a smooth, projective curve over κ = ℂ or ℝ.

1. For κ = ℂ and ∀ q ≥ 0 one has

H^{q}(X̃^{∗})^{N=0} = ⊕_{p∈ℤ}gr_{2p}^{w}H^{q}(X̃^{∗})^{N=0} = ⊕_{q≥2p} gr_{2p}^{w}H^{q}(X^{∗})

2. In particular: H^{q}(X^{∗}) = 0 for q ∉ [0, 3] and the following description holds

H^{0}(X^{∗}) = H^{0}(X̃^{∗})^{N=0} = ⨁_{p≤−1} gr_{2p}^{w}H^{0}(Y) ⊕ gr_{0}^{w}H^{0}(X^{∗}) = ⨁_{p≤0} H^{0}(X_{/ℂ}, ℝ(p))
H^{1}(X^{∗}) = H^{1}(X̃^{∗})^{N=0} ⊕ (gr_{2}^{w}H^{1}(X^{∗}) ⊕ ⨁_{p≥2} gr_{2p}^{w}H_{Y}^{2} (X)) ≃

≃ ⨁_{p≤0} H^{1}(X_{/ℂ}, ℝ(p)) ⊕ ⨁_{p≥1} H^{0}(X_{/ℂ}, ℝ(p − 1))

H^{2}(X^{∗}) = H^{2}(X̃^{∗})^{N=0} ⊕ ⨁_{p≥2} gr_{2p}^{w}H_{Y}^{3} (X) ≃

≃ ⨁_{p≤1} H^{2}(X_{/ℂ}, ℝ(p)) ⊕ ⨁_{p≥2} H^{1}(X_{/ℂ}, ℝ(p − 1))

H^{3}(X^{∗}) = (gr_{4}^{w}H^{3}(X^{∗}) ⊕ ⨁_{p≥3} gr_{2p}^{w}H_{Y}^{4} (X)) ≃ ⨁_{p≥2} H^{2}(X_{/ℂ}, ℝ(p − 1)).

When κ = ℝ similar results hold by taking F̄_{∞}-invariants on both sides.

Proof. 1. follows from Corollary 2.13. The first statement in 2. is a consequence of dim X_{/κ} = 1. For q ∈ [0, 3], the description of the graded groups H^{q}(X^{∗}) follows from Proposition 2.10, Proposition 2.19, Proposition 2.21. For p, q ≥ 2, the isomorphisms H_{D}^{q}(X_{/ℂ}, ℝ(p)) ≃ H^{q−1}(X_{/ℂ}, ℝ(p − 1)) are a consequence of Remark 2.20, whereas the isomorphisms: gr_{2}^{w}H^{1}(X^{∗}) ≃ H_{D}^{1}(X_{/ℂ}, ℝ(1)) ≃ H_{D}^{0}(X_{/ℂ}, ℝ) ≃ H^{0}(X_{/ℂ}, ℝ) and gr_{2}^{w}H^{2}(X^{∗}) ≃ H_{D}^{2}(X_{/ℂ}, ℝ(1)) ≃ H_{D}^{3}(X_{/ℂ}, ℝ(2)) ≃ H^{2}(X_{/ℂ}, ℝ(1)) follow from Proposition 2.21. In particular the last isomorphism holds because dim X = 1. Finally, the case κ = ℝ is a consequence of the fact that the F̄_{∞}-invariants of the homology of the complex (2.23) gives H_{D}^{∗}(X_{/ℝ}, ℝ(p)). ⋄

2.5 Archimedean Frobenius and regularized determinants

On the infinite dimensional real vector space gr_{2p}^{w}H^{∗}(X̃^{∗}) (cf. (2.9)) one defines a linear operator

Φ : gr_{2p}^{w}H^{∗}(X̃^{∗}) → gr_{2p}^{w}H^{∗}(X̃^{∗}),  Φ(x) = p · x  (2.28)

and then extend this definition to the whole group H^{∗}(X̃^{∗}) according to the decomposition H^{q}(X̃^{∗}) = ⊕_{p}gr_{2p}^{w}H^{q}(X̃^{∗}).

In this section we will consider the operator Φ restricted to the subspace H^{∗}(X̃^{∗})^{N=0}. Following the description of this space given in Proposition 2.23, we write Φ = ⊕_{q=0}^{2}Φ_{q}, where

Φ_{q} : H^{q}(X̃^{∗})^{N=0} → H^{q}(X̃^{∗})^{N=0}.

Given a self-adjoint operator T with pure point spectrum, the zeta-regularized determinant is defined by (cf. [20])

det_{∞}(s − T) = exp (−\frac{d}{dz} ζ_{T} (s, z)_{|z=0}) ,  (2.29)

where

ζ_{T} (s, z) = ∑_{λ∈Spec(T)} m_{λ}(s − λ)^{−z}.  (2.30)

Here, Spec(T) denotes the spectrum of T and m_{λ} is the multiplicity of the eigenvalue λ. By the spectral theorem m_{λ} = dim E_{λ}(T), where E_{λ}(T) is the related eigenspace.

Φ_{q} is a self-adjoint operator with respect to the inner product induced by (2.8), with spectrum

Spec(Φ_{q}) = \{ \begin{matrix}\{n ∈ ℤ, n ≤ 0\} & q = 0, 1 \\ \{n ∈ ℤ, n ≤ 1\} & q = 2. \end{matrix}

The eigenspaces E_{n}(Φ_{q}) = gr_{2n}^{w}H^{q}(X̃^{∗})^{N=0} have dimensions dim E_{n}(Φ_{q}) = b_{q}, the q-th Betti number of X_{/ℂ}.

Proposition 2.24 The zeta regularized determinant of Φ_{q} is given by

det_{∞} (\frac{s}{2π} − \frac{Φ_{q}}{2π} ) = Γ_{ℂ}(s)^{−b_{q}},  (2.31)

for q = 0, 1 and

det_{∞} (\frac{s}{2π} − \frac{Φ_{2}}{2π} ) = Γ_{ℂ}(s − 1)^{−b_{2}},  (2.32)

with Γ_{ℂ}(s) and Γ_{ℝ}(s) as in (2.21).

Proof. We write explicitly the zeta function for the operator Φ_{q}/(2π). When q = 0, 1, this has spectrum {n/(2π)}_{n≤0}, hence we have

ζ_{Φ_{q}/(2π)}(s/(2π), z) = ∑_{n≤0} b_{q}(s/(2π) − n/(2π))^{−z} = b_{q}(2π)^{z}ζ(s, z).

ζ(s, z) is the Hurwitz zeta function

ζ(s, z) = ∑_{n≥0} \frac{1}{(s + n)^{z}} .

For q = 2, similarly we have

ζ_{Φ_{2}/(2π)}(s/(2π), z) = b_{2}(2π)^{z}(ζ(s, z) + (s − 1)^{−z}).

It is well known that the Hurwitz zeta function satisfies the following properties:

ζ(s, 0) = \frac{1}{2} − s, \frac{d}{dz} ζ(s, z)_{|_{z=0}} = log Γ(s) − \frac{1}{2} log(2π).  (2.33)

When q = 0, 1, the computation of \frac{d}{dz} ζ_{Φ_{q}/(2π)}(s/(2π), z)_{|z=0} yields

\frac{d}{dz} ζ_{Φ_{q}/(2π)}(s/(2π), z) = b_{q} (log(2π)(2π)^{z}ζ(s, z) + (2π)^{z} \frac{d}{dz} ζ(s, z)) .

At z = 0, this gives

\frac{d}{dz} ζ_{Φ_{q}/(2π)}(s/(2π), z)_{|_{z=0}} = b_{q} (log(2π)ζ(s, 0) + \frac{d}{dz} ζ(s, z)_{|_{z=0}})

= b_{q} (log(2π)(\frac{1}{2} − s) + log Γ(s) − \frac{1}{2} log(2π)) = b_{q}(−s log(2π) + log Γ(s)). Taking the exponential we get

exp (−\frac{d}{dz} ζ_{Φ_{q}/(2π)}(s/(2π), z)_{|_{z=0}}) = exp(−b_{q}(−s log(2π) + log Γ(s))) =
= ((2π)^{−s}Γ(s))^{−b_{q}} = Γ_{ℂ}(s)^{−b_{q}}.

When q = 2, one has similarly

\frac{d}{dz} ζ_{Φ_{2}/(2π)}(s/(2π), z) = b_{2} ((log(2π)(2π)^{z}ζ(s, z) + (2π)^{z} \frac{d}{dz} ζ(s, z)) + b_{2} \frac{d}{dz} \frac{(2π)^{z}}{(s − 1)^{z}} .

At z = 0 this gives

b_{2} (−s log(2π) + log Γ(s) + log(2π) − log(s − 1)) .

Thus, we have

exp (−\frac{d}{dz} ζ_{Φ_{2}/(2π)}(s/(2π), z)_{|_{z=0}}) = ((2π)^{−s+1} \frac{Γ(s)}{(s − 1)})^{−b_{2}} = ((2π)^{−(s−1)}Γ(s − 1))^{−b_{2}} =

= Γ_{ℂ}(s − 1)^{−b_{2}}.

⋄

Remark 2.25 When X_{/ℂ} is a smooth complex algebraic curve, that is, when X_{/ℂ} = X_{α(K)} for an archimedean prime that corresponds to a complex (non-real) embedding α : K ↪ ℂ, the description of the complex Euler factor is given by (cf. (2.20))

L_{ℂ}(H^{q}(X_{/ℂ}, ℂ), s) = \{ \begin{matrix}Γ_{ℂ}(s)^{b_{q}} & q = 0, 1 \\ Γ_{ℂ}(s − 1)^{b_{2}} & q = 2, \end{matrix}

where H^{q}(X_{/ℂ}, ℂ) is the Betti cohomology. The relation to the determinants (2.31) (2.32) is then

det_{∞} (\frac{s}{2π} − \frac{Φ_{q}}{2π} )^{−1} = L_{ℂ}(H^{q}(X_{/ℂ}, ℂ), s).  (2.34)

This result was proved in [14]: §5, via comparison to Deninger’s pair (H_{ar}^{∗}, Θ).

Assume now that X_{/ℝ} is a smooth real algebraic curve; that is X_{/ℝ} = X_{α(K)} for an archimedean prime that corresponds to a real embedding α : K ↪ ℝ. In this case X_{/ℝ} is a symmetric Riemann surface, namely a compact Riemann surface with an involution ι : X_{/ℝ} → X_{/ℝ} induced by complex conjugation. Such involution on the manifold induces an action of the real Frobenius F̄_{∞} on H^{q}(X̃^{∗})^{N=0}: we refer to Remark 2.17 for the description of this operator.

For instance, following the decomposition given in Proposition 2.23, H^{1}(X̃^{∗})^{N=0} = ⊕_{p≤0}gr_{2p}^{w}H^{1}(X^{∗}) splits as the sum of two eigenspaces for F̄_{∞} with eigenvalues ±1:

H^{1}(X̃^{∗})^{N=0} = E^{+} ⊕ E^{−},

where

E^{+} := H^{1}(X̃^{∗})^{N=0,F̄_{∞}=id} = ⨁_{p≤0} E_{1}(2p) ⊕ ⨁_{p≤−1} E_{−1}(2p + 1),  (2.35)
E^{−} := H^{1}(X̃^{∗})^{N=0,F̄_{∞}=−id} = ⨁_{p≤−1} E_{1}(2p + 1) ⊕ ⨁_{p≤0} E_{−1}(2p).

We consider once more the operator Φ acting on H^{∗}(X̃^{∗})^{N=0}, and we denote by Φ̂_{q} the restriction of this operator to the subspace H^{q}(X̃^{∗})^{N=0,F̄_{∞}=id}.

Proposition 2.26 The regularized determinant for the operator Φ̂_{q} = Φ_{|_{H^{q}(X̃^{∗})^{N=0,F̄∞=id}}} is given by (g = genus of X_{/κ})

det_{∞} (\frac{s}{2π} − \frac{Φ̂_{0}}{2π} ) = Γ_{ℝ}(s)^{−b_{0}}  (2.36)

det_{∞} (\frac{s}{2π} − \frac{Φ̂_{1}}{2π} ) = Γ_{ℝ}(s)^{−b_{1}/2}Γ_{ℝ}(s + 1)^{−b_{1}/2} = Γ_{ℂ}(s)^{−g}  (2.37)

det_{∞} (\frac{s}{2π} − \frac{Φ̂_{2}}{2π} ) = Γ_{ℝ}(s − 1)^{−b_{2}}.  (2.38)

Proof. We write explicitly the zeta function for the operators Φ̂_{q} on H^{q}(X̃^{∗})^{N=0,F̄_{∞}=id}. The spectrum of Φ̂_{q} is given by {n ∈ ℤ, n ≤ 0} for q = 0, 1 and {n ∈ ℤ, n ≤ 1} for q = 2. Because complex conjugation is the identity on H^{0}(X_{/ℝ}, ℝ(2n)), the eigenspaces where F̄_{∞} acts as identity are E_{n}(Φ̂_{0}) = gr_{4n}^{w}H^{0}(X^{∗}). On the other hand, F̄_{∞} acts as the identity on H^{2}(X_{/ℝ}, ℝ(2n + 1)), hence E_{n}(Φ̂_{2}) = gr_{2(2n+1)}^{w}H^{2}(X^{∗}). The action of F̄_{∞} on H^{1}(X_{/ℝ}, ℝ(n)) is the identity precisely on the eigenspaces E_{n}(Φ̂_{1}) = E_{1}(2n) ⊕ E_{−1}(2n + 1) as in (2.35).

The zeta function of Φ̂_{0}/(2π) is therefore of the form

ζ_{Φ̂_{0}/(2π)}(s/(2π), z) = ∑_{n≥0} b_{0} (\frac{s + 2n}{2π})^{−z} = b_{0}(2π)^{z} ∑_{n≥0} \frac{1}{(s + 2n)^{z}} = b_{0}(π)^{z}ζ(s/2, z),

where ζ(s, z) is the Hurwitz zeta function. Using the identities (2.33), we obtain

\frac{d}{dz} ζ_{Φ̂_{0}/(2π)}(s/(2π), z)_{|_{z=0}} = b_{0}(log(π)(1/2 − s/2) + log Γ(s/2) − 1/2 log(2π))
= b_{0}(−s/2 log(π) − 1/2 log(2) + log Γ(s/2)).

Hence, using the equalities (2.21), we obtain

exp (−\frac{d}{dz} ζ_{Φ̂_{0}/(2π)}(s/(2π), z)_{|_{z=0}}) = exp(−b_{0}(−s/2 log(π) − 1/2 log(2) + log Γ(s/2)))

= (2^{−1/2}π^{−s/2}Γ(s/2))^{−b_{0}} = Γ_{ℝ}(s)^{−b_{0}}.

The determinant for Φ̂_{1}/(2π) is given by the product

det_{∞} (\frac{s}{2π} − \frac{Φ̂_{1}}{2π} ) =

det_{∞} (\frac{s}{2π} − \frac{Φ̂_{1}}{2π} |_{⊕_{n}E_{1}(2n)}) · det_{∞} (\frac{s}{2π} − \frac{Φ̂_{1}}{2π} |_{⊕_{n}E_{−1}(2n+1)}) .

The zeta function for the first operator is given by

ζ_{\frac{Φ̂_{1}}{2π} |_{⊕nE_{1}(2n)}}(s/(2π), z) = \frac{b_{1}}{2} π^{z}ζ(s/2, z)

while the for the second operator is

ζ_{\frac{Φ̂_{1}}{2π} |_{⊕nE_{−1}(2n+1)}}(s/(2π), z) = \frac{b_{1}}{2} (2π)^{z} ∑_{n≥0} \frac{1}{(s + 1 + 2n)^{z}} = \frac{b_{1}}{2} π^{z}ζ((s + 1)/2, z).

Thus, we obtain

det_{∞} (\frac{s}{2π} − \frac{Φ̂_{1}}{2π} |_{⊕_{n}E_{1}(2n)}) = Γ_{ℝ}(s)^{−b_{1}/2}

and

det_{∞} (\frac{s}{2π} − \frac{Φ̂_{1}}{2π} |_{⊕_{n}E_{−1}(2n+1)}) = Γ_{ℝ}(s + 1)^{−b_{1}/2}.

Then, (2.37) follows using the equality Γ_{ℝ}(s)Γ_{ℝ}(s + 1) = Γ_{ℂ}(s).

Finally, for Φ̂_{2}/(2π), we have

ζ_{\frac{Φ̂_{2}}{2π}} (\frac{s}{2π} , z) = b_{2}(2π)^{z} ∑_{n≥0} \frac{1}{(s − 1 + 2n)^{z}} = b_{2}π^{z}ζ((s − 1)/2, z),

hence

\frac{d}{dz} ζ_{\frac{Φ̂_{2}}{2π}} (\frac{s}{2π} , z)_{|_{z=0}} = b_{2} (−(s − 1)/2 log π − 1/2 log 2 + log Γ((s − 1)/2)) .

Therefore

det_{∞} (\frac{s}{2π} − \frac{Φ̂_{2}}{2π} ) = (2^{−1/2}π^{−(s−1)/2}Γ((s − 1)/2))^{−b_{2}} = Γ_{ℝ}(s − 1)^{−1}.

⋄

Remark 2.27 When X_{/ℝ} is a smooth, real algebraic curve of genus g, that is, when X_{/ℝ} = X_{α(K)} for an archimedean prime that corresponds to a real embedding α : K ↪ ℝ, the description of the real Euler factor is given by (cf. (2.20))

L_{ℝ}(H^{q}(X_{/ℝ}, ℝ), s) = \{ \begin{matrix}Γ_{ℝ}(s) & q = 0 \\ Γ_{ℂ}(s)^{g} & q = 1 \\ Γ_{ℝ}(s − 1) & q = 2, \end{matrix}

As for the complex case, this result was proved in [14]: §5, via comparison to Deninger’s pair (H_{ar}^{∗}, Θ)

3 Arithmetic spectral triple.

In this Section we show that the choice of a Schottky uniformization of a Riemann surface X_{/ℂ} enriches the cohomological construction of Section 2, with the structure of a Connes spectral triple (cf. [10] §VI).

For a discrete group Γ, the (reduced) C^{∗}–algebra C^{∗}(Γ) is the norm completion of the group ring ℂ[Γ] in the algebra of bounded operators on the Hilbert space ℓ^{2}(Γ) of square summable functions on Γ.

If Γ ⊂ PSL(2, ℂ) is a Schottky group, the algebra C_{0}(𝔛_{Γ}) of continuous functions vanishing at infinity on the handlebody 𝔛_{Γ} = Γ\ℍ^{3} is strongly Morita equivalent to a crossed product algebra (cf. Proposition 5.17), which by a result of [22] has the same Ktheory of C^{∗}(Γ) ⊗ C_{0}(ℍ^{3}). Since ℍ^{3} is contractible, the non-commutative space C^{∗}(Γ) carries the complete topological information on the handlebody: in the isomorphisms K_{i}(C_{0}(𝔛_{Γ})) ≅ K_{i+1}(C^{∗}(Γ)) the contribution of C_{0}(ℍ^{3}) is simply given by a degree shift (cf. [22]).

In this Section, we will only use real coefficients. In fact, in order to construct a spectral triple compatible with the arithmetic construction of Section 2, we need to preserve the structure of real vector spaces. For this reason, we consider the real (reduced) C^{∗}–algebra of a Schottky group Γ. This is a norm completion of ℝ[Γ]. The relation between the real C^{∗}(Γ) and the ring of real functions on 𝔛_{Γ} holds, with suitable modifications, as in the complex case (cf. [45], [3]).

Let (H^{·}(X^{∗}), Φ) be the cohomological theory of the fiber at the archimedean prime introduced in Section 2, endowed with the structure of polarized Hodge–Lefschetz module. Let Γ ⊂ PSL(2, ℂ) be a choice of Schottky uniformization of the Riemann surface X_{/ℂ}. In the following we often replace Γ with a lift to SL(2, ℂ), which, with a slight abuse of notation, we still denote by Γ.

In Theorem 3.7 we show that there is a normal subgroup Γ̃ ⊂ Γ such that C^{∗}(Γ̃) has a representation

ρ : C^{∗}(Γ̃) → B(H^{·}(X̃^{∗})),  (3.1)

where B(H^{·}(X̃^{∗})) is the algebra of bounded operators on a real Hilbert space completion of H^{·}(X̃^{∗}) (in the inner product determined by the polarization of K^{·,·}). The representation ρ is compatible with the structure of Hodge–Lefschetz module on H^{·}(X̃^{∗}) = ℍ^{·}(K, d).

The main result of this section is Theorem 3.11, where we prove that the inner product on H^{·}(X̃^{∗}) and the representation (3.1) induce an inner product on H^{·}(X^{∗}) = ℍ^{·}(Cone^{·}(N)) and a representation

ρ_{N} : C^{∗}(Γ̃) → B(H^{·}(X^{∗})),  (3.2)

in such a way that the data (C^{∗}(Γ̃), H^{·}(X^{∗}), Φ) define a spectral triple for which Φ is the Dirac operator.

In non-commutative geometry, the notion of a spectral triple provides the correct generalization of the classical structure of a Riemannian manifold. The two notions agree on a commutative space. In the usual context of Riemannian geometry, the definition of the infinitesimal element ds on a smooth spin manifold can be expressed in terms of the inverse of the classical Dirac operator D. This is the key remark that motivates the theory of spectral triples. In particular, the geodesic distance between two points on the manifold is defined in terms of D^{−1} (cf. [10] §VI). The spectral triple that describes a classical Riemannian spin manifold is (A, H, D), where A is the algebra of complex valued smooth functions on the manifold, H is the Hilbert space of square integrable spinor sections, and D is the classical Dirac operator (a square root of the Laplacian). These data determine completely and uniquely the Riemannian geometry on the manifold. It turns out that, when expressed in this form, the notion of spectral triple extends to more general non-commutative spaces, where the data (A, H, D) consist of a C^{∗}-algebra A (or more generally of some smooth subalgebra of a C^{∗}-algebra) with a representation as bounded operators on a Hilbert space H, and an operator D on H that verifies the main properties of a Dirac operator. Under this perspective, our Theorem 3.11 states that the spectral triple (C^{∗}(Γ̃), H^{·}(X^{∗}), Φ) endows the non-commutative space C^{∗}(Γ̃) with the structure of a non-commutative Riemannian manifold, with the Riemannian metric determined by the Dirac operator Φ. We will refer to (C^{∗}(Γ̃), H^{·}(X^{∗}), Φ) as the arithmetic spectral triple.

The compatibility of the representation (3.1) with the polarized Hodge–Lefschetz module structure is obtained by applying Bers simultaneous uniformization (cf. [5] [7]) to the Schottky group Γ. Given the group Γ and the hyperbolic handlebody satisfying 𝔛_{Γ} ∪X_{/ℂ} = Γ\(ℍ^{3} ∪ℙ^{1}(ℂ)), this process determines a pair of Fuchsian Schottky groups G_{1}, G_{2} ⊂ SL(2, ℝ), which geometrically correspond to a decomposition of the Riemann surface X_{/ℂ} as the union of two Riemann surfaces with boundary. The Fuchsian Schottky groups act on the complex K^{·,·} and on the cohomology via the restriction of the representation of SL(2, ℝ) × SL(2, ℝ) of the Hodge-Lefschetz module structure. The condition of compatibility with the arithmetic structure produces a natural geometric choice of a normal subgroup Γ̃ ⊂ Γ, with G_{1} ≃ Γ̃ ≃ G_{2}. Geometrically, this group corresponds to the choice of a covering 𝔛Γ̃ → 𝔛_{Γ} of 𝔛_{Γ} by a handlebody 𝔛Γ̃. The non-commutative space C^{∗}(Γ̃) describes the topology of 𝔛Γ̃.

In the interpretation of the tangle of bounded geodesics in the handlebody 𝔛Γ̃ as the dual graph of the closed fiber at arithmetic infinity, the covering 𝔛Γ̃ → 𝔛_{Γ} produces a corresponding covering of the dual graph by geodesics in 𝔛Γ̃. Passing to the covering 𝔛Γ̃ may be regarded as an analog, at the archimedean primes, of the refinement of the dual graph of a Mumford curve that corresponds to a minimal resolution (cf. [33] §3).

When the archimedean prime corresponds to a real embedding K ↪ ℝ, so that the corresponding Riemann surface X_{/ℝ} acquires a real structure, Proposition 3.13 shows that, if the set of real points X_{/ℝ}(ℝ) is non-empty, then we can construct the arithmetic spectral triple with the algebra C^{∗}(Γ), with Γ the Fuchsian Schottky group determined by the real structure.

The theory of spectral triples encodes important arithmetic information on the underlying non-commutative space, expressed via an associated family of zeta functions. By studying the zeta functions attached to our triple (C^{∗}(Γ), H^{·}(X^{∗}), Φ) we find a natural one whose associated Ray-Singer determinant is the alternating product of the Γ-factors for the real Hodge structure over ℂ given by the Betti cohomology H^{q}(X_{/ℂ}, ℂ).

3.1 Spectral triples

We recall the basic setting of Connes theory of spectral triples. For a more complete treatment we refer to [12], [10], [13].

Definition 3.1 a spectral triple (A, H, D) consists of an involutive algebra A with a representation

ρ : A → B(H)

as bounded operators on a Hilbert space H, and an operator D (called the Dirac operator) on H, which satisfies the following properties:

1. D is self–adjoint.

2. For all λ ∉ ℝ, the resolvent (D − λ)^{−1} is a compact operator on H.

3. For all a ∈ A, the commutator [D, a] is a bounded operator on H.

Remark 3.2 The property 2. of Definition 3.1 generalizes ellipticity of the standard Dirac operator on a compact manifold. Usually, the involutive algebra A satisfying property 3. can be chosen to be a dense subalgebra of a C^{∗}–algebra. This is the case, for instance, when we consider smooth functions on a manifold as a subalgebra of the commutative C^{∗}-algebra of continuous functions. In the classical case of Riemannian manifolds, property 3. is equivalent the Lipschitz condition, hence it is satisfied by a larger class than that of smooth functions.

We review those aspects of the theory of spectral triples which are of direct interest to us. For a more general treatment we refer to [12], [10], [13]. Volume form. A spectral triple (A, H, D) is said to be of dimension n, or n–summable if the operator |D|^{−n} is an infinitesimal of order one, which means that the eigenvalues λ_{k}(|D|^{−n}) satisfy the estimate λ_{k}(|D|^{−n}) = O(k^{−1}).

For a positive compact operator T such that

∑_{j=0}^{k−1} λ_{j}(T) = O(log k),

the Dixmier trace Tr_{ω}(T) is the coefficient of this logarithmic divergence, namely

Tr_{ω}(T) = lim_{ω} \frac{1}{log k} ∑_{j=1}^{k} λ_{j}(T).  (3.3)

Here the notation lim_{ω} takes into account the fact that the sequence

S(k, T) := \frac{1}{log k} ∑_{j=1}^{k} λ_{j}(T)

is bounded though possibly non-convergent. For this reason, the usual notion of limit is replaced by a choice of a linear form lim_{ω} on the set of bounded sequences satisfying suitable conditions that extend analogous properties of the limit. When the sequence S(k, T) converges (3.3) is just the ordinary limit Tr_{ω}(T) = lim_{k→∞} S(k, T). So defined, the Dixmier trace (3.3) extends to any compact operator that is an infinitesimal of order one, since any such operator is the difference of two positive ones. The operators for which the Dixmier trace does not depend on the choice of the linear form lim_{ω} are called measurable operators.

On a non-commutative space the operator |D|^{−n} generalizes the notion of a volume form. The volume is defined as

V = Tr_{ω}(|D|^{−n}).  (3.4)

More generally, consider the algebra Ã generated by A and [D, A]. Then, for a ∈ Ã, integration with respect to the volume form |D|^{−n} is defined as

∫ a := \frac{1}{V} Tr_{ω}(a|D|^{−n}).  (3.5)

The usual notion of integration on a Riemannian spin manifold M can be recovered in this context (cf. [10], [23]) through the formula (n even):

∫_{M} fdv = (2^{n−[n/2]−1}π^{n/2}nΓ(n/2)) Tr_{ω}(f|D|^{−n}).

Here D is the classical Dirac operator on M associated to the metric that determines the volume form dv, and f in the right hand side is regarded as the multiplication operator acting on the Hilbert space of square integrable spinors on M.

Zeta functions. An important function associated to the Dirac operator D of a spectral triple (A, H, D) is its zeta function

ζ_{D}(z) := Tr(|D|^{−z}) = ∑_{λ} Tr(Π(λ, |D|))λ^{−z},  (3.6)

where Π(λ, |D|) denotes the orthogonal projection on the eigenspace E(λ, |D|).

An important result in the theory of spectral triples ([10] §IV Proposition 4) relates the volume (3.4) with the residue of the zeta function (3.6) at s = 1 through the formula

V = lim_{s→1+}(s − 1)ζ_{D}(s) = Res_{s=1}Tr(|D|^{−s}).  (3.7)

There is a family of zeta functions associated to a spectral triple (A, H, D), to which (3.6) belongs. For an operator a ∈ Ã, we can define the zeta functions

ζ_{a,D}(z) := Tr(a|D|^{−z}) = ∑_{λ} Tr(a Π(λ, |D|))λ^{−z}  (3.8)

and

ζ_{a,D}(s, z) := ∑_{λ} Tr(a Π(λ, |D|))(s − λ)^{−z}.  (3.9)

These zeta functions are related to the heat kernel e^{−t|D|} by Mellin transform

ζ_{a,D}(z) = \frac{1}{Γ(z)} ∫_{0}^{∞} t^{z−1}Tr(a e^{−t|D|}) dt  (3.10)

where

Tr(a e^{−t|D|}) = ∑_{λ} Tr(a Π(λ, |D|))e^{−tλ} =: θ_{a,D}(t).  (3.11)

Similarly,

ζ_{a,D}(s, z) = \frac{1}{Γ(z)} ∫_{0}^{∞} θ_{a,D,s}(t) t^{z−1} dt  (3.12)

with

θ_{a,D,s}(t) := ∑_{λ} Tr(a Π(λ, |D|))e^{(s−λ)t}.  (3.13)

Under suitable hypothesis on the asymptotic expansion of (3.13) (cf. Theorem 2.7-2.8 of [25] §2), the functions (3.8) and (3.9) admit a unique analytic continuation (cf. [13]) and there is an associated regularized determinant in the sense of Ray–Singer (cf. [39]):

det_{∞}_{a,D}(s) := exp (−\frac{d}{dz} ζ_{a,D}(s, z)|_{z=0})  (3.14)

The family of zeta functions (3.8) also provides a refined notion of dimension for a spectral triple (A, H, D), called the dimension spectrum. This is a subset Σ = Σ(A, H, D) in ℂ with the property that all the zeta functions (3.8), as a varies in Ã, extend holomorphically to ℂ \ Σ.

3.2 Hodge-Lefschetz modules and simultaneous uniformization

We begin by recalling the following elementary fact of hyperbolic geometry. Let Γ be a Kleinian group acting on ℙ^{1}(ℂ). Let Ω ⊂ ℙ^{1}(ℂ) be a Γ-invariant domain. A subset Ω_{0} ⊂ Ω is Γ-stable if, for every γ ∈ Γ, either γ(Ω_{0}) = Ω_{0} or γ(Ω_{0}) ∩ Ω_{0} = ∅. The Γ-stabilizer of Ω_{0} is the subgroup Γ_{0} of those γ ∈ Γ such that γ(Ω_{0}) = Ω_{0}. Let π_{Γ} denote the quotient map π_{Γ} : Ω → Γ\Ω.

Claim 3.3 (cf. Theorem 6.3.3 of [4]). Let Ω_{0} ⊂ Ω be an open subdomain and let Γ_{0} be the Γ-stabilizer of Ω_{0}. Then the quotient map π_{Γ} induces a conformal equivalence

Γ_{0}\Ω_{0} ≃ π_{Γ}(Ω_{0}).

If Γ is a Kleinian group, a quasi circle for Γ is a Jordan curve C in ℙ^{1}(ℂ) which is invariant under the action of Γ. In particular, such curve contains the limit set Λ_{Γ}.

In the case of Schottky groups, the following theorem shows that Bowen’s construction of a quasi–circle for Γ (cf. [7]) determines a pair of Fuchsian Schottky groups G_{1}, G_{2} ⊂ PSL(2, ℝ) associated to Γ ⊂ PSL(2, ℂ). The theorem describes the simultaneous uniformization by Γ̃ of the two Riemann surfaces with boundary X_{i} = G_{i}\ℍ^{2}.

Theorem 3.4 Let Γ ⊂ PSL(2, ℂ) be a Schottky group of rank g ≥ 2. Then the following properties are satisfied:

1. There exists a quasi–circle C for Γ.

2. There is a collection of curves Ĉ on the compact Riemann surface X_{/ℂ} = Γ\Ω_{Γ}

such that

X_{/ℂ} = X_{1} ∪_{∂X_{1}=Ĉ=∂X_{2}} X_{2},

where X_{i} = G_{i}\ℍ^{2} are Riemann surfaces with boundary, and the G_{i} ⊂ PSL(2, ℝ) are Fuchsian Schottky groups. The G_{i} are isomorphic to Γ̃ ⊂ PSL(2, ℂ), the Γ–stabilizer of the two connected components Ω_{i} of ℙ^{1}(ℂ)\C.

Proof. 1. For the construction of a quasi–circle we proceed as in [7]. The choice of a set of generators {g_{i}}_{i=1}^{g} for Γ determines 2g Jordan curves γ_{i}, i = 1 … 2g in ℙ^{1}(ℂ) with pairwise disjoint interiors D_{i} such that, if we write g_{i+g} = g_{i}^{−1} for i = 1 … g, the fractional linear transformation g_{i} maps the interior of γ_{i} to the exterior of γ_{i+g mod 2g}. Now fix a choice of 2g pairs of points ρ_{i}^{±} on the curves γ_{i} in such a way that g_{i} maps the two points ρ_{i}^{±} to the two points ρ_{i+g mod 2g}^{∓}. Choose a collection C_{0} of pairwise disjoint oriented arcs in ℙ^{1}(ℂ) with the property that they do not intersect the interior of the disks D_{i}. Also assume that the oriented boundary of C_{0} as a 1-chain is given by ∂C_{0} = ∑_{i} ρ_{i}^{+} − ∑_{i} ρ_{i}^{−}. Then the curve

C := Λ_{Γ} ∪ ⋃_{γ∈Γ} γC_{0}  (3.15)

is a quasi–circle for Γ.

2. The image of the curves γ_{i} in the quotient X_{/ℂ} = Γ\Ω_{Γ} consists of g closed curves, whose homology classes a_{i}, i = 1 … g, span the kernel Ker(I_{∗}) of the map I_{∗} : H_{1}(X_{/ℂ}, ℤ) → H_{1}(𝔛_{Γ}, ℤ) induced by the inclusion of X_{/ℂ} as the boundary at infinity in the compactification of 𝔛_{Γ}. The image under the quotient map of the collection of points {γρ_{i}^{±}}_{γ∈Γ,i=1...2g} consists of two points on each curve a_{i}, and the image of C ∩ Ω_{Γ} consists of a collection Ĉ of pairwise disjoint arcs on X_{/ℂ} connecting these 2g points. By cutting the surface X_{/ℂ} along Ĉ we obtain two surfaces X_{i}, i = 1, 2, with boundary ∂X_{i} = Ĉ.

Since C is Γ-invariant, the two connected components Ω_{i}, i = 1, 2, of ℙ^{1}(ℂ)\C are Γ-stable. Let Γ_{i} denote the Γ-stabilizer of Ω_{i}. Notice that Γ_{1} = Γ_{2}. In fact, suppose there is γ ∈ Γ such that γ ∈ Γ_{1} and γ ∉ Γ_{2}. Then γ(ℙ^{1}(ℂ)) ⊂ Ω_{1}∪C, so that the image γ(ℙ^{1}(ℂ)) is contractible in ℙ^{1}(ℂ). This would imply that γ has topological degree zero, but an orientation preserving fractional linear transformation has topological degree one.

We denote by Γ̃ the Γ-stabilizer Γ̃ = Γ_{1} = Γ_{2}. Since the components Ω_{i} are open subdomains of the Γ-invariant domain Ω_{Γ}, Claim 3.3 implies that the quotients Γ̃\Ω_{i} are conformally equivalent to the image π_{Γ}(Ω_{i}) ⊂ X_{/ℂ}. By the explicit description of the surfaces with boundary X_{i}, it is easy to see that π_{Γ}(Ω_{i}) = X_{i}.

The quasi-circle C is a Jordan curve in ℙ^{1}(ℂ), hence by the Riemann mapping theorem there exist conformal maps α_{i} of the two connected components Ω_{i} to the two hemispheres U_{i} of ℙ^{1}(ℂ) ∖ ℙ^{1}(ℝ),

α_{i} : Ω_{i} ⟶^{≃} U_{i} U_{1} ∪ U_{2} = ℙ^{1}(ℂ) ∖ ℙ^{1}(ℝ).  (3.16)

Consider the two groups

G_{i} := {α_{i}γα_{i}^{−1} : γ ∈ Γ̃}.

These are isomorphic as groups to Γ̃, G_{i} ≃ Γ̃. Moreover, the G_{i} preserve the upper/lower hemisphere U_{i}, hence they are Fuchsian groups, G_{i} ⊂ PSL(2, ℝ).

The conformal equivalence Γ̃\Ω_{i} ≃ X_{i} implies that the G_{i} provide the Fuchsian uniformization of X_{i} = G_{i}\ℍ^{2}, where ℍ^{2} is identified with the upper/lower hemisphere U_{i} in ℙ^{1}(ℂ) ∖ ℙ^{1}(ℝ).

The group Γ̃ ⊂ Γ is itself a discrete purely loxodromic subgroup of PSL(2, ℂ) isomorphic to free groups, hence a Schottky group, so that the G_{i} are Fuchsian Schottky groups. ⋄

Let X_{/ℝ} be an orthosymmetric smooth real algebraic curve. In this case, we can apply the following refinement of the result of Theorem 3.4. We refer to [1], [46] for a proof.

Proposition 3.5 Let X_{/ℝ} be a smooth real orthosymmetric algebraic curve of genus g ≥ 2. Then the following holds.

1. X_{/ℝ} has a Schottky uniformization such that the domain of discontinuity Ω_{Γ} ⊂ ℙ^{1}(ℂ) is symmetric with respect to ℙ^{1}(ℝ) ⊂ ℙ^{1}(ℂ).

2. The reflection about ℙ^{1}(ℝ) gives an involution on Ω_{Γ} that induces the involution ι : X_{/ℝ} → X_{/ℝ} of the real structure.

3. The circle ℙ^{1}(ℝ) ⊂ ℙ^{1}(ℂ) is a quasi-circle for the Schottky group Γ, such that the image in X_{/ℝ} of ℙ^{1}(ℝ) ∩ Ω_{Γ} is the fixed point set X_{ι} of the involution.

4. The Schottky group Γ is a Fuchsian Schottky group.

The choice of a lifting Γ ⊂ SL(2, ℂ) of the Schottky group determines corresponding lifts of Γ̃ ⊂ SL(2, ℂ) and G_{i} ⊂ SL(2, ℝ).

We consider the polarized bigraded Hodge-Lefschetz module (K^{·,·}, N, ℓ, ψ) associated to X_{/ℂ} at the archimedean prime, as described in Section 2. We have the following result.

Lemma 3.6 Let σ : SL(2, ℝ) × SL(2, ℝ) → Aut(K) be the representation associated to the bigraded Lefschetz module structure on the complex K^{·,·}. Let Γ ⊂ SL(2, ℂ) be a Schottky group that determines a Schottky uniformization of X_{/ℂ}. Let Γ̃ is the corresponding lift to SL(2, ℂ) of the Γ-stabilizer of the components Ω_{i} in the complement of a quasi-circle C. Then (K^{·,·}, N, ℓ, ψ) carries a left and a right action of Γ̃,

σ_{1}(γ) := σ{α_{1}γα_{1}^{−1}, 1}  (3.17)
σ_{2}(γ) := σ{1, α_{2}γα_{2}^{−1}},  (3.18)

where α_{i} are the conformal maps (3.16) of Ω_{i} to the two hemispheres in ℙ^{1}(ℂ)∖ℙ^{1}(ℝ).

Proof. By Theorem 3.4 we obtain Fuchsian Schottky groups G_{i} = {α_{i}γα_{i}^{−1}, γ ∈ Γ̃} in SL(2, ℝ). We consider the restriction of the representation σ : SL(2, ℝ) × SL(2, ℝ) → Aut(K) to G_{1} × {1} and {1} × G_{2} as in (3.17) and (3.18). ⋄

We set

Φ̃ : K^{i,j,k} → K^{i,j,k}  Φ̃(x) = \frac{(1 + j − i)}{2}x.  (3.19)

The operator Φ̃ induces the operator Φ of (2.28) on the cohomology H^{·}(X̃^{∗})^{N=0}. We have the following result.

Theorem 3.7 Let (K^{·,·}, d, N, ℓ, ψ) be the polarized bigraded Lefschetz module associated to a Riemann surface X_{/ℂ} of genus g ≥ 2, and let Γ ⊂ SL(2, ℂ) be a choice of Schottky uniformization for X_{/ℂ}. Let Γ̃ be a lift to SL(2, ℂ) of the Γ–stabilizer of the two connected components of ℙ^{1}(ℂ) \ C as in Theorem 3.4. Then the following holds.

1. The representation σ_{2} of Lemma 3.6 extends to a representation of C^{∗}(Γ̃)

ρ : C^{∗}(Γ̃) → B(ℍ^{·}(K, d))  (3.20)

as bounded operators on a real Hilbert space completion of the cohomology ℍ^{·}(K, d) with the inner product defined by the polarization ψ.

2. The operator Φ̃ defined in (3.19) has bounded commutators with all the elements in ρ(C^{∗}(Γ̃)).

Proof. 1. The representations σ_{1} and σ_{2} of Lemma 3.6 extend by linearity to representations σ_{i} : ℝ[Γ̃] → Aut(K).

By Theorem 2.6 and Corollary 2.7, the cohomology ℍ^{·}(K, d) has an induced Lefschetz module structure, thus we obtain induced representations of the real group ring ℝ[Γ̃] on ℍ^{·}(K, d). We complete H^{·}(X̃^{∗}) = ℍ^{·}(K, d) to a real Hilbert space with respect to the inner product induced by the polarization ψ. Consider operators of the form (2.5) with b = 1,

U_{a}(x) := σ \{( \begin{matrix}a & 0 \\ 0 & a^{−1} \end{matrix} ) , ( \begin{matrix}1 & 0 \\ 0 & 1 \end{matrix} )\} (x) = a^{i}x for x ∈ K^{i,j}.

A direct calculation shows that the U_{a} are in general unbounded operators: since the index i varies over a countable set, it is not hard to construct examples of infinite sums x = ∑_{i} x_{i} that are in the Hilbert space completion of H^{·}(X̃^{∗}) but such that U_{a}(x) is no longer contained in this space.

On the other hand, the index j in the complex K^{i,j} varies subject to the constraint j + 1 = q, where q is the degree of the differential forms (cf. Remark 2.1). Thus, expressions of the form

σ \{( \begin{matrix}1 & 0 \\ 0 & 1 \end{matrix} ) , ( \begin{matrix}b & 0 \\ 0 & b^{−1} \end{matrix} )\} (x) = b^{j}x for x ∈ K^{i,j}  (3.21)

give rise to bounded operators. Thus the representation σ_{2} of ℝ[Γ̃] extends to a representation of the real (reduced) C^{∗}–algebra C^{∗}(Γ̃), by taking the norm closure of σ_{2}(ℝ[Γ̃]) in the algebra of bounded operators B(ℍ^{·}(K, d)).

(2) We can compute explicitly all the commutators with the operator Φ̃. Elements of the form (2.5) commute with Φ̃. Moreover, we have:

[N, Φ̃] (x) = \frac{1}{(2π\sqrt{−1})} (\frac{(1 + j − i)}{2} − \frac{(1 + j − i + 2)}{2}) x = −N(x),

[σ_{1}(w_{1}), Φ̃](x) = (\frac{(1 + j − i)}{2} − \frac{(1 + j + i)}{2}) σ_{1}(w_{1})(x) = −iσ_{1}(w_{1})(x).

[ℓ, Φ̃] (x) = (\frac{(1 + j − i)}{2} − \frac{(1 + j + 2 − i)}{2}) (2π\sqrt{−1})^{−1}x ∧ ω = −ℓ(x)

and

[σ_{2}(w_{2}), Φ̃](x) = (\frac{(1 + j − i)}{2} − \frac{(1 − j − i)}{2}) σ_{2}(w)(x) = j σ_{2}(w)(x).

In particular, it follows that all the commutators that arise from the right representation are bounded operators (cf. Remark 2.1). ⋄

Corollary 3.8 The action of C^{∗}(Γ̃) on the odd cohomology ℍ^{2q+1}(K^{·}, d) is independent of the lift of Γ̃ ⊂ PSL(2, ℂ) to SL(2, ℂ).

Proof. It is sufficient to check that −id acts trivially on the odd cohomology. For x ∈ K^{i,j} we have

σ \{1, ( \begin{matrix}−1 & 0 \\ 0 & −1 \end{matrix} )\} (x) = (−1)^{j}x.

Since j + 1 = q, where q is the degree of the differential forms, we obtain that the induced action is trivial on odd cohomology. Thus, the induced representation σ_{2} of SL(2, ℝ) on the odd cohomology descends to a representation of PSL(2, ℝ). ⋄

Remark 3.9 The data (C^{∗}(Γ̃), ℍ^{·}(K, d), Φ̃) of Theorem 3.7 do not define a spectral triple: the operator Φ̃ does not satisfy all the properties of a Dirac operator. In fact, the eigenspaces of Φ̃, which coincide with the graded pieces gr_{2p}^{w}H^{q}(X̃^{∗}) of the cohomology, are not finite dimensional as the condition on the resolvent in Definition 3.1 would imply. Therefore, it is necessary to restrict the structure (C^{∗}(Γ̃), ℍ^{·}(K, d), Φ̃) to a suitable subspace of ℍ^{·}(K, d), which still carries all the arithmetic information.

Definition 3.10 The operator Φ on ℍ^{·}(Cone(N)^{·}) = H^{·}(X^{∗}) is obtained by extending the action on its graded pieces

Φ|_{gr_{2p}^{w}H^{q}(X^{∗})} := \{ \begin{matrix}p & q ≥ 2p \\ p − 1 & q ≤ 2p − 1. \end{matrix}  (3.22)

according to the decomposition H^{q}(X^{∗}) = ⊕_{p∈ℤ}gr_{2p}^{w}H^{q}(X^{∗}).

Notice that this definition is compatible with the operator Φ̃ defined in (3.19), acting on the complex K^{·,·} and with the induced operator Φ on ℍ^{·}(K, d) = H^{·}(X̃^{∗}). In fact, from the Wang exact sequence (2.13) and Corollary 2.13 we know that, for q ≥ 2p − 1 gr_{2p}^{w}H^{q}(X^{∗}) is identified with a subspace of gr_{2p}^{w}H^{q}(X̃^{∗}), hence the restriction of the operator Φ on H^{·}(X̃^{∗}) acts on gr_{2p}^{w}H^{q}(X^{∗}) as multiplication by p. In the case when q ≤ 2(p − 1), again using the exact sequence (2.13) (cf. (2.16) Corollary 2.13), we can define Φ of an element in gr_{2p}^{w}H^{q}(X^{∗}) as Φ of a preimage in gr_{2(p−1)}^{w}H^{q−1}(X̃^{∗}), hence as multiplication by p − 1. This is obviously well defined, hence the definition (3.22) is compatible with the exact sequences and the duality isomorphisms. Moreover, the operator Φ of (3.22) agrees with the operator (2.28) on the subspace H^{·}(X̃^{∗})^{N=0} of H^{·}(X^{∗}).

Theorem 3.11 Let X be an arithmetic surface with 1-dimensional fibers of genus g ≥ 2. Let Γ be a Schottky group uniformizing the Riemann surface X_{/ℂ} at a given archimedean prime, and let Γ̃ ⊂ Γ be defined as in Theorem 3.4. Let H^{·}(X^{∗}) be the hyper-cohomology of Cone(N). The inner product (2.8) defined by the polarization ψ induces an inner product on H^{·}(X^{∗}). Moreover, the representation (3.20) induces a representation of C^{∗}(Γ̃) on the resulting real Hilbert space completion of H^{q}(X^{∗}). The data (C^{∗}(Γ̃), H^{·}(X^{∗}), Φ), with Φ as in (3.22), determine a 1–summable spectral triple.

Proof. The Wang exact sequence (2.13) and Corollary 2.13 imply that the hypercohomology H^{·}(X^{∗}) of Cone(N) injects or is mapped upon surjectively by the hypercohomology ℍ^{·}(K, d) of the complex, in a way which is compatible with the grading. Thus, we obtain an induced inner product and Hilbert space completion on H^{·}(X^{∗}). Consider Ker(N) ⊂ gr_{2p}^{w}H^{q}(X̃^{∗}). By Corollary 2.13, we know that this is non-trivial only if q ≥ 2p, and in that case it is given by gr_{2p}^{w}H^{q}(X^{∗}). Thus, we can show that there is an induced representation on ⊕_{2p≤q}gr_{2p}^{w}H^{q}(X^{∗}) by showing that the representation of C^{∗}(Γ̃) on H^{q}(X̃^{∗}) preserves Ker(N).

In the definition of the complex K^{i,j,k} in (2.1), the indices i, j, k and the integers p, q are related by 2p = j +1−i and q = j +1. Thus, the condition q ≥ 2p corresponds to i ≥ 0. The representation σ_{2} of SL(2, ℝ) on K^{i,j} preserves the subspace with i ≥ 0. Similarly, by construction, the representation σ_{2} preserves the subspaces ⊕_{j}K^{i,j,k} of ⊕_{j,k≥i≥0}K^{i,j,k}. This implies that the induced representation σ_{2} on ℍ^{·}(K, d) preserves the summands of gr_{2p}^{w}H^{q}(X̃^{∗}) as in Remark 2.14, and in particular it preserves Ker(N). Thus we obtain a representation

ρ_{Ker(N)} : C^{∗}(Γ̃) → B(Ker(N)).

The duality isomorphisms N^{q−2p} of Proposition 2.15 determine duality isomorphisms between pieces of the hyper-cohomology H^{·}(X^{∗}) of the cone:

\begin{matrix}δ_{0} : gr_{2p}^{w}H^{0}(X^{∗}) →^{≃} gr_{2r}^{w}H^{1}(X^{∗}), & p ≤ 0, r = −p + 1 ≥ 1 \\ δ_{1} : gr_{2p}^{w}H^{1}(X^{∗}) →^{≃} gr_{2r}^{w}H^{2}(X^{∗}), & p ≤ 0, r = −p + 2 ≥ 2 \\ δ_{2} : gr_{2p}^{w}H^{2}(X^{∗}) →^{≃} gr_{2r}^{w}H^{3}(X^{∗}), & p ≤ 1, r = −p + 3 ≥ 2. \end{matrix}  (3.23)

We set δ = ⊕_{q=0}^{2}δ_{q} and we obtain an action of C^{∗}(Γ̃) on H^{·}(X^{∗}) by extending the representation ρ_{Ker(N)} by δ ∘ ρ_{Ker(N)} ∘ δ^{−1} on the part of H^{·}(X^{∗}) dual to Ker(N).

The operator Φ̃ of (3.19) induces the operator Φ of (3.22) on ℍ^{·}(Cone(N)) = H^{·}(X^{∗}). This has the properties of a Dirac operator: the eigenspaces are all finite dimensional by the result of Proposition 2.22, and the commutators are bounded by Theorem 3.7. The spectrum of Φ is given by ℤ, so that Φ^{−1} on the complement of the zero modes is an infinitesimal of order one. ⋄

Remark 3.12 Notice that the spectral triple we obtain is somewhat reminiscent of the spectral triple for the Hodge-Dirac operator (signature). In that case the Hilbert space of spinors H consists of L^{2}-differential forms, whereas in our setting we have a space of forms harmonic with respect to the harmonic theory defined by □ on the complex K^{·,·} (cf. Theorem 2.6 and Corollary 2.7) that are “square integrable” with respect to the inner product (2.8) given by the polarization. Instead of the algebra of functions on a manifold, we have the algebra C^{∗}(Γ̃) which provides a noncommutative version of the manifold 𝔛_{Γ}.

If X is an arithmetic surface over Spec(O_{𝕂}), where O_{𝕂} is the ring of integers of a number field 𝕂 with n = [𝕂 : ℚ], the above result can be applied at each of the n archimedean primes, by choosing at each prime α : 𝕂 ↪ ℂ a Schottky uniformization of the corresponding Riemann surface X_{α(𝕂)}. At the primes that correspond to the r real embeddings, X_{α(𝕂)} has a real structure.

We have the following refinement of Theorems 3.7 and 3.11. The proof is completely analogous, but we use Proposition 3.5 instead of Theorem 3.4.

Proposition 3.13 Let X be an arithmetic surface over Spec(O_{𝕂}), with the property that, at all the real archimedean primes, the Riemann surface X_{α(𝕂)} is an orthosymmetric smooth real algebraic curve of genus g ≥ 2. Let (K^{·,·}, d, N, ℓ, ψ) be the polarized bigraded Lefschetz module associated to X_{/ℝ} = X_{α(𝕂)}. Then the representation σ_{2} extends to representations

ρ : C^{∗}(Γ) → B(ℍ^{·}(K, d)),
ρ_{N} : C^{∗}(Γ) → B(ℍ^{·}(Cone(N)))

with the properties as in Theorems 3.7 and 3.11, where Γ ⊂ SL(2, ℂ) is the Fuchsian Schottky uniformization for X_{/ℝ}.

Remark 3.14 In [24], the condition dim_{H}(Λ_{Γ}) < 1 on the limit set was necessary in order to ensure convergence of the Poincaré series that gives the abelian differentials on X_{/ℂ}, hence in order to express the Green function on X_{/ℂ} in terms of geodesics in the handlebody 𝔛_{Γ}. Notice that this condition is always satisfied for an orthosymmetric smooth real algebraic curve X_{/ℝ}, with the choice of Schottky uniformization described above, since the limit set Λ_{Γ} is contained in the rectifiable circle ℙ^{1}(ℝ).

3.3 Some zeta functions and determinants

The duality isomorphisms N^{q−2p} of Proposition 2.15 and the induced isomorphisms δ_{q} of (3.23) give some further structure to the spectral triple.

Define subspaces H^{±}(X^{∗}) of H^{·}(X^{∗}) in the following way:

\begin{matrix}H^{−}(X^{∗}) = ⊕_{p≤0}gr_{2p}^{w}H^{0}(X^{∗}) ⊕ ⊕_{p≤0}gr_{2p}^{w}H^{1}(X^{∗}) ⊕ ⊕_{p≤1}gr_{2p}^{w}H^{2}(X^{∗}), \\ H^{+}(X^{∗}) := ⊕_{p≥1}gr_{2p}^{w}H^{1}(X^{∗}) ⊕ ⊕_{p≥2}gr_{2p}^{w}H^{2}(X^{∗}) ⊕ ⊕_{p≥2}gr_{2p}^{w}H^{3}(X^{∗}), \end{matrix}  (3.24)

Let δ = ⊕_{q=0}^{2}δ_{q} be the duality isomorphism of (3.23) and set

ω = ( \begin{matrix}0 & δ^{−1} \\ δ & 0 \end{matrix} ) .

The map ω interchanges the subspaces H^{±}(X^{∗}).

Lemma 3.15 The map ω has the following properties:

• ω^{2} = id, ω^{∗} = ω.

• [ω, a] = 0, for all a ∈ C^{∗}(Γ̃).

• (Φω + ωΦ)|_{H^{q}(X^{∗})} = q · id.

Proof. By construction (cf. Theorem 3.11) the action of C^{∗}(Γ̃) commutes with ω. By (3.22), for q ≥ 2p the operator Φ on gr_{2p}^{w}H^{2}(X^{∗}) acts as multiplication by p. The duality isomorphism, mapping gr_{2p}^{w}H^{2}(X^{∗}) to gr_{2(q−p+1)}H^{q+1}(X^{∗}) (cf. Proposition 2.15), and Φ acts on gr_{2(q−p+1)}H^{q+1}(X^{∗}) as multiplication by (q − p). Thus, we obtain

(Φω + ωΦ)|_{gr_{2p}^{w}H^{2}(X^{∗})}(x) = (q − p) · x + p · x = q · x.

⋄

Remark 3.16 Recall that a spectral triple (A, H, D) is called even if there is an operator ω such that ω^{2} = id and ω^{∗} = ω; the commutator [ω, a] = 0, for all a ∈ A and the Dirac operator satisfies Dω +ωD = 0 (cf. [23]). The conditions of Lemma 3.15 provide a weaker version of this notion, depending on the degree of the cohomology H^{q}(X^{∗}).

Due to the presence of this further structure on the spectral triple, determined by the duality isomorphisms, in addition to the family of zeta functions (3.8), (3.9), we can consider zeta functions of the form

ζ_{a,P_{±}Φ}(s, z) := ∑_{λ∈Spec(P_{±}Φ)} Tr(a Π(λ, P_{±}Φ))(s − λ)^{−z},  (3.25)

where P_{±} are the projections on H^{±}(X^{∗}). In this setting we can now recover the Γ-factors.

Theorem 3.17 Consider a = σ_{2}(−id) as an element in Aut(K^{·,·}), acting on H^{·}(X^{∗}) via the induced representation (cf. Theorem 3.11). Then the zeta function (3.25)

ζ_{a,P_{−}Φ}(s, z) := ∑_{λ∈Spec(P_{−}Φ)} Tr(aΠ(λ, P_{−}Φ))(s − λ)^{−z}

satisfies

exp (−\frac{d}{dz} ζ_{a,P_{−}Φ/(2π)}(s/(2π), z)|_{z=0})^{−1} = \frac{L_{ℂ}(H^{1}(X_{/ℂ}, ℂ), s)}{L_{ℂ}(H^{0}(X_{/ℂ}, ℂ), s) · L_{ℂ}(H^{2}(X_{/ℂ}, ℂ), s)}. (3.26)

Proof. Notice that we have P_{−}Φ = Φ|_{H^{·}(X̃^{∗})^{N=0}}. Moreover, recall that the element a = σ_{2}(−id) acts as (−1)^{q−1} on differential forms of degree q. We have

exp (−\frac{d}{dz} ζ_{a,P_{−}Φ/(2π)}(s/(2π), z)|_{z=0}) = ∏_{q=0}^{2} exp (−\frac{d}{dz} ζ_{a,P_{−} \frac{Φ}{2π} |_{H^{q}(X^{∗})}}(s/(2π), z)|_{z=0})

= ∏_{q=0}^{2} exp ((−1)^{q} \frac{d}{dz} ζ_{Φ_{q}}(s/(2π), z)|_{z=0}) ,

where Φ_{q} = Φ|_{H^{q}(X̃^{∗})^{N=0}}. The result then follows by Proposition 2.24. ⋄

We give a few more examples of computations with zeta functions related to the arithmetic spectral triple.

Example 3.18 For Re(s) >> 0, the zeta function (3.6) of the Dirac operator Φ is given by

ζ_{Φ}(s) = Tr(|Φ|^{−s}) = (4g + 4)ζ(s) + 1 + \frac{1}{2^{s}} ,  (3.27)

where ζ(s) is the Riemann zeta function

ζ(s) = ∑_{n≥1} \frac{1}{n^{s}} .

Proof. We compute explicitly Tr(|Φ|^{−s}). On the complement of the zero modes, the operator |Φ| has eigenvalues the positive integers, and the corresponding eigenspaces E_{n}(|Φ|) are described as follows:

\begin{aligned}E_{1}(|Φ|) = & gr_{−2}H^{0}(X^{∗})⊕ \\ & gr_{−2}H^{1}(X^{∗})⊕ & & gr_{4}H^{1}(X^{∗})⊕ \\ & gr_{−2}H^{2}(X^{∗})⊕ & gr_{2}H^{2}(X^{∗})⊕ & gr_{4}H^{2}(X^{∗})⊕ \\ & & gr_{4}H^{3}(X^{∗}), \end{aligned}  (3.28)

\begin{aligned}E_{2}(|Φ|) = & gr_{−4}H^{0}(X^{∗})⊕ \\ & gr_{−4}H^{1}(X^{∗})⊕ & & gr_{6}H^{1}(X^{∗})⊕ \\ & gr_{−4}H^{2}(X^{∗})⊕ & gr_{4}H^{2}(X^{∗})⊕ & gr_{6}H^{2}(X^{∗})⊕ \\ & & & gr_{6}H^{3}(X^{∗}), \end{aligned}  (3.29)

\begin{aligned}E_{n}(|Φ|) = & gr_{−2n}H^{0}(X^{∗})⊕ \\ & gr_{−2n}H^{1}(X^{∗})⊕ & gr_{2(n+1)}H^{1}(X^{∗})⊕ \\ & gr_{−2n}H^{2}(X^{∗})⊕ & gr_{2(n+1)}H^{2}(X^{∗})⊕ \\ & & gr_{2(n+1)}H^{3}(X^{∗}), \end{aligned}  (3.30)

for n ≥ 3.

Using then the result of Proposition 2.22 to compute the dimension of these eigenspaces, we have

dim gr_{2p}H^{0}(X^{∗}) = 1 (p ≤ 0)
dim gr_{2p}H^{1}(X^{∗}) = 2g (p ≤ 0) dim gr_{2p}H^{1}(X^{∗}) = 1 (p ≥ 1)
dim gr_{2p}H^{2}(X^{∗}) = 1 (p ≤ 1) dim gr_{2p}H^{2}(X^{∗}) = 2g (p ≥ 2)
dim gr_{2}H^{2}(X^{∗}) = 1
dim gr_{2p}H^{3}(X^{∗}) = 1 (p ≥ 2)

We obtain

dim E_{n}(|Φ|) = \{ \begin{matrix}(4g + 4) & n ≥ 3 \\ (4g + 5) & n = 1, 2 \end{matrix}  (3.31)

This completes the calculation. Thus, we obtain

Tr(|Φ|^{−s}) = ∑_{n≥1} dim E_{n}(|Φ|)n^{−s} = (4g + 5)(1 + \frac{1}{2^{s}}) + (4g + 4) ∑_{n≥3} n^{−s}

= (4g + 4)ζ(s) + 1 + \frac{1}{2^{s}} .

⋄

As an immediate consequence of this calculation we obtain the volume of the noncommutative space C^{∗}(Γ̃).

Example 3.19 The volume of the non-commutative space C^{∗}(Γ̃), in the metric determined by Φ is given by V = (4g + 4).

Proof. We compute the volume using the residue formula (3.7). Recall that the Riemann zeta function has residue 1 at s = 1. In fact, the well known formula

lim_{s→1+}(s − 1)ζ_{𝕂}(s) = 2^{r_{1}}(2π)^{r_{2}}|d|^{−1/2}hRw^{−1},

holds for an arbitrary number field 𝕂, with class number h, w roots of unity, discriminant d and regulator R, with r_{1} and r_{2} counting the embeddings of 𝕂 into ℝ and ℂ. Applied to 𝕂 = ℚ this yields the result. This implies that, for the zeta function computed in (3.27), we obtain

V = Tr_{ω}(|Φ|^{−1}) = Res_{s=1}Tr(|Φ|^{−s}) = (4g + 4)Res_{s=1}ζ(s) = (4g + 4).  (3.32)

⋄

Notice how, while the handlebody 𝔛Γ̃ in its natural hyperbolic metric has infinite volume, the Dirac operator Φ induces on C^{∗}(Γ̃), which is a non-commutative version of the handlebody, a metric of finite volume.

It is evident from the calculation of the eigenspaces E_{n}(|Φ|) in Example 3.18 that the Dirac operator Φ has a spectral asymmetry (cf. [2]). This corresponds to an eta invariant, which can be computed easily from the dimensions of the eigenspaces in Example 3.18, as follows.

Example 3.20 The eta function of the Dirac operator Φ is given by

η_{Φ}(s) := ∑_{0≠λ∈Spec(Φ)} sign(λ)\frac{1}{|λ|^{s}} = 1 + \frac{1}{2^{s}} ,  (3.33)

The eta invariant η_{Φ}(0) = 2, measuring the spectral asymmetry, is independent of g.

3.4 Zeta function of the special fiber and Reidemeister torsion

In this paragraph we show that the expression (3.26) of Theorem 3.17 can be interpreted as a Reidemeister torsion, and it is related to a zeta function for the fiber at arithmetic infinity.

We begin by giving the definition of a zeta function of the special fiber of a semistable fibration, which motivates the analogous notion at arithmetic infinity.

Let X be a regular, proper and flat scheme over Spec(Λ), for Λ a discrete valuation ring with quotient field K and finite residue field k. Assume that X has geometrically reduced, connected and one-dimensional fibers. Let us denote by η and v resp. the generic and the closed point of Spec(Λ) and by η̄ and v̄ the corresponding geometric points. Assume that the special fiber X_{v} of X is a connected, effective Cartier divisor with reduced normal crossings defined over k = k(v). This degeneration is sometime referred to as a semistable fibration over Spec(Λ).

Let N_{v} denote the cardinality of k. Then, define the zeta-function of the special fiber X_{v} as follows (u is an indeterminate)

Z_{X_{v}}(u) = \frac{P_{1}(u)}{P_{0}(u)P_{2}(u)},  P_{i}(u) = det(1 − f^{∗}u | H^{i}(Xη̄, ℚ_{ℓ})^{Iv̄}),  (3.34)

where f^{∗} is the geometric Frobenius i.e. the map induced by the Frobenius morphism f : Xv̄ → Xv̄ on the cohomological inertia-invariants at v̄.

The polynomials P_{i}(u) are closely related to the characteristic polynomials of the Frobenius F_{i}(u) = det(u · 1 − f^{∗} | H^{i}(Xη̄, ℚ_{ℓ})^{Iv̄}) through the formula

P_{i}(u) = u^{b_{i}}F_{i}(u^{−1}),  b_{i} = degree(F_{i}).  (3.35)

The zeta function Z_{X_{v}}(u) generalizes on a semistable fiber the description of the Hasse-Weil zeta function of a smooth, projective curve over a finite field.

Based on this construction we make the following definition for the fiber at an archimedean prime of an arithmetic surface:

Z_{Φ}(u) := \frac{P_{1}(u)}{P_{0}(u)P_{2}(u)},  (3.36)

where we set

P_{q}(u) := det_{∞} (\frac{1}{2π} − u\frac{Φ_{q}}{2π} ) ,  (3.37)

with Φ_{q} = Φ|_{H^{q}(X̃^{∗})^{N=0}}.

In order to see how this is related to the result of Theorem 3.17, we recall briefly a simple observation of Milnor (cf. §3 [32]). Suppose given a finite complex L and an infinite cyclic covering L̃, with H_{∗}(L̃, κ) finitely generated over the coefficient field κ. Let h : π_{1}L → κ(s) be the composition of the homomorphism π_{1}L → Π associated to the cover with the inclusion Π ⊂ Units(κ(s)). The Reidemeister torsion for this covering is given (up to multiplication by a unit of κΠ) by the alternating product of the characteristic polynomials F_{q}(s) of the κ–linear map

s_{∗} : H_{q}(L̃, κ) → H_{q}(L̃, κ),
τ(s) ≃ F_{0}(s)F_{1}(s)^{−1}F_{2}(s) ⋯ F_{n}(s)^{±1}.  (3.38)

Moreover, for a map T : L → L, let ζ_{T} (u) be the Weil zeta

ζ_{T} (u) = P_{0}(u)^{−1}P_{1}(u)P_{2}(u)^{−1} ⋯ P_{n}(u)^{±1},

where the polynomials P_{q}(u) of the map T_{∗} are related to the characteristic polynomials F_{q}(s) by (3.35) and b_{q} are the q-the Betti number of the complex L. By analogy with (3.38), Milnor writes the Reidemeister torsion τ_{T} (s) (up to multiplication by a unit) as

τ_{T} (s) := F_{0}(s)F_{1}(s)^{−1}F_{2}(s) ⋯ F_{n}(s)^{∓1},

where F_{q}(s) are the characteristic polynomials of the map T_{∗}. Then the relation between zeta function and Reidemeister torsion is given by:

ζ_{T} (s^{−1})τ_{T} (s) = s^{χ(L)},  (3.39)

where χ(L) is the Euler characteristic of L.

Similarly, we can derive the relation between the zeta function of the fiber at infinity defined as in (3.36) and the alternating product of Gamma factors in (3.26).

First notice that the expression (3.26) is of the form (3.38). Namely, we write

\frac{L_{ℂ}(H^{1}(X_{/ℂ}, ℂ), s)}{L_{ℂ}(H^{0}(X_{/ℂ}, ℂ), s) · L_{ℂ}(H^{2}(X_{/ℂ}, ℂ), s)} = \frac{F_{0}(s) · F_{2}(s)}{F_{1}(s)},  (3.40)

where we set

F_{q}(s) := det_{∞} (\frac{s}{2π} − \frac{Φ_{q}}{2π} ) ,  (3.41)

with Φ_{q} = Φ|_{H^{q}(X̃^{∗})^{N=0}}. For this reason we may regard (3.40) as the Reidemeister torsion of the fiber at arithmetic infinity:

τ_{Φ}(s) := \frac{F_{0}(s) · F_{2}(s)}{F_{1}(s)}.  (3.42)

The relation between zeta function and Reidemeister torsion is then given as follows.

Proposition 3.21 The zeta function Z_{Φ} of (3.36) and the Reidemeister torsion τ_{Φ} of (3.42) are related by

Z_{Φ}(s^{−1})τ_{Φ}(s) = s^{g−2}e^{χ s log s},

with g is the genus of the Riemann surface X_{/ℂ} and χ = 2−2g its Euler characteristic.

Proof. The result follows by direct calculation of the regularized determinants as in Section 2.5. Namely, we compute (in the case q = 0, 1)

P_{q}(u) = det_{∞} (\frac{1}{2π} − u\frac{Φ_{q}}{2π} ) = exp (−b_{q} \frac{d}{dz} ((2π)^{z} ∑_{n≥0}(1 + un)^{−z})|_{z=0})

= exp (b_{q} (log Γ (\frac{1}{u}) + \frac{log 2π}{u} + \frac{log u}{2} + \frac{log u}{u})) = u^{b_{q}/2}e^{−b_{q} \frac{log u}{u}} (2π)^{−1/u}Γ(1/u),

where b_{q} are the Betti numbers of X_{/ℂ}. The case q = 2 is analogous, but for the presence of the +1 eigenvalue in the spectrum of Φ_{2}, hence we obtain

P_{2}(u) = exp (−b_{2} \frac{d}{dz} ((2π)^{z}u^{−z}ζ(1/u, z) − (\frac{1}{u} − 1)^{−z})_{z=0})

= Γ_{ℂ} (\frac{1}{u} − 1)^{−1} u^{−3/2}e^{\frac{log u}{u}} .

Thus, we obtain

Z_{Φ}(s^{−1}) = \frac{L_{ℂ}(H^{0}(X_{/ℂ}, ℂ), s) · L_{ℂ}(H^{2}(X_{/ℂ}, ℂ), s)}{L_{ℂ}(H^{1}(X_{/ℂ}, ℂ), s)}s^{g−2}e^{χ s log s}.

⋄

4 Shift operator and dynamics.

In this section we consider an arithmetic surface X over Spec(O_{𝕂}), with 𝕂 a number field, and a fixed archimedean prime which corresponds to a real embedding α : 𝕂 ↪ ℝ. We also assume that the corresponding Riemann surface X_{/ℝ} is an orthosymmetric smooth real algebraic curve of genus g ≥ 2.

We consider a Schottky uniformization of the Riemann surface X_{/ℝ} and the hyperbolic filling given by the handlebody 𝔛_{Γ} that has X_{/ℝ} as the conformal boundary at infinity, 𝔛_{Γ} ∪ X_{/ℝ} = Γ\(ℍ^{3} ∪ ℙ^{1}(ℂ)).

Geodesics in 𝔛_{Γ} can be lifted to geodesics in ℍ^{3} with ends on ℙ^{1}(ℂ). Among these, geodesics with one or both ends on Ω_{Γ} ⊂ ℙ^{1}(ℂ) correspond to geodesics in 𝔛_{Γ} that reach the boundary at infinity X_{/ℝ} = Γ\Ω_{Γ} in infinite time. The geodesics in ℍ^{3} with both ends on Λ_{Γ} ⊂ ℙ^{1}(ℂ) project in the quotient to geodesics contained in the convex core ℭ_{Γ} = Γ\ Hull (Λ_{Γ}) of 𝔛_{Γ}. Since the Schottky group Γ is geometrically finite, ℭ_{Γ} is a bounded region inside 𝔛_{Γ} (cf. [31]). For this reason, geodesics in 𝔛_{Γ} that lift to geodesics in ℍ^{3} with both ends on Λ_{Γ} are called bounded geodesics.

We denote by Ξ ⊂ 𝔛_{Γ} the image under the quotient map of all geodesics in ℍ^{3} with endpoints on Λ_{Γ} ⊂ ℙ^{1}(ℂ), endowed with the induced topology. Similarly, we denote by Ξ_{c} ⊂ Ξ the image in 𝔛_{Γ} of all geodesics in ℍ^{3} with endpoints of the form {z^{−}(h), z^{+}(h)} for some primitive h ∈ Γ. Here z^{±}(h) are the attractive and repelling fixed points of h, and the element h is primitive in Γ if it is not a power of some other element of Γ.

Definition 4.1 We denote by Ξ̃ the orientation double cover of Ξ and we refer to it as the infinite tangle of bounded geodesics. Similarly, we define Ξ̃_{c} to be the orientation double cover of Ξ_{c} and we refer to it as the tangle of primitive closed geodesics.

Since Ξ is orientable, Ξ̃ ≅ Ξ × ℤ/2, where the second coordinate is the choice of an orientation on each geodesic. So, for instance, the geodesics in ℍ^{3} with endpoints {z^{−}(h), z^{+}(h)} and {z^{+}(h), z^{−}(h)} = {z^{−}(h^{−1}), z^{+}(h^{−1})} correspond to the same closed geodesic in Ξ_{c}, while in the double cover Ξ̃_{c} they give rise to the two different lifts of the geodesic in Ξ_{c}.

More precisely, let L_{{a,b}} denote the geodesic in ℍ^{3} ≅ ℂ × ℝ^{+} with endpoints {a, b} in the complement of the diagonal in ℙ^{1}(ℂ) × ℙ^{1}(ℂ),

L_{\{a,b\}}(s) = (\frac{ae^{s} + be^{−s}}{e^{s} + e^{−s}} , \frac{|a − b|}{e^{s} + e^{−s}} ) s ∈ ℝ  (4.1)

The parameter s in (4.1) becomes the parameterization by arc length on the corresponding geodesic π_{Γ}(L_{{a,b}}) in Ξ̃, where π_{Γ} : ℍ^{3} → 𝔛_{Γ} is the quotient map. Then we have

Ξ̃ = {π_{Γ}(L_{{a,b}}(s)) : s ∈ ℝ, (a, b) ∈ (Λ_{Γ} × Λ_{Γ})^{0}},

where

(Λ_{Γ} × Λ_{Γ})^{0} := (Λ_{Γ} × Λ_{Γ}) ∖ Δ

denotes the complement of the diagonal in Λ_{Γ} × Λ_{Γ}.

Remark 4.2 The ℤ/2 involution on Ξ̃ that has Ξ as quotient corresponds to the involution that exchanges the two factors in Λ_{Γ} × Λ_{Γ}.

Remark 4.3 Notice that most constructions and results presented in this section are topological in nature, hence they do not require any assumption on the conformal structure of the Riemann surface X_{/ℝ}. In particular they are not sensitive to whether X is a complex or real smooth algebraic curve. However, when we refer to the results of the previous sections, by comparing the dynamical and archimedean cohomology in Theorem 5.5, we need to know that the ℤ/2 cover Ξ̃ is compatible with the conformal structure on the Riemann surface. This requires the presence of the real structure, so that complex conjugation z ↦ z̄ on ℙ^{1}(ℂ), which induces the change of orientation on the geodesics L_{{a,b}} in ℍ^{3}, determines an involution on the Γ–quotient.

According to [24], the tangle of bounded geodesics Ξ̃ provides a geometric realization of the dual graph G_{∞} of the maximally degenerate closed fiber at arithmetic infinity.

In this section, we consider a resolution of Ξ̃ by the mapping torus (suspension flow) S_{T} of a dynamical system T (cf. Proposition 4.11), with a surjection S_{T} → Ξ̃.

In Theorem 4.12 we give an explicit description of the cohomology H^{1}(S_{T}). Such cohomology is endowed with a filtration whose graded pieces depend uniquely on the coding of geodesics in Ξ̃ (cf. Proposition 4.14 and Proposition 4.15). We also show that the C^{∗}-algebra describing the quotient Λ_{Γ}/Γ acts on the cohomology H^{1}(S_{T})

We recall the construction and basic properties of the Cuntz-Krieger algebra O_{A} associated to the shift of finite type (S, T). In Theorem 4.17 we show that this algebra describes, as a non-commutative space, the quotient of the limit set Λ_{Γ} by the action of the Schottky group Γ. Moreover, in Proposition 4.19, we show that the algebra O_{A} acts on a suitable Hilbert space completion of the cohomology H^{1}(S_{T}).

4.1 The limit set and the shift operator

Given a choice of a set of generators {g_{i}}_{i=1}^{g} for the Schottky group Γ, there is a bijection between the elements of Γ and the set of all reduced words in the {g_{i}}_{i=1}^{2g}, where we use the notation g_{i+g} := g_{i}^{−1}, for i = 1, …, g. Here by reduced words we mean all finite sequences w = a_{0} … a_{ℓ} in the g_{i}, for any ℓ ∈ ℕ, satisfying a_{i+1} ≠ a_{i} for all i = 0 …, ℓ − 1.

We also consider the set S^{+} of all right–infinite reduced sequences in the {g_{i}}_{i=1}^{2g},

S^{+} = {a_{0}a_{1} … a_{ℓ} … |a_{i} ∈ {g_{i}}_{i=1}^{2g}, a_{i+1} ≠ a_{i}, ∀i ∈ ℕ},  (4.2)

and the set S of doubly infinite reduced sequences in the {g_{i}}_{i=1}^{2g},

S = {… a_{−m} … a_{−1}a_{0}a_{1} … a_{ℓ} … |a_{i} ∈ {g_{i}}_{i=1}^{2g}, a_{i+1} ≠ a_{i}, ∀i ∈ ℤ}.  (4.3)

On the space S we consider the topology generated by the sets W^{s}(x, ℓ) = {y ∈ S|x_{k} = y_{k}, k ≥ ℓ}, and the W^{u}(x, ℓ) = {y ∈ S|x_{k} = y_{k}, k ≤ ℓ} for x ∈ S and ℓ ∈ ℤ. This induces a topology with analogous properties on S^{+} by realizing it as a subset of S, for instance, by extending each sequence to the left as a constant sequence.

We define the one-sided shift operator T on S^{+} as the map

T (a_{0}a_{1}a_{2} … a_{ℓ} …) = a_{1}a_{2} … a_{ℓ} …  (4.4)

We also define a two-sided shift operator T on S as the map

\begin{aligned}T ( & … & a_{−m} & … & a_{−1} & a_{0} & a_{1} & … & a_{ℓ} & … & ) = \\ & … & a_{−m+1} & … & a_{0} & a_{1} & a_{2} & … & a_{ℓ+1} & … \end{aligned}  (4.5)

Given a choice of a base point x_{0} ∈ ℍ^{3} ∪ Ω_{Γ} we can define a map Z : S^{+} → Λ_{Γ} in the following way. For an eventually periodic sequence w\overline{a_{0} … a_{N}} ∈ S^{+}, with an initial word w, we set

Z(w\overline{a_{0} … a_{N}}) = wz^{+}(a_{0} … a_{N}).  (4.6)

Here we identify the finite reduced word w with an element in Γ, hence wz^{+}(a_{0} … a_{N}) is the image under w ∈ Γ of the attractive fixed point of the element a_{0} … a_{N} of Γ. For sequences a_{0} … a_{ℓ} … that are not eventually periodic, we set

Z(a_{0} … a_{ℓ} …) = lim_{ℓ→∞}(a_{0} … a_{ℓ})x_{0},  (4.7)

where again we identify a finite reduced word a_{0} … a_{ℓ} with an element in Γ.

We also introduce the following notation. We denote by S^{+}(w) ⊂ S^{+} the set of right infinite reduced sequences in the {g_{i}}_{i=1}^{2g} that begin with an assigned word w = a_{0} … a_{ℓ}. If g ∈ Γ is expressed as a reduced word w in the g_{i}, we write Λ_{Γ}(g) := Z(S^{+}(w)).

In the following we denote by Λ_{Γ} ×_{Γ} Λ_{Γ} the quotient by the diagonal action of Γ of the complement of the diagonal (Λ_{Γ} × Λ_{Γ})^{0}.

The group Γ acts on ℙ^{1}(ℂ) by fractional linear transformations, hence on Λ_{Γ}, which is a Γ-invariant subset of ℙ^{1}(ℂ). The group Γ also acts on S^{+}: an element γ ∈ Γ, identified with a reduced word γ = c_{0} … c_{k} in the g_{i}, maps a sequence a_{0} … a_{ℓ} … to the sequence obtained from c_{0} … c_{k}a_{0} … a_{ℓ} … by making the necessary cancellations that yield a reduced sequence.

Lemma 4.4 The following properties are satisfied.

1. The spaces S^{+} and S are topologically Cantor sets. The one-sided shift T of (4.4) is a continuous surjective map on S^{+}, while the two-sided shift T of (4.5) is a homeomorphism of S.

2. The limit set Λ_{Γ} with the topology induced by the embedding in ℙ^{1}(ℂ) is also a Cantor set, and the map Z of (4.6) and (4.7) is a homeomorphism.

The shift operator T on S^{+} induces the map ZT Z^{−1} : Λ_{Γ} → Λ_{Γ} of the form

ZT Z^{−1}|_{Λ_{Γ}(g_{i})}(z) = g_{i}^{−1}(z).

3. The map Z is Γ-equivariant.

Proof. 1. The first claim can be verified easily.

2. It is not hard to see that the correspondence a_{0} … a_{ℓ} … ↦ lim_{ℓ→∞}(a_{0} … a_{ℓ})x_{0} gives a bijection between the complement of eventually periodic sequences in S^{+} and the complement of the fixed points {z^{−}(h), z^{+}(h)}_{h∈Γ} in Λ_{Γ} (cf. [17] Prop. 1.2). To see that the correspondence w\overline{a_{0} … a_{N}} ↦ w z^{+}(a_{0} … a_{N}) = z^{+}(wa_{0} … a_{N}w^{−1}) is a bijection between the set of eventually periodic sequences and the set of fixed points {z^{−}(h), z^{+}(h)}_{h∈Γ} in Λ_{Γ}, we proceed as in [17]. For any h ∈ Γ, h can be written as a reduced word h = a_{0} … a_{ℓ} in the g_{i}’s. If this word satisfies a_{ℓ} ≠ a_{0}^{−1}, then Z^{−1}(z^{+}(h)) = \overline{a_{0} … a_{ℓ}} and Z^{−1}(z^{−}(h)) = \overline{a_{ℓ}^{−1} … a_{0}^{−1}}. If a_{ℓ} = a_{0}^{−1}, then there is an element γ ∈ Γ, such that h = γa_{i_{k}} … a_{i_{k+N}} γ^{−1}, with a_{i_{k+N}} ≠ a_{i_{k}}^{−1}. In this case Z^{−1}(z^{+}(h)) = γ\overline{a_{i_{k}} … a_{i_{k+N}}} and Z^{−1}(z^{−}(h)) = γ\overline{a_{i_{k+N}}^{−1} … a_{i_{k}}^{−1}}. As a continuous bijection from a compact to a Hausdorff space, Z is a homeomorphism. The expression for ZT Z^{−1} is then immediate.

3. By continuity it is sufficient to check that the map Z restricted to the dense subset {z^{−}(h), z^{+}(h)}_{h∈Γ} ⊂ Λ_{Γ} is Γ-equivariant. This was proved already in 2., since Z(γ\overline{a_{0} … a_{N}}) = γ z^{+}(a_{0} … a_{N}). ⋄

We recall two notions that will be useful in the following.

Let A = (A_{ij}) be an N × N elementary matrix. The subshift of finite type with transition matrix A is the subset of the set of all doubly infinite sequences in the alphabet {1, …, N} of the form

S_{A} := {… i_{−m} … i_{−1}i_{0}i_{1} … i_{ℓ} … |1 ≤ i_{k} ≤ N, A_{i_{k}i_{k+1}} = 1, ∀k ∈ ℤ}.  (4.8)

A double sided shift operator of the form (4.5) can be defined on any subshift of finite type.

In the following we will consider the case where the elementary matrix A is the the symmetric 2g × 2g matrix with A_{ij} = 0 for |i − j| = g and A_{ij} = 1 otherwise.

Definition 4.5 The pair (S, T) of a space and a homeomorphism is a Smale space if locally S can be decomposed as the product of expanding and contracting directions for T. Namely, the following properties are satisfied.

1. For every point x ∈ S there exist subsets W^{s}(x) and W^{u}(x) of S, such that W^{s}(x) × W^{u}(x) is homeomorphic to a neighborhood of x.

2. The map T is contracting on W^{s}(x) and expanding on W^{u}(x), and W^{s}(T x) and T (W^{s}(x)) agree in some neighborhood of x, and so do W^{u}(T x) and T (W^{u}(x)).

Lemma 4.6 The following properties are satisfied.

1. The map Q : S → Λ_{Γ} × Λ_{Γ}

Q(… a_{−m} … a_{−1}a_{0}a_{1} … a_{ℓ} …) = (Z(a_{−1}^{−1}a_{−2}^{−1} … a_{−m}^{−1} …) , Z(a_{0}a_{1}a_{2} … a_{ℓ} …))

is an embedding of the space S in the Cartesian product Λ_{Γ} × Λ_{Γ}. The image of the embedding Q is given by Im(Q) = ⋃_{i≠j} Λ_{Γ}(g_{i}) × Λ_{Γ}(g_{j}). On Im(Q) the two-sided shift operator T of (4.5) induces the map QT Q^{−1}

\begin{matrix}QT Q^{−1}|_{Λ_{Γ}(g_{i})×Λ_{Γ}(g_{j})}( & Z(g_{i}b_{1}b_{2} … b_{m} …), & & Z(g_{j}a_{1} … a_{ℓ} …) & ) = \\ ( & Z(g_{j}^{−1}g_{i}b_{1} … b_{m} …), & g_{j}^{−1}g_{j} & Z(a_{1}a_{2} … a_{ℓ} …) & ). \end{matrix}

2. The map Q : S → Λ_{Γ} × Λ_{Γ} descends to a homeomorphism of the quotients

Q̄ : S/T →^{≃} Λ_{Γ} ×_{Γ} Λ_{Γ}.

3. The space S can be identified with the subshift of finite type S_{A} with the symmetric 2g × 2g matrix A = (A_{ij}) with A_{ij} = 0 for |i − j| = g and A_{ij} = 1 otherwise.

The two-sided shift operator T on S of (4.5) decomposes S in a product of expanding and contracting directions, so that (S, T) is a Smale space.

Proof. 1. The first claim follows directly from the definitions and Lemma 4.4.

2. Notice that Im(Q) intersects each Γ orbit in the complement of the diagonal (Λ_{Γ} × Λ_{Γ})^{0}. Moreover, if (a, b) = Q(x), then for γ ∈ Γ the element (γa, γb) is in Im(Q) iff b ∈ Λ_{Γ}(γ^{−1}), and in that case (γa, γb) = Q(T^{n}x), for n = length (γ) as a reduced word in the g_{i}. The statement on Q̄ then follows easily.

3. We write S as the shift of finite type S_{A}. Namely, we identify a reduced sequence … a_{−m} … a_{−1}a_{0}a_{1} … a_{ℓ} … where each a_{k} = g_{i_{k}} ∈ {g_{i}}_{i=1}^{2g} with the sequence … i_{−m} … i_{−1}i_{0}i_{1} … i_{ℓ} … satisfying A_{i_{k}i_{k+1}} = 1, for all k ∈ ℤ.

Then, for subshifts of finite type with the double sided shift (4.5), the sets

W^{u}(x) = ∪_{ℓ∈ℤ}W^{u}(x, ℓ),

with

W^{u}(x, ℓ) := {y ∈ S|x_{k} = y_{k}, k ≤ ℓ}

and

W^{s}(x) = ∪_{ℓ∈ℤ}W^{s}(x, ℓ).

with

W^{s}(x, ℓ) := {y ∈ S|x_{k} = y_{k}, k ≥ ℓ}

give the expanding and contracting directions, so that (S, T) satisfies the properties of a Smale space (cf. [37]). ⋄

Remark 4.7 It is well known that one can associate different C^{∗}–algebras to Smale spaces (cf. [42], [37], [38]). For the Smale space (S, T) we consider, there are four possibilities: the crossed product algebra C(S) ⋊_{T} ℤ and the C^{∗}–algebras C^{∗}(G^{s}) ⋊_{T} ℤ, C^{∗}(G^{u}) ⋊_{T} ℤ, C^{∗}(G^{a}) ⋊_{T} ℤ obtained by considering the action of the shift T on the groupoid C^{∗}–algebra (cf. [40] for the definition of such algebra) associated to the groupoids G^{s}, G^{u}, G^{a} of the stable, unstable, and asymptotic equivalence relations on (S, T).

In the following we consider the algebras C(S) ⋊_{T} ℤ and C^{∗}(G^{u}) ⋊_{T} ℤ and show that the first is a non-commutative space describing the quotient Λ_{Γ} ×_{Γ} Λ_{Γ} and the second is a non-commutative space describing the quotient Λ_{Γ}/Γ.

4.2 Coding of geodesics

Since the Schottky group Γ is a free group consisting of only loxodromic elements, the coding of geodesics in 𝔛_{Γ} in terms of the dynamical system (S, T) is particularly simple. The following facts are well known. We recall them briefly for convenience.

We denote by S^{p} ⊂ S the set of periodic reduced sequences in the g_{i}, i.e. the set of periodic points of the shift T. We define

Ξ̂ := {π_{Γ}(L_{{a,b}}) : (a, b) ∈ (Λ_{Γ} × Λ_{Γ})^{0}} ,

and

Ξ̂_{c} := {π_{Γ}(L_{{z^{+}(h),z^{−}(h)}}) : h ∈ Γ \ id} ,

where L_{{a,b}} denotes the geodesic as in (4.1).

The following Lemma gives a coding of the primitive closed geodesics in Ξ̃_{c} by the quotient S^{p}/T.

Lemma 4.8 The correspondence L_{c} : w\overline{a_{0} … a_{N}} ↦ π_{Γ}(L_{\{w z^{+}(a_{0}...a_{N}),w z^{−}(a_{0}...a_{N})\}}) induces a bijection between S^{p}/T and Ξ̂_{c}

Proof. Arguing as in Lemma 4.4, we see that every closed geodesic in Ξ̃_{c} is of the form π_{Γ}(L_{{z^{+}(a_{0}...a_{N}),z^{−}(a_{0}...a_{N})}}) for some reduced sequence a_{0} … a_{N} with a_{N} ≠ a_{0}^{−1}. If two elements \overline{a_{0}, a_{2}, …, a_{N}} and \overline{b_{0}, b_{1}, …, b_{M}} of S^{p} represent the same primitive closed geodesic, then the elements h_{a} = a_{0}a_{2} ⋯ a_{N} and h_{b} = b_{0}b_{1} ⋯ b_{M} are conjugate in Γ, h_{a} = gh_{b}g^{−1}, by an element g = c_{1}c_{2} ⋯ c_{k}. It is easy to see that this implies c_{k} = b_{0}^{−1}, c_{k−1} = b_{1}^{−1}, etc. so that, for some 1 ≤ N_{0} ≤ N, we have

\overline{b_{0}, b_{1}, …, b_{M}} = T^{N_{0}}(\overline{a_{0}, a_{1}, …, a_{N}}),

that is, the two sequences are in the same equivalence class modulo the action of T. We refer to [17] for details. ⋄

Via the map L_{c} of Lemma 4.8 we can define on Ξ̂_{c} a topology which makes it homeomorphic to the quotient S^{p}/T.

Similarly, we obtain a coding of geodesics in Ξ̃ by the quotient space S/T.

Lemma 4.9 The map L : S → Ξ̂, L : x ↦ π_{Γ}(L_{{a,b}}), for x ∈ S and (a, b) = Q(x) in Λ_{Γ} × Λ_{Γ}, induces a bijection between S/T and Ξ̂.

Proof. Any geodesic in Ξ̃ lifts to a geodesic in ℍ^{3} with ends (a, b) in the complement of the diagonal (Λ_{Γ} × Λ_{Γ})^{0}. Notice that in ℍ^{3} we have γL_{{a,b}} = L_{{γa,γb}}. The claim then follows easily. ⋄

Via the map L of Lemma 4.9 we can define on Ξ̂ a topology which makes it homeomorphic to the quotient S/T.

We introduce a topological space defined in terms of the Smale space (S, T), which we consider as a graph associated to the fiber at arithmetic infinity. This maps onto the dual graph Ξ̃ considered in [24].

Definition 4.10 The mapping torus (suspension flow) of the dynamical system (S, T) is defined as

S_{T} := S × [0, 1]/(x, 0) ∼ (T x, 1)  (4.9)

Consider the map Q̃ : S_{T} → Ξ̃, defined by

Q̃([x, t]) = π_{Γ}L_{{a,b}}(s(t)),  (4.10)

where (a, b) = Q(x) in (Λ_{Γ} × Λ_{Γ})^{0}. The smooth change of parameterization s(t) from (4.1) is introduced so that the geodesic line L_{{a,b}}(s(t)) in ℍ^{3} crosses a fundamental domain for the action of Γ in time t ∈ [0, 1].

Proposition 4.11 The map Q̃ : S_{T} → Ξ̃ of (4.10) is a continuous surjection. It is a bijection away from the intersection points of different geodesics in Ξ̃.

Proof. The parameterization s(t) is chosen in such a way that the map (4.10) is well defined on equivalence classes. By 2. of Lemma 4.6 and Lemma 4.9 the map is a continuous surjection. ⋄

4.3 Cohomology and homology of S_{T}

We give an explicit description of the cohomology H^{1}(S_{T} , ℤ).

Theorem 4.12 The cohomology H^{1}(S_{T} , ℤ) satisfies the following properties.

1. There is an identification of H^{1}(S_{T} , ℤ) with the K_{0}-group of the crossed product C^{∗}-algebra for the action of T on S,

H^{1}(S_{T} , ℤ) ≅ K_{0}(C(S) ⋊_{T} ℤ).  (4.11)

2. The identification (4.11) endows H^{1}(S_{T} , ℤ) with a filtration by free abelian groups F_{0} ↪ F_{1} ↪ ⋯ F_{n} ↪ ⋯, with rankF_{0} = 2g and rankF_{n} = 2g(2g − 1)^{n−1}(2g − 2) + 1, for n ≥ 1, so that

H^{1}(S_{T} , ℤ) = lim_{n} F_{n}.

Proof. 1. The shift T acting on S induces an automorphism of the C^{∗}–algebra of continuous functions C(S). With an abuse of notation we still denote it by T. Consider the crossed product C^{∗}–algebra C(S)⋊_{T} ℤ. This is a suitable norm completion of C(S)[T, T^{−1}] with product (V ∗ W)_{k} = ∑_{r∈ℤ} V_{k} · (T^{r}W_{r+k}), for V = ∑_{k} V_{k}T^{k}, W = ∑_{k} W_{k}T^{k}, and V ∗ W = ∑_{k}(V ∗ W)_{k}T^{k}.

The K–theory group K_{0}(C(S) ⋊_{T} ℤ) is described by the co–invariants of the action of T (cf. [8],[34]). Namely, let C(S, ℤ) be the set of continuous functions from S to the integers. This is an abelian group generated by characteristic functions of clopen sets of S. The invariants and co–invariants are given respectively by C(S, ℤ)^{T} := {f ∈ C(S, ℤ) |f − f ∘ T = 0} and C(S, ℤ)_{T} := C(S, ℤ)/B(S, ℤ), with B(S, ℤ) := {f − f ∘ T |f ∈ C(S, ℤ)}. We have the following result (cf. [8]):

• The C^{∗}–algebra C(S) is a commutative AF–algebra (approximately finite dimensional), obtained as the direct limit of the finite dimensional commutative C^{∗}–algebras generated by characteristic functions of a covering of S. Thus, K_{0}(C(S)) ≅ C(S, ℤ), being the direct limit of the K_{0}-groups of the finite dimensional commutative C^{∗}–algebras, and K_{1}(C(S)) = 0 for the same reason.

• The Pimsner–Voiculescu exact sequence (cf. [35]) then becomes of the form

0 → K_{1}(C(S) ⋊_{T} ℤ) → C(S, ℤ) →^{I−T_{∗}} C(S, ℤ) → K_{0}(C(S) ⋊_{T} ℤ) → 0, (4.12)

with K_{0}(C(S) ⋊_{T} ℤ) ≅ C(S, ℤ)_{T}. Since the shift T is topologically transitive, i.e. it has a dense orbit, we also have K_{1}(C(S) ⋊_{T} ℤ) ≅ C(S, ℤ)^{T} ≅ ℤ.

Now consider the cohomology group H^{1}(S_{T} , ℤ). Via the identification with Čech cohomology, we can identify H^{1}(S_{T} , ℤ) with the group of homotopy classes of continuous maps of S_{T} to the circle. The isomorphism

C(S, ℤ)_{T} ≅ H^{1}(S_{T} , ℤ)  (4.13)

is then given explicitly by

f ↦ [exp(2πitf(x))],  (4.14)

for f ∈ C(S, ℤ) and with [·] the homotopy class. The map is well defined on the equivalence class of f mod B(S, ℤ) since, for an element f − h + h ∘ T the function

exp(2πit(f − h + h ∘ T)(x)) = exp(2πitf(x)) exp(2πi((1 − t)h(x) + th(T (x)))),

since h is integer valued, but exp(2πi((1 − t)h(x) + th(T (x)))) is homotopic to the constant function equal to 1. It is not hard to see that (4.14) gives the desired isomorphism (4.13) (cf. §4-5 [8]). This proves the first statement.

2. There is a filtration on the set of coinvariants C(S, ℤ)_{T} (cf. Theorem 19 §4 of [34]). It is obtained in the following way. First, it is possible to identify

C(S, ℤ)_{T} = C(S, ℤ)/B(S, ℤ) ≅ P/δP,  (4.15)

where P ⊂ C(S, ℤ) is the set of functions that depend only on future coordinates, and δ is the operator δ(f) = f − f ∘ T. In fact, since characteristic functions of clopen sets in S depend only on finitely many coordinates, any function in C(S, ℤ), when composed with a sufficiently high power of T becomes a function only of the future coordinates, i.e. of S^{+}, the set of (right) infinite reduced sequences in the generators of Γ and their inverses, {g_{i}}_{i=1}^{2g}. Since all the f ∘ T^{k}, k ≥ 0, define the same equivalence class in C(S, ℤ)_{T}, we have the identification of (4.15).

Then P can be identified with C(S^{+}, ℤ) viewed as the submodule of the ℤ-module C(S, ℤ) of functions that only depend on future coordinates. As such, it is generated by characteristic functions of clopen subsets of S^{+}. A basis of clopen sets for the topology of S^{+} is given by the sets S^{+}(w) ⊂ S^{+}, where w = a_{0} … a_{N} is a reduced word in the g_{i} and S^{+}(w) is the set of reduced right infinite sequences b_{0}b_{1}b_{2} … b_{n} … such that b_{k} = a_{k} for k = 0 … N. Thus, P has a filtration P = ∪_{n=0}^{∞}P_{n}, where P_{n} is generated by the characteristic functions of S^{+}(w) with w of length at most n + 1. Taking into account the relations between these, we obtain that P_{n} is a free abelian group generated by the characteristic functions of S^{+}(w) with w of length exactly n+1. The number of such words is 2g(2g − 1)^{n}, hence rankP_{n} = 2g(2g − 1)^{n}.

The map δ satisfies δ : P_{n} → P_{n+1}, with a 1-dimensional kernel given by the constant functions. More precisely, if we write f(a_{0}...a_{n}) for a function in P_{n}, then

(δf)(a_{0} … a_{n}a_{n+1}) = f(a_{0} … a_{n}) − f(a_{1} … a_{n+1}).

The resulting quotients

F_{n} = P_{n}/δP_{n−1}

are torsion free (cf. Theorem 19 §4 of [34]) and have ranks

rankF_{n} = 2g(2g − 1)^{n−1}(2g − 2) + 1

for n ≥ 1, while F_{0} ≅ P_{0} is of rank 2g. There is an injection F_{n} ↪ F_{n+1} induced by the inclusion P_{n} ⊂ P_{n+1}, and P/δP is the direct limit of the F_{n} under these inclusions. Thus we obtain the filtration on H^{1}(S_{T} , ℤ):

H^{1}(S_{T} , ℤ) = lim_{n} F_{n}.

This proves the second statement. ⋄

Remark 4.13 There is an interesting degree shift between K–group K_{0}(C(S)⋊_{T} ℤ) of the crossed product algebras associated to the Smale space (S, T) and the cohomology H^{1}(S_{T} , ℤ). There are other examples in non-commutative geometry where K_{0} of a noncommutative space replaces the H^{1} of its commutative counterpart. This happens, for instance, in the study of of non-commutative elliptic curves, where the Hodge filtration on H^{1} has its non-commutative analog as a filtration on K_{0}, but not on K_{1}, [9] p. 132– 139, [11] §XIII.

The following result computes the first homology of S_{T}.

Proposition 4.14 The homology group H_{1}(S_{T} , ℤ) has a filtration by free abelian groups K_{N},

H_{1}(S_{T} , ℤ) = lim_{N} K_{N},  (4.16)

with

K_{N} = rank(K_{N}) = \{ \begin{matrix}(2g − 1)^{N} + 1 & N even \\ (2g − 1)^{N} + (2g − 1) & N odd \end{matrix}

The group H_{1}(S_{T} , ℤ) can also be written as

H_{1}(S_{T} , ℤ) = ⊕_{N=0}^{∞}R_{N}

where R_{n} is a free abelian group of ranks R_{1} = 2g and

R_{N} = rank(R_{N}) = \frac{1}{N} ∑_{d|N} μ(d) (2g − 1)^{N/d},

for N > 1, with μ the Möbius function. This is isomorphic to the free abelian group on countably many generators, ℤ⟨S^{p}/T ⟩.

Proof. The lines {[x, t] : t ∈ ℝ} are pairwise disjoint in the mapping torus S_{T}, hence H_{1}(S_{T} , ℤ) is generated by closed curves {[x, t] : t ∈ ℝ}. By construction, a closed curve in S_{T} corresponds to a point x ∈ S such that T^{N}x = x for some N ≥ 1. More precisely, if x = \overline{a_{0} … a_{N}} is a doubly infinite periodic sequence in S, obtained by repeating the word a_{0} … a_{N}, so that T^{N}x = x, then the map c_{x} : S^{1} → S_{T} of the form

c_{x} : e^{2πit} ↦ [(x, Nt)], T^{N}x = x.  (4.17)

defines a closed curve in S_{T}, which is a non-trivial homology class in H_{1}(S_{T} , ℤ). Since all non-trivial homology classes can be obtained this way, the homology H_{1}(S_{T} , ℤ) is generated by all c_{x} as in (4.17), for x a reduced word a_{0} … a_{N} such that a_{N} ≠ a_{0}^{−1}.

Let K_{N} be the free abelian groups generated by all the reduced words a_{0} … a_{N} of length N + 1 satisfying a_{N} ≠ a_{0}^{−1}. When we identify the elements of the K_{N} with homology classes via (4.17), we introduce relations between the K_{N} for different N, namely we have embeddings kK_{N} ↪ K_{kN},

k⟨a_{0} … a_{N}⟩ ↦ ⟨\underbrace{a_{0} … a_{N}a_{0} … a_{N} … a_{0} … a_{N}}_{k−times}⟩.  (4.18)

Thus, the homology H_{1}(S_{T} , ℤ) is computed as the limit H_{1}(S_{T} , ℤ) = lim_{N} K_{N} with respect to the maps J_{k} : K_{N} ⟶ K_{kN} that send the c_{x} of (4.17) to the composite

S^{1} ⟶^{z↦z^{k}} S^{1} →^{c_{x}} S_{T} .

Thus, the homology H_{1}(S_{T} , ℤ) can be identified with the ℤ-module generated by the elements of S^{p}/T. This quotient can be written as a disjoint union S^{p}/T = ∪_{n=0}^{∞}S_{n}^{p}/T, where S_{n}^{p} ⊂ S^{p} is the subset of (primitive) periodic sequences with period of length n + 1. This gives the description of H_{1}(S_{T} , ℤ) as direct sum of the R_{n} = ℤ⟨S_{n}^{p}/T ⟩.

The computation of the ranks of the K_{N} and of the R_{n} is obtained as follows. For a fixed a_{0}, denote by p(N, k) the number of reduced sequences a_{0} … a_{N}, such

that the last k terms are all equal to a_{0}^{−1}. Then 2g · p(N, 0) = rankK_{N}. Moreover, since the total number of all reduced sequences of length N + 1 is 2g(2g − 1)^{N}, we have ∑_{k=0}^{N} p(N, k) = (2g − 1)^{N}. It is not hard to see from the definition that the p(N, k) satisfy p(N, k) = p(N + 1, k + 1), p(N, N) = 0, p(1, 0) = 2g − 1, p(N, 0) = p(N + 1, 1) + p(N − 1, 0).

The calculation of p(N, 0) then follows inductively, using p(N, 1) = p(N − 1, 0) − p(N − 2, 0), and the sum ∑_{k=0}^{N} p(N, k) = (2g − 1)^{N}, where p(N, k) = p(N − k + 1, 1).

The rank of the R_{N} can be computed by first considering that K_{N} = ∑_{d|N} d R_{d}, since the total number of reduced sequences a_{0} … a_{N} with a_{N} ≠ a_{0}^{−1} is the sum of the cardinalities of the S_{d}^{p} over all d dividing N. These satisfy #S_{d}^{p} = d R_{d}. Then we obtain

R_{N} = \frac{1}{N} ∑_{d|N} μ(d)K_{N/d} = \frac{1}{N} ∑_{d|N} μ(d)(2g − 1)^{N/d},

using the Möbius inversion formula in the first equality and the fact that ∑_{d|N} μ(d) = δ_{N,1} in the second. ⋄

Combining Theorem 4.12 with Proposition 4.14, we can compute explicitly the pairing of homology and cohomology for S_{T}. This relates the filtration by the F_{n} on the cohomology of S_{T} to the coding of closed geodesics in Ξ̃_{c}.

Proposition 4.15 Let F_{n} and K_{N} be the filtrations defined, respectively, in Theorem 4.12 and Proposition 4.14. There is a pairing

⟨·, ·⟩ : F_{n} × K_{N} → ℤ ⟨[f], x⟩ = N · f(x̄),  (4.19)

with x = a_{0} … a_{N}. Here the representative f ∈ [f] is a function that depends on the first n + 1 terms a_{0} … a_{n} of sequences in S, and x̄ is the truncation of the periodic sequence \overline{a_{0} … a_{N}} after the first n terms. This pairing descends to the direct limits of the filtrations, where it agrees with the classical cohomology/homology pairing

⟨·, ·⟩ : H^{1}(S_{T} , ℤ) × H_{1}(S_{T} , ℤ) → ℤ.  (4.20)

Proof. First, it is not hard to check that the pairing (4.19) is compatible with the maps F_{n} ↪ F_{n+1} and J_{k} : K_{N} → K_{kN}. In fact, (4.19) is invariant under the maps F_{n} ↪ F_{n+1}, while under the map J_{k} : K_{N} ↪ K_{kN} we have

⟨[f], J_{k}(x)⟩ = kNf(x̄) = k ⟨[f], x⟩.

Thus, (4.19) induces a pairing of the direct limits

⟨·, ·⟩ : lim_{n} F_{n} × lim_{N} K_{N} → ℤ.  (4.21)

In order to check that (4.21) agrees with the cohomology/homology pairing (4.20), notice that a class c in the homology H_{1}(S_{T} , ℤ) is realized as a finite linear combination of oriented circles in S_{T}, where each such circle is described by a map c_{x} : S^{1} → S_{T} of the form (4.17).

The pairing ⟨u, c⟩ of an element u of the cohomology H^{1}(S_{T} , ℤ) with a generator c of the homology H_{1}(S_{T} , ℤ) is given by the homotopy class [u ∘ c] of

u ∘ c : S^{1} ⟶ S^{1}.

We write u(x, t) = [2πitf(x)], for a generator of H^{1}(S_{T} , ℤ) in F_{n}, where f is an element in P_{n}, which depends only on the first n+1 terms in the sequences a_{0} … a_{n} … in S_{T} (cf. Theorem 4.12). If y = a_{0} … a_{d} is an element in R_{d} of period d, and c is the corresponding generator of H_{1}(S_{T} , ℤ) of the form c_{x}(t) = [(x, d · t) : x = \overline{a_{0} … a_{d}}], then the homotopy class [u ∘ c_{x}] ∈ π_{1}(S^{1}) = ℤ is equal to d · f(x) and this proves the claim. ⋄

4.4 Cuntz-Krieger algebra

A partial isometry is a linear operator S satisfying the relation S = SS^{∗}S. The Cuntz– Krieger algebra O_{A} (cf. [15] [16]) is defined as the universal C^{∗}–algebra generated by partial isometries S_{1}, …, S_{2g}, satisfying the relations

∑_{j} S_{j}S_{j}^{∗} = I  (4.22)

S_{i}^{∗}S_{i} = ∑_{j} A_{ij} S_{j}S_{j}^{∗},  (4.23)

where A = (A_{ij}) is the 2g × 2g transition matrix of the subshift of finite type (S, T), namely the matrix whose entries are A_{ij} = 1 whenever |i − j| ≠ g, and A_{ij} = 0 otherwise.

We give a more explicit description of the generators of the Cuntz–Krieger algebra O_{A}.

Consider the following operators, acting on the Hilbert space L^{2}(Λ_{Γ}, μ), where μ is the Patterson–Sullivan measure on the limit set (cf. [51]):

(T_{γ}f)(x) := f(γ^{−1}x), and (P_{γ}f)(x) := χ_{γ}(x)f(x),  (4.24)

where γ ∈ Γ is identified with a reduced word in the generators {g_{j}}_{j=1}^{g} and their inverses, and χ_{γ} is the characteristic function of the cylinder Λ_{Γ}(γ) of all (right) infinite reduced words that begin with the word γ. Then, for all γ ∈ Γ, T_{γ} is a unitary operator with T_{γ}^{∗} = T_{γ^{−1}} and P_{γ} is a projector. In particular, with the usual notation g_{j+g} = g_{j}^{−1}, for j = 1, …, g, we write

T_{j} := T_{g_{j}} and P_{j} := P_{g_{j}} for j = 1, …, 2g.

Proposition 4.16 The operators

S_{i} := ∑_{j} A_{ij}T_{i}^{∗}P_{j}  (4.25)

are bounded operators on L^{2}(Λ_{Γ}, μ) satisfying the relations (4.22) and (4.23). Thus, the Cuntz-Krieger algebra O_{A} can be identified with the subalgebra of bounded operators on the Hilbert space L^{2}(Λ_{Γ}, μ) generated by the S_{i} as in (4.25).

Proof. The operators P_{i} are orthogonal projectors, i.e. P_{i}P_{j} = δ_{ij}P_{j}. The composite T_{i}^{∗}P_{j}T_{i} satisfies

∑_{j} A_{ij}(T_{i}^{∗}P_{j}T_{i} f)(x) = \{ \begin{matrix}f(x) & if P_{i}(x) = x \\ 0 & otherwise. \end{matrix}

In fact, for x = g_{i}y, we have

\begin{matrix}∑_{j} A_{ij}(T_{i}^{∗}P_{j}T_{i}f)(x) = ∑_{j} A_{ij}T_{i}^{∗}P_{j}f(g_{i}^{−1}g_{i}y) = \\ T_{i}^{∗}(∑_{j} A_{ij}P_{j})f(y) = T_{i}^{∗}(I − P_{i+g})f(y) = f(g_{i}y) = f(x). \end{matrix}

In the remaining cases with x ≠ g_{i}y, we have

∑_{j} A_{ij}(T_{i}^{∗}P_{j}T_{i}f)(x) = ∑_{j} A_{ij}T_{i}^{∗}P_{j}f(g_{i}^{−1}x) = T_{i}^{∗} ∑_{j} A_{i i+g}P_{i+g}f(g_{i+g}x) = 0,

where the last equality follows from the fact that the transition matrix A satisfies A_{i i+g} = 0 (cf. Lemma 4.6.2). This implies that the S_{i} satisfy

S_{i}S_{i}^{∗} = ∑_{j} A_{ij}T_{i}^{∗}P_{j}T_{i} = P_{i}.

Since the projectors P_{i} satisfy ∑_{i} P_{i} = I, we obtain the relation (4.22). Moreover, since T_{i}T_{i}^{∗} = 1, and the entries A_{ij} are all zeroes and ones, we also obtain

S_{i}^{∗}S_{i} = ∑_{j,k} A_{ij}A_{ik}P_{k}T_{i}T_{i}^{∗}P_{j} = ∑_{j}(A_{ij})^{2} P_{j} = ∑_{j} A_{ij}P_{j}.

Replacing P_{j} = S_{j}S_{j}^{∗} from (4.22) we then obtain (4.23). ⋄

The Cuntz–Krieger algebra O_{A} can be described in terms of the action of the free group Γ on its limit set Λ_{Γ} (cf. [41], [48]), so that we can regard O_{A} as a noncommutative space replacing the classical quotient Λ_{Γ}/Γ.

In fact (cf. Lemma 4.4), the action of Γ on Λ_{Γ} ⊂ ℙ^{1}(ℂ) determines a unitary representation

Γ → Aut(C(Λ_{Γ})) (T_{γ}f)(x) = f(γ^{−1}x),  (4.26)

where C(Λ_{Γ}) is the C^{∗}–algebra of continuous functions on Λ_{Γ}. Thus, we can form the (reduced) crossed product C^{∗}–algebra C(Λ_{Γ}) ⋊ Γ.

In the following Theorem we construct an explicit identification between the algebra C(Λ_{Γ}) ⋊ Γ and the subalgebra of bounded operators on L^{2}(Λ_{Γ}, μ) generated by the S_{i}, which is isomorphic to O_{A}.

Theorem 4.17 The Cuntz–Krieger algebra O_{A} satisfies the following properties.

1. There is an injection C(Λ_{Γ}) → O_{A} which identifies C(Λ_{Γ}) with the maximal commutative subalgebra of O_{A} generated by the P_{γ} as in (4.24).

2. The generators S_{i} of O_{A} given in (4.25) realize O_{A} as a subalgebra of C(Λ_{Γ}) ⋊ Γ.

3. The operators T_{γ} of (4.24) are elements in O_{A}, hence the injection of O_{A} inside C(Λ_{Γ}) ⋊ Γ is an isomorphism,

O_{A} ≅ C(Λ_{Γ}) ⋊ Γ.  (4.27)

Proof. 1. The algebra C(Λ_{Γ}) acts on L^{2}(Λ_{Γ}, μ) as multiplication operators. We identify the characteristic function χ_{Λ_{Γ}(γ)} of the subset Λ_{Γ}(γ) ⊂ Λ_{Γ} with the projector P_{γ} defined in (4.24). A direct calculation shows that, for any γ ∈ Γ, the projector P_{γ} satisfies P_{γ} = S_{i_{1}} ⋯ S_{i_{k}}S_{i_{k}}^{∗} ⋯ S_{i_{1}}^{∗}, for γ = g_{i_{1}} ⋯ g_{i_{k}}, hence it is in the algebra O_{A}.

For a multi-index μ = {i_{1}, …, i_{k}}, the range projection P_{μ} is the element P_{μ} = S_{μ}S_{μ}^{∗} in O_{A}. We have identified χ_{Λ_{Γ}(γ)} in C(Λ_{Γ}) with the range projection P_{μ}, for μ = {i_{1}, …, i_{k}} the multi-index of γ = g_{i_{1}} ⋯ g_{i_{k}}. Thus, we have identified C(Λ_{Γ}) with the maximal commutative subalgebra of O_{A} generated by the range projections (cf.

[15] [16]).

2. The operators S_{i} defined in (4.25) determine elements in C(Λ_{Γ})⋊Γ, by identifying the projectors P_{j} with χ_{Λ_{Γ}(g_{j})} as in (1), and the operators T_{i} with the corresponding elements in the unitary representation (4.26). These elements still satisfy the relations (4.22) (4.23), hence the algebra generated by the S_{i} can be regarded as a subalgebra of C(Λ_{Γ}) ⋊ Γ.

3. Given 1. and 2., in order to prove the isomorphism (4.27), it is enough to show that, for any γ ∈ Γ, the operators T_{γ} in the unitary representation (4.26) are in the subalgebra of C(Λ_{Γ}) ⋊ Γ generated by the S_{i}. In fact, since this subalgebra contains C(Λ_{Γ}), if we know it also contains the T_{γ}, it has to be the whole of C(Λ_{Γ}) ⋊ Γ. Again this follows by a direct calculation: T_{γ} = T_{i_{1}} ⋯ T_{i_{k}} and T_{i} = P_{i}S_{i}^{∗}. ⋄

Remark 4.18 It is shown in [15], [16], and [37] that the Cuntz–Krieger algebra O_{A} has a second description as a crossed product algebra. Namely, up to stabilization (i.e. tensoring with compact operators) we have

O_{A} ≃ F_{A} ⋊_{T} ℤ,  (4.28)

where F_{A} is an approximately finite dimensional (AF) algebra stably isomorphic to the groupoid C^{∗}–algebra C^{∗}(G^{u}) of Remark 4.7. It can be shown that the shift operator T induces an action by automorphisms on F_{A}, so that the crossed product algebra (4.28) corresponds to C^{∗}(G^{u}) ⋊_{T} ℤ associated to the Smale space (S, T). (cf. Remark 4.7).

Finally we show that the Cuntz-Krieger algebra acts on (a completion of) the dynamical cohomology.

Proposition 4.19 1. The operators S_{i} defined in (4.25) determine endomorphisms of C(S^{+}, ℤ). These induce endomorphisms S̄_{i} of H^{1}(S_{T} , ℤ).

2. Let H^{1}(S_{T} , ℂ) be the dynamical cohomology with complex coefficients and denote by O_{A}^{alg} the ∗–algebra generated algebraically by the S_{i} satisfying (4.22) (4.23). Then, the endomorphisms S̄_{i} of H^{1}(S_{T} , ℤ) determine a representation

ρ : O_{A}^{alg} → End(H^{1}(S_{T} , ℂ)),

which induces an action of O_{A} on a suitable Hilbert space completion of H^{1}(S_{T} , ℂ), in the inner product induced by L^{2}(Λ_{Γ}, μ), with μ the Patterson-Sullivan measure.

Proof. 1. We write (4.25) equivalently as S_{i} = ∑_{j} A_{ij}T_{i+g mod 2g} P_{j}, where

T_{i}f(a_{0} … a_{ℓ} …) = f(g_{i}^{−1}a_{0} … a_{ℓ} …)

and

P_{i}f(a_{0} … a_{ℓ} …) = χ_{S^{+}(g_{i})}(a_{0} … a_{ℓ} …) · f(a_{0} … a_{ℓ} …)

are well defined as endomorphisms of C(S^{+}, ℤ). It is then sufficient to show that the endomorphisms S_{i} commute with the endomorphism δ : f ↦ f − f ∘ T. A direct calculation shows that this is the case. In fact, we have S_{i} : P_{n} → P_{n}, and for any f ∈ P_{n}, we obtain S_{i}(δf) = δ(P_{i}f). Thus, there are induced endomorphisms S̄_{i} : F_{n} ↦ F_{n} compatible with the inclusions F_{n} ↪ F_{n+1}, which in the limit induce endomorphisms of H^{1}(S_{T} , ℤ). The claim 2. then follows directly by change of coefficients and norm completion (cf. Proposition 4.16). ⋄

5 Dynamical spectral triple.

We define the dynamical cohomology H_{dyn}^{1} as the graded vector space given by the sum of the graded pieces of the filtration of H^{1}(S_{T}), introduced in Theorem 4.12. These graded pieces Gr_{p} are considered with coefficients in the p-th Hodge–Tate twist ℝ(p). The resulting graded space carries an action of an involution, which we denote F̄_{∞} by analogy with the real Frobenius acting on the cohomologies of §2. In Theorem 5.5 we show that there is an embedding of H^{1}(X̃^{∗})^{N=0} in H_{dyn}^{1}, which is compatible with the graded structure. This embedding is equivariant with respect to the action of the real Frobenius.

Similarly, we define the dynamical homology H_{1}^{dyn} as the graded vector space given by the sum of the terms in the filtration of H_{1}(S_{T}), introduced in Proposition 4.14. These vector spaces are again considered with twisted ℝ(p)-coefficients. The resulting graded vector space also has an action of a real Frobenius F̄_{∞}. Theorem 5.10 shows that there is an identification of the dual of H^{1}(X̃^{∗})^{N=0}, under the duality isomorphisms in ℍ^{·}(X^{∗}) of (2.18), with a subspace of H_{1}^{dyn}. The identification is compatible with the F̄_{∞} action induced by the change of orientation z ↦ z̄ on X_{/ℝ}.

The pair

H_{dyn}^{1} ⊕ H_{1}^{dyn}  (5.1)

provides a geometric setting, defined in terms of the dynamics of the shift operator T, which contains a copy of the archimedean cohomology H^{1}(X^{∗})^{N=0} and of its dual under the duality isomorphisms acting on ℍ^{∗}(Cone(N)). In Theorem 5.10 we also prove that, under these identifications, the duality isomorphism corresponds to the homology/cohomology pairing between H^{1}(S_{T}) and H_{1}(S_{T}).

This construction also shows that the map 1 − T plays, in this dynamical setting, a role dual to the monodromy map N of the arithmetic construction of Section 2 (cf. Remark 5.11).

We then construct a spectral triple (A, H, D), where H is a Hilbert space obtained from a completion of H_{dyn}^{1}⊕H_{1}^{dyn}, the algebra A is a refinement of both C_{0}(𝔛_{Γ}) and O_{A}, and the Dirac operator D, restricted to the subspace of H isomorphic to H^{1}(X̃^{∗})^{N=0}, agrees with the Frobenius-type operator Φ of Section 2. We refer to this as dynamical spectral triple.

The algebra A provides an analog, in the ∞-adic case, of the p-adic reduction map obtained by considering the reductions mod p^{k} (cf. [24] [33]).

In order to construct the dynamical spectral triple, we first introduce the Hilbert space completions. Then, in Propositions 5.13 and 5.16, we construct representations of two Morita equivalent C^{∗}-algebras A and Â acting as bounded operators on the completion of H_{dyn}^{1} and H_{1}^{dyn}, respectively. Finally, in Theorem 5.22 we obtain the spectral triple for the algebra A by tensoring H_{1}^{dyn} with the Â− A-bimodule implementing the Morita equivalence.

5.1 Dynamical cohomology of the fiber at infinity

In this paragraph we relate the archimedean cohomology of Section 2 to the sum of graded pieces of the filtration of Theorem 4.12. Notice that the terms F_{n} in this filtration can be regarded as real vector spaces of dimension dim F_{n} = 2g(2g−1)^{n−1}(2g− 2) + 1 for n ≥ 1 and dim F_{0} = 2g, since, as ℤ-modules they are torsion free. We make the following definition.

Definition 5.1 Let H^{1}(S_{T} , ℝ) = lim_{n} F_{n}, for a filtration F_{n} as in Theorem 4.12, with real coefficients. Let Gr_{n} = F_{n}/F_{n−1} be the corresponding graded pieces, with Gr_{0} = F_{0}. We define the dynamical cohomology as

H_{dyn}^{1} := ⊕_{p≤0}gr_{2p}^{Γ}H_{dyn}^{1},  (5.2)

where we set

gr_{2p}^{Γ}H_{dyn}^{1} := Gr_{−p} ⊗_{ℝ} ℝ(p)  (5.3)

with ℝ(p) = (2π\sqrt{−1})^{p}ℝ.

Let ι_{Ξ} : Ξ̃ → Ξ̃ be the involution on the orientation double cover Ξ̃ of Ξ given by the ℤ/2-action. This corresponds to an involution ι_{S} : S → S and an induced map ι_{S}^{∗} : H^{1}(S_{T} , ℝ) → H^{1}(S_{T} , ℝ) which preserves the F_{n}, hence it induces a map ῑ^{∗} : H_{dyn}^{1} → H_{dyn}^{1}.

Definition 5.2 We define the action of the real Frobenius F̄_{∞} on H_{dyn}^{1} as the composition of the involution ῑ^{∗} induced by the ℤ/2-action on Ξ̃ and the action by (−1)^{p} on ℝ(p).

We now proceed to identify a subspace of H_{dyn}^{1} which is isomorphic, as a graded vector space, to the archimedean cohomology H_{ar}^{1} = H^{1}(X̃^{∗})^{N=0}, in a way that is compatible with the action of the real Frobenius F̄_{∞}.

Let χ_{S^{+}(w_{n,k})} be the characteristic functions of the sets S^{+}(w_{n,k}) ⊂ S^{+}, where w_{n,k} is a word in the {g_{j}}_{j=1}^{2g} of the form

w_{n,k} = \underbrace{g_{k}g_{k} ⋯ g_{k}}_{n−times} .

Lemma 5.3 The functions χ_{S^{+}(w_{n,k})} define classes

χ_{n,k} := [χ_{S^{+}(w_{n,k})}] ∈ Gr_{n−1}.  (5.4)

The χ_{n,k} are all linearly independent, hence they space a 2g dimensional subspace in each Gr_{n−1}.

Proof. It is not hard to see that the characteristic functions χ_{S^{+}(w_{n,k})} for n ≥ 1 and k = 1, …, 2g are all linearly independent in P and no linear combination of the χ_{S^{+}(w_{n,k})} in P_{n−1} lies in the image of δ = 1 − T. Thus, passing to equivalence classes modulo the image of the map δ = 1 − T, we obtain linearly independent elements

(χ_{S^{+}(w_{n,k})} mod (1 − T)) ∈ F_{n−1} ⊂ H^{1}(S_{T}).  (5.5)

Again it is not difficult to check that, for any fixed n, no linear combination of the classes (5.5) lies in F_{n−2}. Thus, by further taking the equivalence classes of the (5.5) modulo F_{n−2}, we obtain 2g linearly independent elements (5.4) in each Gr_{n−1}. ⋄

We obtain elements in H_{dyn}^{1} by considering

(2π\sqrt{−1})^{p}χ_{−p+1,k} ∈ gr_{2p}^{Γ}H_{dyn}^{1}.  (5.6)

Definition 5.4 We denote by V ⊂ H_{dyn}^{1} the graded subspace

V := ⊕_{p≤0}gr_{2p}^{Γ}V

where gr_{2p}^{Γ}V is the subspace of gr_{2p}^{Γ}H_{dyn}^{1} spanned by the elements of the form (5.6), for k = 1, …, 2g.

For k = 1, …, g, let η_{k} denote a basis of holomorphic differentials on the Riemann surface X_{/ℝ} satisfying the normalization

∫_{a_{j}} η_{k} = δ_{jk},  (5.7)

where the a_{k} are a basis of Ker(I_{∗}) for I_{∗} : H_{1}(X_{/ℝ}, ℤ) → H_{1}(𝔛_{Γ}, ℤ) induced by the inclusion of X_{/ℝ} as the boundary at infinity of the handlebody 𝔛_{Γ}. We refer to this basis as the canonical basis of holomorphic differentials.

Recall that, since we are considering an orthosymmetric smooth real algebraic curve, X_{/ℝ} has a Schottky uniformization by a Fuchsian Schottky group as in Proposition 3.5. It is known then (cf. [24] [46]) that a holomorphic differential η on X_{/ℝ} can be obtained as Poincaré series with exponent 1, η = Θ^{1}(f), where f is a meromorphic function on ℙ^{1}(ℂ) with divisor D(f) ⊂ Ω_{Γ}, and

Θ^{m}(f)(z) := ∑_{γ∈Γ} f(γ(z)) (\frac{∂γ(z)}{∂z})^{m}  (5.8)

is the Poincaré series with exponent m. The fact that the Hausdorff dimension of the limit set satisfies dim_{H}(Λ_{Γ}) < 1 ensures absolute convergence on compact sets in Ω_{Γ} (cf. Remark 3.14). In particular, consider the automorphic series

ω_{k} = ∑_{h∈C(·|g_{k})} d_{z} log⟨hz^{+}(g_{k}), hz^{−}(g_{k}), z, z_{0}⟩,  (5.9)

where ⟨a, b, c, d⟩ is the cross ratio of points in ℙ^{1}(ℂ), and C(·|g_{k}) denotes a set of representatives of the coset classes Γ/ℤ⟨g_{k}⟩, for {g_{k}}_{k=1}^{g} the generators of Γ, and z_{0} a base point in Ω_{Γ}. By Lemma 8.2 of [24] (cf. Proposition 1.5.2 of [28]), the expression (5.9) gives the canonical basis of holomorphic differentials satisfying the normalization condition (5.7), in the form

η_{k} = \frac{1}{2π\sqrt{−1}}ω_{k}, so that ∫_{a_{j}} ω_{k} = (2π\sqrt{−1})δ_{jk}.  (5.10)

The formula (5.9) gives the explicit correspondence between the set of generators of Γ and the canonical basis of holomorphic differentials g_{k} ↦ ω_{k}, for k = 1, …, g, which we use in order to produce the following identification.

Theorem 5.5 Consider the map

U : gr_{2p}^{w}H^{1}(X̃^{∗})^{N=0} ⟶ gr_{2p}^{Γ}V,  (5.11)

given by

U((2π\sqrt{−1})^{p−1}φ_{k}) := (2π\sqrt{−1})^{p} \frac{χ_{−p+1,k} − χ_{−p+1,k+g}}{2}  (5.12)

U((2π\sqrt{−1})^{p−1}φ_{k+g}) := (2π\sqrt{−1})^{p} \frac{χ_{−p+1,k} + χ_{−p+1,k+g}}{2} ,  (5.13)

for k = 1, …, g and p ≤ 0, where we set

φ_{k} = (ω_{k} + ω̄_{k})/2 and φ_{g+k} = −i(ω_{k} − ω̄_{k})/2.  (5.14)

The map U is an isomorphism of H^{1}(X̃^{∗})^{N=0} and V as graded vector spaces, which is equivariant with respect to the action of the real Frobenius F̄_{∞}.

Proof. The elements (5.14) give a basis of H_{DR}^{1}(X_{/ℝ}, ℝ(1)), where the twist is due to the choice of normalization (5.7) and the relation (5.10). A basis for H^{1}(X_{/ℝ}, ℝ(p)) is then given by the (2π\sqrt{−1})^{p−1}φ_{k}, k = 1, …, 2g.

By (2.26), we have H^{1}(X̃^{∗})^{N=0} = ⊕_{p≤0}gr_{2p}^{w}H^{1}(X̃^{∗})^{N=0}, with gr_{2p}^{w}H^{1}(X̃^{∗})^{N=0} = H^{1}(X_{/ℝ}, ℝ(p)). Thus, we obtain a basis for H^{1}(X̃^{∗})^{N=0}, of the form

\{(2π\sqrt{−1})^{p−1}φ_{k} : k = 1, …, 2g, p ≤ 0\}.

It is then clear from the construction that the map (5.11) is an isomorphism of graded vector spaces. We need to check that it is equivariant with respect to the action of the real Frobenius.

In the case of a real X_{/ℝ}, the action of complex conjugation z ↦ z̄ corresponds geometrically to a change of orientation on X_{/ℝ}, which induces a change of orientation on the handlebody 𝔛_{Γ}. If L_{{a,b}} is the geodesic in Ξ̃ such that the orientation of the geodesic at the endpoint b ∈ ℙ^{1}(ℂ) agrees with the outward pointing normal vector, then under the change of orientation induced by z ↦ z̄ the geodesic L_{{a,b}} is exchanged with L_{{b,a}}, which is exactly the effect of the involution on Ξ̃. The induced involution on P exchanges χ_{S^{+}(w_{n,k})} and χ_{S^{+}(w_{n,g+k})}, so that the corresponding involution on V is given by

F̄_{∞}((2π\sqrt{−1})^{p}χ_{−p+1,k}) = (−1)^{p}(2π\sqrt{−1})^{p}χ_{−p+1,g+k}.

On the other hand, under the action of the real Frobenius F̄_{∞} we have H^{1}(X_{/ℝ}, ℝ) = E_{1}⊕E_{−1} with dim E_{±1} = g (cf. Remark 2.17), generated respectively by the (2π\sqrt{−1})^{−1}φ_{k} and (2π\sqrt{−1})^{−1}φ_{g+k}, for k = 1, …, g. This gives the corresponding splitting into eigenspaces H^{1}(X̃^{∗})^{N=0} = E^{+} ⊕ E^{−} as in (2.35). Thus, we see that (5.12) and (5.13) are F̄_{∞}-equivariant, since we have

F̄_{∞} ((2π\sqrt{−1})^{p} \frac{1}{2}(χ_{−p+1,k} ± χ_{−p+1,k+g})) = ±1 · (−1)^{p} ·

(2π\sqrt{−1})^{p} \frac{1}{2}(χ_{−p+1,k} ± χ_{−p+1,k+g})

F̄_{∞} ((2π\sqrt{−1})^{p−1}φ_{k}) = +1 · (−1)^{p−1}(2π\sqrt{−1})^{p−1}φ_{k}
F̄_{∞} ((2π\sqrt{−1})^{p−1}φ_{k+g}) = −1 · (−1)^{p−1}(2π\sqrt{−1})^{p−1}φ_{k+g}.

⋄

Remark 5.6 The characteristic function χ_{S^{+}(w_{n,k})} can be regarded as the “best approximation” within P_{n} (cf. Theorem 4.12.2.) to a function (non-continuous) supported on the periodic sequence of period g_{k},

f(g_{k}g_{k}g_{k}g_{k}g_{k} …) = 1 and f(a_{0}a_{1}a_{2}a_{3} …) = 0 otherwise.

The periodic sequence g_{k}g_{k}g_{k}g_{k}g_{k} … represents under the correspondence of Lemma 4.8 the closed geodesic in Ξ̃ that is an oriented core handle of the handlebody 𝔛_{Γ}. Thus, the elements (5.6) that span the subspace gr_{2p}^{Γ}V ⊂ gr_{2p}^{Γ}H_{dyn}^{1} can be regarded as the “best approximations” within gr_{2p}^{Γ}H_{dyn}^{1} to cohomology classes supported on the core handles of the handlebody. In other words, we may regard the index p ≤ 0 in the graded structure V = ⊕_{p}gr_{2p}^{Γ}V as measuring a way of “zooming in”, with increasing precision for larger |p|, on the core handles of the handlebody 𝔛_{Γ}.

Notice that, while the archimedean cohomology H^{1}(X̃^{∗})^{N=0} is identified with the kernel of the monodromy map, the dynamical cohomology is constructed by considering the cokernel of the map δ = 1 − T. This suggests a duality between the monodromy N and the map 1 − T. This will be made more precise in the next paragraph.

5.2 Dynamical homology and duality isomorphisms

Consider the ℝ-vector space K_{N} generated by all reduced sequences a_{0} … a_{N} in the {g_{i}}_{i=1}^{2g} with the condition a_{0} ≠ a_{N}^{−1} (cf. Proposition 4.14).

Definition 5.7 We define the dynamical homology H_{1}^{dyn} as

H_{1}^{dyn} := ⊕_{p≥1}gr_{2p}^{Γ}H_{1}^{dyn},  (5.15)

where we set

gr_{2p}^{Γ}H_{1}^{dyn} := K_{p−1} ⊗_{ℝ} ℝ(p)  (5.16)

for ℝ(p) = (2π\sqrt{−1})^{p}ℝ. The action of F̄_{∞} on H_{1}^{dyn} is given by

F̄_{∞}((2π\sqrt{−1})^{p} a_{0} … a_{p−1}) = (−1)^{p}(2π\sqrt{−1})^{p} a_{p−1}^{−1} … a_{0}^{−1}.  (5.17)

We also define W ⊂ H_{1}^{dyn} to be the graded subspace W = ⊕_{p≥1}gr_{2p}^{Γ}W, where gr_{2p}^{Γ}W is generated by the 2g elements

(2π\sqrt{−1})^{p} \underbrace{g_{k}g_{k} … g_{k}}_{p−times} .

Remark 5.8 Notice that the generators of H_{1}^{dyn} are periodic sequences \overline{a_{0} … a_{N}}, hence elements in Ker(1 − T^{d}) for d the period length, d|N. Notice in particular that the subspace W can be identified with the part of H_{1}^{dyn} that is generated by elements in Ker(1 − T), i.e. periodic sequences with period length d = 1.

The subspace W of the dynamical homology is related both to the subspace V of the dynamical cohomology, and to the space

⊕_{r≥2}gr_{2r}^{w}H_{Y}^{3} (X) ≅ δ_{1}(⊕_{p≤0}gr_{2p}^{w}H^{1}(X̃^{∗})^{N=0}),

for r = −p + 2, and with δ_{1} the duality isomorphism of (3.23) (cf. Proposition 2.15).

Lemma 5.9 The homology/cohomology pairing (4.19) induces an identification

D : gr_{2p}^{Γ}V ⟶^{≃} gr_{2(−p+1)}^{Γ}W

of the vector spaces gr_{2p}^{Γ}W and gr_{2p}^{Γ}V, for all p ≤ 0. The map D satisfies D ∘ F̄_{∞} + F̄_{∞} ∘ D = 0.

Proof. The pairing (4.19) of the class of the characteristic function χ_{S^{+}(w_{−p+1,k})} with

the element \underbrace{g_{j}g_{j} … g_{j}}_{(−p+1)−times} is

⟨χ_{S^{+}(w_{−p+1,k})}, \underbrace{g_{j}g_{j} … g_{j}}_{(−p+1)−times}⟩ = (−p + 1) δ_{jk},

for j, k = 1 … 2g. This induces a pairing

⟨·, ·⟩ : gr_{2p}^{Γ}V × gr_{2(−p+1)}^{Γ}W → ℝ(1)  (5.18)

⟨(2π\sqrt{−1})^{p}χ_{−p+1,k}, (2π\sqrt{−1})^{(−p+1)} \underbrace{g_{j}g_{j} … g_{j}}_{(−p+1)−times}⟩ = (2π\sqrt{−1})(−p + 1)δ_{jk}.

Via this pairing, we obtain an identification

D : gr_{2p}^{Γ}V ⟶^{≃} gr_{2(−p+1)}^{Γ}W

of the form

\begin{matrix}D : & (2π\sqrt{−1})^{p} \frac{1}{2}(χ_{−p+1,k} ± χ_{−p+1,k+g}) \\ ↦ & \frac{(2π\sqrt{−1})^{(−p+1)}}{(−p+1)} \frac{1}{2} ( \underbrace{g_{k}g_{k} … g_{k}}_{(−p+1)−times} ± \underbrace{g_{k}^{−1}g_{k}^{−1} … g_{k}^{−1}}_{(−p+1)−times}) . \end{matrix}  (5.19)

The relation D ∘ F̄_{∞} + F̄_{∞} ∘ D = 0 follows by construction. ⋄

We then obtain the following result.

Theorem 5.10 For p ≤ 0, consider the map

Ũ : gr_{2(−p+2)}^{w}H^{2}(X^{∗}) ⟶ gr_{2(−p+1)}^{Γ}W,  (5.20)

given by

Ũ((2π\sqrt{−1})^{−p}φ_{k}) := (2π\sqrt{−1})^{−p+1} \frac{1}{2(−p + 1)} ( \underbrace{g_{k} … g_{k}}_{(−p+1)−times} − \underbrace{g_{k}^{−1} … g_{k}^{−1}}_{(−p+1)−times}) (5.21)

U((2π\sqrt{−1})^{−p}φ_{k+g}) := (2π\sqrt{−1})^{−p+1} \frac{1}{2(−p + 1)} ( \underbrace{g_{k} … g_{k}}_{(−p+1)−times} + \underbrace{g_{k}^{−1} … g_{k}^{−1}}_{(−p+1)−times}) , (5.22)

for k = 1, …, g, with φ as in (5.14). The map Ũ is a F̄_{∞}-equivariant isomorphism of the graded vector spaces ⊕_{r=−p+2≥2}gr_{2r}^{w}H^{2}(X^{∗}) and W. Moreover, the following diagram commutes and is compatible with the action of F̄_{∞}:

\xymatrix{gr_{2p}^{w}H^{1}(X̃^{∗})^{N=0} \ar[d]^{U} \ar[r]^{δ_{1}} & gr_{2(−p+2)}^{w}H^{2}(X^{∗}) \ar[d]^{Ũ} \\ gr_{2p}^{Γ}V \ar[r]^{D} & gr_{2(−p+1)}^{Γ}W}  (5.23)

Proof. We have gr_{2(−p+2)}^{w}H^{2}(X^{∗}) ≅ H^{1}(X_{/ℝ}, ℝ(−p + 1)) by Propositions 2.22 and 2.23. The duality isomorphism (3.23) (cf. Proposition 2.15)

δ_{1} : gr_{2p}^{w}H^{1}(X̃^{∗})^{N=0} →^{≃} gr_{2(−p+2)}^{w}H^{2}(X^{∗}),

is given by

N^{2p−1} : H^{1}(X_{/ℝ}, ℝ(p)) →^{≃} H^{1}(X_{/ℝ}, ℝ(−p + 1)),

(2π\sqrt{−1})^{p−1}φ_{k} ↦ (2π\sqrt{−1})^{−p}φ_{k}.

Notice that the duality isomorphism δ_{1} also satisfies δ_{1} ∘ F̄_{∞} + F̄_{∞} ∘ δ_{1} = 0. The result then follows immediately. ⋄

Remark 5.11 There is an intrinsic duality in the identifications of diagram (5.23). In fact, as discussed in the Remark 5.8, the space H^{1}(X̃^{∗})^{N=0} corresponds to Ker(N) is identified with V, which is obtained by considering the Coker(1 − T), while the image of H^{1}(X̃^{∗})^{N=0} under the duality isomorphism δ_{1} is obtained by taking the Coker(N) in ℍ^{·}(X^{∗}) and is identified with W, which can be identified with Ker(1 − T). Modulo this duality, we have a correspondence between the monodromy map N and the dynamical map 1 − T. The presence of this duality is not surprising, considering that the cohomological construction of Section 2 is a theory of the special fiber, while the dynamical construction of Section 4 and [24] is a theory of the dual graph.

5.3 Hilbert completions

In order to construct a spectral triple associated to the dynamical homology and cohomology, we first need to choose a suitable Hilbert space completions of the vector space H_{dyn}^{1} ⊕ H_{1}^{dyn}.

Since we want to preserve the information given by the twisted coefficients ℝ(p), we choose to work with real coefficients.

We consider the real Hilbert space L^{2}(Λ_{Γ}, μ), of real valued functions of Λ_{Γ} that are square integrable with respect to the Patterson–Sullivan measure μ. Since the operator δ = 1 − T is a bounded operator on L^{2}(Λ_{Γ}, μ), its range is a closed subspace of L^{2}(Λ_{Γ}, μ), hence the quotient space Coker(1 − T) inherits the structure of Hilbert space.

With a slight abuse of notation we denote by P_{n} the finite dimensional linear subspace of L^{2}(Λ_{Γ}, μ) spanned by functions that are constant on Λ_{Γ}(γ) ⊂ Λ_{Γ}, for all γ ∈ Γ of word length |γ| > n + 1. Since we have δ : P_{n} → P_{n+1} as in the proof of Theorem (4.12), we realize the F_{n} of §5.1 as finite dimensional linear subspaces of the Hilbert space Coker(1 − T). This induces an inner product on each Gr_{n}. We denote the corresponding norm by ∥·∥_{Gr_{n}}. This defines a real Hilbert space structure on H_{dyn}^{1}, with norm

∥ ∑_{p} x_{p}∥ := (∑_{p} ∥x_{p}∥_{Gr_{−p}}^{2})^{1/2} .  (5.24)

We denote the Hilbert space completion of H_{dyn}^{1} in this norm by H_{dyn}^{1}.

A real Hilbert space structure on H_{1}^{dyn} is obtained in the following way. Each summand K_{N} can be regarded as a finite dimensional linear subspace of the real Hilbert space ℓ^{2}(Γ), by identifying the generators a_{0} … a_{N} with a_{0} ≠ a_{N}^{−1} with a subset of the set of all finite reduces words, which is a complete basis of ℓ^{2}(Γ). This determines the inner product on each K_{N}. We denote the corresponding norm by ∥ · ∥_{K_{N}}. We obtain a real Hilbert space structure on H_{1}^{dyn} with norm

∥ ∑_{p} x_{p}∥ := (∑_{p} ∥x_{p}∥_{K_{N}}^{2} )^{1/2} .  (5.25)

We denote the Hilbert space completion of H_{1}^{dyn} in this norm by H_{1}^{dyn}.

5.4 Representations

We now construct representations on the Hilbert spaces H_{dyn}^{1} and H_{1}^{dyn}.

There is an action of the algebra O_{A} on H_{dyn}^{1}, which is described as follows.

Proposition 5.12 Let O_{A}^{alg} be the ∗–algebra over ℝ generated algebraically by the S_{i} satisfying (4.22) (4.23). The action S_{i} : F_{n} ↦ F_{n} of Proposition 4.19 induces a representation

ρ : O_{A}^{alg} → End(H_{dyn}^{1}).

This extends to a representation

ρ : O_{A} → B(H_{dyn}^{1})  (5.26)

as bounded operators on the real Hilbert space completion H_{dyn}^{1}, where O_{A} is considered here as a real C^{∗}-algebra.

Proof. Recall that by Proposition 4.19 the generators S_{i} of O_{A} act on each F_{n}. In fact, we can write explicitly the action as

S_{i} : f(a_{0} … a_{n}) ↦ u(a_{0} … a_{n−1}) = \{ \begin{matrix}f(g_{i}^{−1}a_{0} … a_{n−1}) & a_{0} ≠ g_{i} \\ 0 & a_{0} = g_{i}. \end{matrix}  (5.27)

Thus, under the action of the S_{i} we have S_{i} : F_{0} → F_{0} and S_{i} : F_{n} → F_{n−1} for each n ≥ 1, hence we can consider the induced action

S̄_{i} : Gr_{n} → Gr_{n−1} S̄_{i} : Gr_{0} → Gr_{0}.

The action of the S_{i} can be defined over real coefficients (cf. Proposition 4.19) hence we obtain an action of the algebra O_{A}^{alg} over ℝ on the dynamical cohomology H_{dyn}^{1}. By Proposition 4.19 the elements of O_{A}^{alg} act as bounded operators with respect to the norm (5.24) on H_{dyn}^{1}, hence by taking the closure in the operator norm we obtain an action of the real O_{A} on H_{dyn}^{1}. ⋄

On the space H_{dyn}^{1} we can also define an action of the C^{∗}-algebra (C(Λ_{Γ})⊗C_{0}(ℍ^{3}))⋊ Γ. Here C_{0}(ℍ^{3}) denotes the C^{∗}-algebra of continuous functions that vanish on the sphere at infinity ℙ^{1}(ℂ) of ℍ^{3}.

The algebraic crossed product (C(Λ_{Γ})⊗C_{0}(ℍ^{3}))⋊^{alg} Γ is the algebra whose elements consist of finite sums ∑_{γ} f_{γ}T_{γ} with f_{γ} in C(Λ_{Γ}) ⊗ C_{0}(ℍ^{3}) and the product

(f_{γ}T_{γ})(f_{γ′}T_{γ′}) = f_{γ}(T_{γ}f_{γ′})T_{γγ′},  (5.28)

with T_{γ}f_{γ′} = f_{γ′} ∘γ^{−1}. The (reduced) crossed product (C(Λ_{Γ})⊗C_{0}(ℍ^{3}))⋊Γ is obtained by taking the closure of (C(Λ_{Γ}) ⊗ C_{0}(ℍ^{3})) ⋊^{alg} Γ in the algebra of bounded operators.

Proposition 5.13 Let x_{0} ∈ ℍ^{3} be a fixed base point. For f ∈ C_{0}(ℍ^{3}), we define an operator ρ̃(f) : F_{n} → F_{n} as follows. For h ∈ F_{n}, we set

(ρ̃(f)h)(a_{0} … a_{n}) = f((a_{0} … a_{n}) · x_{0}) · h(a_{0} … a_{n}),  (5.29)

where we identify the reduced word a_{0} … a_{n} with an element of Γ. This extends the representation of O_{A} on H_{dyn}^{1} given in Proposition 5.12 to a representation

ρ̃: (C(Λ_{Γ}) ⊗ C_{0}(ℍ^{3})) ⋊ Γ → B(H_{dyn}^{1}).  (5.30)

Proof. We first show that, using the operators ρ̃(f) of (5.29), we can construct a representation of the algebraic crossed product (C(Λ_{Γ}) ⊗ C_{0}(ℍ^{3})) ⋊^{alg} Γ as linear operators on H_{dyn}^{1}. Consider an element ∑_{γ} f_{γ}T_{γ} in the algebraic crossed product. This is a finite sum, hence in order to define ρ̃(∑_{γ} f_{γ}T_{γ}) it suffices to define ρ̃(f_{γ}T_{γ}) for each term. The operators T_{γ} act on H_{dyn}^{1} as prescribed by Proposition 5.12, by viewing T_{γ} as an element of O_{A} (cf. Theorem 4.17). Thus, we set

ρ̃(f_{γ}T_{γ}) := ρ̃(f)ρ(T_{γ}),

where ρ is the representation of O_{A} as in (5.26). Notice that the composition rule (5.28) is satisfied by construction.

Next, we check that the operators ρ̃(f)ρ(T_{γ}) acting on H_{dyn}^{1} are all bounded operators. It is sufficient to check that (5.29) are bounded operators. This follows by an estimate

∥ρ(f)h∥ ≤ ( sup_{x∈ℍ^{3}} |f(x)|) ∥h∥,

where the supremum is finite, since f is continuous and vanishing at infinity. Thus, by taking the closure in the operator norm in B(H_{dyn}^{1}) we obtain a representation ρ̃ of the (reduced) crossed product (C(Λ_{Γ}) ⊗ C_{0}(ℍ^{3})) ⋊ Γ. ⋄

On the space H_{1}^{dyn} we construct an action of the C^{∗}-algebra C(Λ_{Γ}) in the following way.

Lemma 5.14 For f a real valued continuous functions on Λ_{Γ}, consider the operator

ρ(f) : (2π\sqrt{−1})^{p} a_{0} … a_{p} ↦ f(\overline{a_{0} … a_{p}}) (2π\sqrt{−1})^{p} a_{0} … a_{p}.  (5.31)

These operators define a representation of the algebra of real valued continuous functions C(Λ_{Γ}, ℝ) as bounded operators on H_{1}^{dyn},

ρ : C(Λ_{Γ}, ℝ) → B(H_{1}^{dyn}).

Proof. We write an element in H_{1}^{dyn} as an infinite sum

X = ∑_{p} X_{p} X_{p} ∈ gr_{2p}^{Γ}H_{1}^{dyn},

with the condition that ∑_{p} ∥X_{p}∥_{K_{p−1}}^{2} < ∞. Each term X_{p} in turn can be written as a finite linear combination with real coefficients of basis elements (2π\sqrt{−1})^{p}a_{0} … a_{p}, where a_{p} ≠ a_{0}^{−1}. Given f ∈ C(Λ_{Γ}, ℝ), we define the operator ρ(f) as in (5.31), where the (2π\sqrt{−1})^{p} a_{0} … a_{p} are a complete basis of H_{1}^{dyn}. The action extends by linearity. The elements ρ(f) defined this way are bounded operators. In fact, the definition (5.31) implies that the following estimate holds:

∥ρ(f) (X)∥ ≤ ( sup_{x∈Λ_{Γ}} |f(x)|) · ∥X∥.

⋄

For a C^{∗}-algebra A, we denote by C_{b}(ℍ^{3}, A) the C^{∗}-algebra of bounded continuous functions f : ℍ^{3} → A.

We extend the representation of C(Λ_{Γ}) on the space H_{1}^{dyn}, given in Lemma 5.14, to a representation of the algebra C_{b}(ℍ^{3}, C(Λ_{Γ})).

An element f ∈ C_{b}(ℍ^{3}, C(Λ_{Γ})) is a collection f_{u} of elements in C(Λ_{Γ}), which vary continuously in the parameter u ∈ ℍ^{3}, and converge to zero in the norm of C(Λ_{Γ}) as u approaches the boundary at infinity ℙ^{1}(ℂ) in ℍ^{3}.

Lemma 5.15 Let x_{0} ∈ ℍ^{3} be a fixed base point. For f ∈ C_{b}(ℍ^{3}, C(Λ_{Γ})) we set:

ρ̃(f) a_{0} … a_{p} = f_{(a_{0}...a_{p}) x_{0}}(\overline{a_{0} … a_{p}}) · a_{0} … a_{p}.  (5.32)

This extends the representation (5.31) to a representation

ρ̃: C_{b}(ℍ^{3}, C(Λ_{Γ})) → B(H_{1}^{dyn}).

Proof. It is sufficient to check that the ρ̃(f) are bounded operators. This again follows from the estimate

∥ρ̃(f) X∥ ≤ ( sup_{u∈ℍ^{3}} sup_{x∈Λ_{Γ}} |f_{u}(x)|) · ∥X∥.

The first supremum is finite since we are considering bounded functions on ℍ^{3}. ⋄

Let 𝔄 be a C^{∗}-algebra with an action of Γ by automorphisms. Let E be the flat 𝔄-bundle

E = (𝔄 × ℍ^{3})/Γ ⟶ 𝔛_{Γ},  (5.33)

where we consider the diagonal action of Γ on 𝔄 × ℍ^{3}. We consider the C^{∗}-algebra C_{0}(𝔛_{Γ}, E) of continuous sections of E that vanish on the boundary at infinity X_{/ℝ} of the handlebody 𝔛_{Γ}.

Proposition 5.16 The representation of C_{b}(ℍ^{3}, C(Λ_{Γ})) defined by (5.32) induces a representation

ρ̃: C_{0}(𝔛_{Γ}, E) → B(H_{1}^{dyn}),

where E is the bundle (5.33) for 𝔄 = C(Λ_{Γ}).

Proof. The algebra C_{0}(𝔛_{Γ}, E) can be identified with a subalgebra of functions in C_{b}(ℍ^{3}, C(Λ_{Γ})) satisfying f_{γx} = T_{γ}f_{x}, where T_{γ} is the operator implementing the action of the element γ on C(Λ_{Γ}), as in (4.24). Thus, by restriction of the representation of Lemma 5.15, we obtain a representation of C_{0}(𝔛_{Γ}, E). ⋄

We introduce the notation A := (C(Λ_{Γ}) ⊗ C_{0}(ℍ^{3})) ⋊ Γ and Â := C_{0}(𝔛_{Γ}, E). In Propositions 5.13 and 5.16 we have obtained in representations

\begin{aligned}ρ̃: A & → B(H_{dyn}^{1}) \\ ρ̃: Â & → B(H_{1}^{dyn}). \end{aligned}  (5.34)

The important observation is that the algebras A and Â represent the same noncommutative space. To understand this point, we recall the following result (cf. [22]).

Proposition 5.17 Let G be a connected Lie group and K a maximal compact subgroup. For Γ ⊂ G a lattice and 𝔄 a C^{∗}-algebra with an action of Γ by automorphisms, the algebras (𝔄 ⊗ C_{0}(G/K)) ⋊ Γ and C_{0}(Γ\G/K, E) are strongly Morita equivalent. Here E is the flat 𝔄-bundle over Γ\G/K,

E = (𝔄 × G/K)/Γ ⟶ Γ\G/K,

where Γ acts diagonally on 𝔄 × G/K.

In our case, the result of Proposition 5.17 implies the following.

Corollary 5.18 Let K be the algebra of compact operators on a separable Hilbert space. Then the algebras A and Â in (5.34) are strongly Morita equivalent.

Proof. In the case of ℍ^{3} and the Schottky group Γ the conditions of Proposition 5.17 are verified, by taking G = SO_{0}(3, 1), so that ℍ^{3} = G/K, and 𝔄 = C(Λ_{Γ}). ⋄

5.5 Reduction mod ∞

We argue that the non-commutative space considered in the previous paragraph, represented by the (Morita equivalence class of the) C^{∗}-algebra A of (5.34), describes what Manin refers to in [24] as the “reduction mod ∞”.

We recall briefly how the reduction map works in the non-archimedean setting of Mumford curves (cf. [24] [33]). Let 𝔎 be a finite extension of ℚ_{p} and let 𝔞 be its ring of integers. The correct analog for the archimedean case is obtained by “passing to a limit’, replacing 𝔎 with its Tate closure in ℂ_{p} (cf. [24] §3.1), however, for our purposes here it is sufficient to illustrate the case of a finite extension.

The role of the hyperbolic space ℍ^{3} in the non-archimedean case is played by the Bruhat-Tits tree T_{BT} with vertices

T_{BT}^{0} = {𝔞 − lattices of rank 2 in a 2-dim 𝔎-space}/𝔎^{∗}.

Vertices in T_{BT} have valence |ℙ^{1}(𝔞/𝔪)|, where 𝔪 is the maximal ideal. Each edge in T_{BT} has length log |𝔞/𝔪|. The set of ends of T_{BT} is identified with X(𝔎) = ℙ^{1}(𝔎). This is the analog of the conformal boundary ℙ^{1}(ℂ) of ℍ^{3}. Geodesics correspond to doubly infinite paths in T_{BT} without backtracking.

Fix a vertex v_{0} on T_{BT}. This corresponds to the closed fiber X_{𝔞}⊗(𝔞/𝔪) for the chosen 𝔞-structure X_{𝔞}. Each x ∈ ℙ^{1}(𝔎) determines a unique choice of a subgraph e(v_{0}, x) in T_{BT} with vertices (v_{0}, v_{1}, v_{2}, …) along the half infinite path without backtracking which has end x. The subgraphs e(v_{0}, x)_{k} with vertices (v_{0}, v_{1}, … v_{k}) correspond to the reduction mod 𝔪^{k}, namely

{e(v_{0}, x)_{k} : x ∈ X(𝔎)} ↭ X_{𝔞}(𝔞/𝔪^{k}).

Thus the finite graphs e(v_{0}, x)_{k} represent 𝔞/𝔪^{k} points, and the infinite graph e(v_{0}, x) represents the reduction of x.

A Schottky group Γ, in this non-archimedean setting, is a purely loxodromic free discrete subgroup of PSL(2, 𝔎) in g generators. The doubly infinite paths in T_{BT} with ends at the pairs of fixed points x^{±}(γ) of the elements γ ∈ Γ produce a copy of the combinatorial tree T of the group Γ in T_{BT}. This is the analog of regarding ℍ^{3} as the union of the translates of a fundamental domain for the action of the Schottky group, which can be thought of as a ‘tubular neighborhood’ of a copy of the Cayley graph T of Γ embedded in ℍ^{3}. The ends of the tree T ⊂ T_{BT} constitute the limit set Λ_{Γ} ⊂ ℙ^{1}(𝔎). The complement Ω_{Γ} = ℙ^{1}(𝔎) ∖ Λ_{Γ} gives the uniformization of the Mumford curve X(𝔎) ≃ Ω_{Γ}/Γ. In turn, X(𝔎) can be identified with the ends of the quotient graph T_{BT} /Γ, just as in the archimedean case the Riemann surface is the conformal boundary at infinity of the handlebody 𝔛_{Γ}.

The reduction map is then obtained by considering the half infinite paths e(v, x) in T_{BT} /Γ that start at a vertex v of the finite graph T /Γ and whose end x is a point of X(𝔎), while the finite graphs e(v, x)_{k} provide the 𝔞/𝔪^{k} points.

This suggests that the correct analog of the reduction map in the archimedean case is obtained by considering geodesics in ℍ^{3} with an end on Ω_{Γ} and the other on Λ_{Γ}, as described in [24]. Arguing as in Lemma 4.9, we see that the set of such geodesics can be identified with the quotient Ω_{Γ} ×_{Γ} Λ_{Γ}. The analog of the finite graphs e(v, x)_{k} is then given by the quotient ℍ^{3} ×_{Γ} Λ_{Γ}.

The C^{∗}-algebra A = (C(Λ_{Γ}) ⊗ C_{0}(ℍ^{3})) ⋊ Γ corresponds to a way of describing this quotient as a non-commutative space.

In particular, in the definition of the action (5.30) and (5.32), we consider elements of the form (a_{0} … a_{N}) · x_{0}, for x_{0} a fixed base point in ℍ^{3}, and the reduced word a_{0} … a_{N} defining an element γ ∈ Γ of word length |γ| = N + 1. Notice that the choice of x_{0} determines an embedding of a Cayley graph T of the group Γ in ℍ^{3}. This has set of vertices T^{0} = Γ · x_{0} and set of edges T^{1} given by the set of geodesic segments connecting each vertex v ∈ T^{0} to the vertices g_{k}v, for g_{k} the g generators of Γ. For a fixed x = a_{0}a_{1} … a_{N} … in Λ_{Γ}, the points (a_{0} … a_{N}) · x_{0}, for N ≤ k correspond to the graph e(v, x)_{k} of the non-archimedean setting. Thus, an archimedean analog of the reductions mod 𝔪^{k} is built in with our choice of representations.

5.6 The Dirac operator

We consider the linear operator D on H_{dyn}^{1} ⊕ H_{1}^{dyn} defined by setting, for p ≤ 0,

D : gr_{2p}^{Γ}H_{dyn}^{1} → gr_{2p}^{Γ}H_{dyn}^{1} D : x ↦ p · x  (5.35)

and, for p ≥ 1,

D : gr_{2p}^{Γ}H_{1}^{dyn} → gr_{2p}^{Γ}H_{1}^{dyn} D : x ↦ p · x.  (5.36)

This defines an unbounded linear operator

D : H_{dyn}^{1} ⊕ H_{1}^{dyn} → H_{dyn}^{1} ⊕ H_{1}^{dyn}.  (5.37)

We show that this operator has the correct properties of a Dirac operator.

Proposition 5.19 The operator D acting on the Hilbert space H = H_{dyn}^{1} ⊕ H_{1}^{dyn} is self adjoint. For all λ ∉ ℝ, the resolvent R_{λ}(D) := (D − λ)^{−1} is a compact operator on H. The restriction of the operator D to V ⊕ W ⊂ H agrees with the Frobenius-type operator Φ of (3.22):

\begin{aligned}U Φ|_{gr_{2p}^{w}H^{1}(X̃^{∗})^{N=0}} U^{−1} & = D|_{gr_{2p}^{Γ}V} & p ≤ 0 \\ Ũ Φ|_{gr_{2p}^{w}H^{2}(X^{∗})} Ũ^{−1} & = D|_{gr_{2(p−1)}^{Γ}W} & p ≥ 2. \end{aligned}

Proof. The operator D as in (5.37) is already given in diagonal form and is clearly symmetric with respect to the inner product of H, hence it is self-adjoint on the domain dom(D) = {X ∈ H : DX ∈ H}. For λ ∉ Spec(D), the operators R_{λ}(D) are bounded, and related by the resolvent equation

R_{λ}(D) − R_{λ′}(D) = (λ′ − λ)R_{λ}(D)R_{λ′}(D).

This implies that, if R_{λ}(D) is compact for one λ ∉ Spec(D), then all the other R_{λ′}(D), λ′ ∉ Spec(D) are also compact. In our case we have Spec(D) = ℤ, hence it is easy to check that, for instance, R_{1/2}(D) is a compact operator with spectrum {(n + 1/2)^{−1} : n ∈ ℤ} ∪ {0}. Furthermore, it is clear that the restriction of D to V agrees with the restriction of Φ to H^{1}(X̃^{∗})^{N=0}. On the other hand, recall that the operator Φ defined as in (3.22) acts on gr_{2p}^{w}H^{2}(X^{∗}), for p ≥ 2, as multiplication by p−1. The map Ũ of (5.20) identifies gr_{2p}^{w}H^{2}(X^{∗}) with gr_{2(p−1)}^{Γ}W, hence we obtain ŨΦ|_{gr_{2p}^{w}H^{2}(X^{∗})}Ũ^{−1} = D|_{gr_{2(p−1)}^{Γ}W}. ⋄

We then consider the commutators of the Dirac operator D with the elements of the representations (5.30) and (5.32).

Proposition 5.20 The operator D of (5.37) satisfies the following conditions:

1. The commutator [D, ρ̃(a)] is a bounded operator on H_{dyn}^{1} for all a ∈ (C(Λ_{Γ}) ⊗ C_{0}(ℍ^{3})) ⋊ Γ, and ρ̃ as in (5.30).

2. The operator D on H_{1}^{dyn} has trivial commutators [D, ρ̃(a)] = 0 with all the elements a ∈ C_{0}(𝔛_{Γ}, E), for ρ̃ as in (5.32).

Proof. 1. First we check that the commutators [D, S̄_{i}] are bounded operators, where the S̄_{i} are the generators of the algebra O_{A} acting on H_{dyn}^{1} as in Lemma 5.12. This follows by the explicit description (5.27), which implies

[D, S̄_{i}]h = −S̄_{i}h h ∈ Gr_{n} n ≥ 1.

This implies, in particular, that the commutators [D, T_{γ}] are bounded operators on H_{dyn}^{1}. Notice then that we have trivial commutators [D, ρ̃(f)] = 0 for f ∈ C_{0}(ℍ^{3}), and ρ̃ as in (5.29). In fact, the operators ρ̃(f) preserve each gr_{2p}^{Γ}H_{dyn}^{1}, where D acts as a scalar multiple of the identity. This proves the result.

2. The argument is analogous to the case of (5.29) discussed above. The operators ρ̃(f) with f ∈ C_{0}(𝔛_{Γ}, E) and ρ̃ as in (5.32) preserve each gr_{2p}^{Γ}H_{dyn}^{1}, where D acts as a scalar multiple of the identity, hence [D, ρ̃(f)] = 0. ⋄

5.7 Morita equivalence and spectral triples

In order to construct a spectral triple from the results stated in Propositions 5.13, 5.16, 5.19, and 5.20, we need to recall how to transfer the structure of spectral triple between strongly Morita equivalent algebras.

We have the following result (cf. [11], [10], [23]).

Proposition 5.21 Let ρ be a representation ρ_{1} : A_{1} → B(H), and let D be an operator on the Hilbert space H, such that (A_{1}, H, D) satisfies the properties of a spectral triple. Let M be a bimodule implementing a Morita equivalence between A_{1} and A_{2}, which is a finite projective right Hilbert module over A_{1}. Let Ω_{D}^{1} ⊂ B(H) be the A_{1}-bimodule generated by operators of the form a_{1}[D, b_{1}], for a_{1}, b_{1} ∈ A_{1}. Let ∇ be a linear map

∇ : M → M ⊗_{A_{1}} Ω_{D}^{1}

such that, for all ξ ∈ M and a_{1} ∈ A_{1},

∇(ξa_{1}) = (∇ξ)a_{1} + ξ ⊗ [D, a_{1}]

and, for all ξ_{1}, ξ_{2} ∈ M,

⟨ξ_{1}, ∇ξ_{2}⟩_{A_{1}} − ⟨∇ξ_{1}, ξ_{2}⟩_{A_{1}} = [D, ⟨ξ_{1}, ξ_{2}⟩_{A_{1}}].

The choice of such ∇ determines a spectral triple (A_{2}, H̃, D̃). Here the Hilbert space is

H̃ = M ⊗_{A_{1}} H,

the representation ρ_{2} : A_{2} → B(H̃) is defined as

ρ_{2}(a_{2}) (ξ ⊗_{A_{1}} x) := (a_{2}ξ) ⊗_{A_{1}} x),  (5.38)

and the Dirac operator

D̃(ξ ⊗ x) = ξ ⊗ D(x) + (∇ξ)x.

Notice that the choice of the Hermitian connection ∇ is necessary, because in general the commutators [D, ρ(a)] for a ∈ A_{1} are non-trivial, hence the operator 1 ⊗ D would not be well defined on H̃ = M ⊗_{A_{1}} H.

However, in our particular case, the condition of Proposition 5.20.2. implies that we are in a special case where we can simply consider the operator D̃ = 1 ⊗ D, with ∇ = 0. We obtain the following result.

Theorem 5.22 Let H = H_{dyn}^{1} ⊕ M ⊗Â H_{1}^{dyn}, with M a bimodule implementing the Morita equivalence between the C^{∗}-algebras Â and A of (5.34). Then (A, H, D̃) is a spectral triple, where D̃|_{H_{dyn}^{1}} = D as defined in (5.35), and D̃|_{M⊗H_{1}^{dyn}} = 1 ⊗ D, with D as defined in (5.36). The representation ρ : A → B(H) is given by ρ = ρ̃ ⊕ ρ̂, with ρ̃ as in (5.30) and ρ̂ obtained as in (5.38).

6 Further structure at arithmetic infinity

We point out some other interesting aspects of the relation we have developed between the two approaches of Manin and Deninger to the theory of the closed fiber at arithmetic infinity. We hope to return to them in future work.

(i) Solenoids. The infinite tangle of bounded geodesics in 𝔛_{Γ} can also be related to generalized solenoids, thus making another connection between Manin’s theory of the closed finer at infinity and Deninger’s theory of “arithmetic cohomology”, for which the importance of solenoids was stressed in the remarks of §5.8 of [19].

There are various ways in which the classical p–adic solenoids can be generalized. One standard way of producing more general solenoids is by considering a map ϕ : C → C, which is a continuous surjection of a Cantor set C, and form the quotient space (C × [0, 1])/ ∼ by the identification (x, 0) ∼ (ϕ(x), 1). The resulting space is a generalized solenoid provided it is an indecomposable compact connected space, where the indecomposable condition means that the image under the quotient map of any D × [0, 1] where D is a clopen subset of C is connected if and only if D = C. It is not hard to check that this condition is satisfied if ϕ is a homeomorphism of a Cantor set that has a dense orbit. Thus, our model for the dual graph of the fiber at arithmetic infinity, given by the mapping torus S_{T}, is a generalized solenoid. This can also be related to generalized solenoids for subshifts of finite type [54].

(ii) Dynamical zeta function. The action of the shift operator T on the limit set Λ_{Γ} provides a dynamical zeta function on the special fiber at arithmetic infinity. In fact, there is a Perron–Frobenius theory for the shift T, with the Ruelle transfer operator

(L_{−sf} g)(z) := ∑_{y:T y=z} e^{−sf(y)}g(y),  (6.1)

depending on a parameter s ∈ ℂ and with the function f(z) = log |T′(z)|. This is the analog of the Gauss–Kuzmin operator studied in [29] [30] in the case of modular curves. For B_{q} a function space of q–forms on U ⊂ ℙ^{1}(ℂ), with U = ∪_{i}D_{i}^{±}, with analytic coefficients and uniformly bounded, we denote by L_{s,q} the operator (6.1) acting on B_{q}. In the case q = 0, for s = δ the Hausdorff dimension of Λ_{Γ}, the operator L_{δ,0} has top eigenvalue 1. This is a simple eigenvalue, with a non–negative eigenfunction which is the density of the invariant measure. The analysis is similar to the case described in [29]. For Re(s) >> 0 the operators L_{s,q} are nuclear. If we denote by {λ_{i,q}(s)}_{i} the eigenvalues of L_{s,q}, we can define

P_{q}(s) = det(1 − L_{s,q}) = ∏_{i}(1 − λ_{i,q}(s)),  (6.2)

so that we have a Ruelle zeta function

Z_{Γ}(s) := \frac{P_{1}(s)}{P_{0}(s)P_{2}(s)}.  (6.3)

By the results of [36] §4, the Ruelle zeta function Z_{Γ}(s) defined above is directly related to the Selberg zeta function of the hyperbolic handlebody 𝔛_{Γ} of the form

Z(s) = ∏_{γ∈Γ}(1 − N(γ)^{−s}),

for N(γ) = exp(ℓ_{γ}) the length of the corresponding primitive closed geodesic (equivalently N(γ) is the norm of the derivative of γ at the repelling fixed point z^{+}(γ) ∈ ℙ^{1}(ℂ)). If we consider the set of s ∈ ℂ where 1 is an eigenvalue of L_{s,q}, these produce possible zeroes and poles of the zeta function Z_{Γ}(s): among these, the Hausdorff dimension s = δ. The set of complex numbers where Z_{Γ}(s) has a pole provides this way another notion of dimension spectrum of the fractal Λ_{Γ} extending the ordinary Hausdorff dimension.

It should be interesting to study the properties of Z_{Γ}(s) in relation to the properties of the arithmetic zeta function (3.36) we defined in §3.4.

(iii) Missing Galois theory. The data of a spectral triple (A, H, D) associated to the special fiber at arithmetic infinity may provide a different approach to what Connes refers to as a missing Galois theory at archimedean places. The idea is derived from the general setting of Hilbert’s twelfth problem, for a local or global field K, and its maximal abelian (separable) extension K^{ab}. While class field theory provides a description of Gal(K^{ab}/K), Hilbert’s twelfth problem addresses the question of providing explicit generators of K^{ab} and an explicit action of the Galois group, much as in the Kronecker–Weber case over ℚ. The approach to Hilbert’s twelfth problem given by Stark’s conjectures (see e.g. [52]) consists of considering a family of zeta functions (e.g. for the Kronecker–Weber case these consist of arithmetic progressions ζ_{n,m}(s) := ∑_{k∈m+nℤ} |k|^{−s}) and generate numbers, which have the form of zeta–regularized determinants (e.g. exp(ζ′_{n,m}(0)) for the Kronecker–Weber case). For more general fields it is a conjecture that such numbers are algebraic and that they provide the desired generators with Galois group action. In [26] [27] Manin conjectured that, in the case of real quadratic fields, noncommutative geometry should produce Stark numbers and Galois action via the theory of noncommutative tori.

In our setting, whenever we associate the structure of a spectral triple to the archimedean places, we obtain, in particular, a family of zeta functions (3.8) determined by the spectral triple. These provide numbers of the form considered by Stark,

exp (\frac{d}{ds}ζ_{a,D}(s)|_{s=0}) ,  (6.4)

as regularized determinants. Of course, in general there is no reason to expect these numbers to be algebraic. However, it may be an interesting question whether there are spectral triples for which they are, and whether the original Stark numbers arise via a spectral triple. Notice that there is an explicit action of the algebra A on the collection of numbers (6.4), which may provide the right framework for a Galois action. We hope to return to this in a future work.

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Caterina Consani, Department of Mathematics, University of Toronto, Canada. email: kc@math.toronto.edu

Matilde Marcolli, Max–Planck–Institut für Mathematik, Bonn Germany. email: marcolli@mpim-bonn.mpg.de