A TWISTED BURNSIDE THEOREM FOR COUNTABLE GROUPS AND REIDEMEISTER NUMBERS

ALEXANDER FEL’SHTYN AND EVGENIJ TROITSKY

Abstract. The purpose of the present paper is to prove for finitely generated groups of type I the following conjecture of A. Fel’shtyn and R. Hill [8], which is a generalization of the classical Burnside theorem.

Let G be a countable discrete group, ϕ one of its automorphisms, R(ϕ) the number of ϕ-conjugacy classes, and S(ϕ) = # Fix(ϕ̂) the number of ϕ-invariant equivalence classes of irreducible unitary representations. If one of R(ϕ) and S(ϕ) is finite, then it is equal to the other.

This conjecture plays a important role in the theory of twisted conjugacy classes (see [12], [6]) and has very important consequences in Dynamics, while its proof needs rather sophisticated results from Functional and Non-commutative Harmonic Analysis.

We begin a discussion of the general case (which needs another definition of the dual object). It will be the subject of a forthcoming paper.

Some applications and examples are presented.

Contents

1. Introduction and formulation of results 1 2. Abelian case 3 3. Compact case 5 4. Extensions and Reidemeister classes 6 5. The case of groups of type I 8 6. Examples and their discussion 10 7. Congruences for Reidemeister numbers of endomorphisms 11 8. Congruences for Reidemeister numbers of a continuous map 12 References 14

1. Introduction and formulation of results

Definition 1.1. Let G be a countable discrete group and ϕ : G → G an endomorphism. Two elements x, x′ ∈ G are said to be ϕ-conjugate or twisted conjugate iff there exists g ∈ G with

x′ = gxϕ(g^{−1}).

We shall write {x}_{ϕ} for the ϕ-conjugacy or twisted conjugacy class of the element x ∈ G. The number of ϕ-conjugacy classes is called the Reidemeister number of an endomorphism

The second author is partially supported by RFFI Grant 02-01-00574, Grant for the support of leading scientific schools HIII-619.203.1 and Grant “Universities of Russia.”

ϕ and is denoted by R(ϕ). If ϕ is the identity map then the ϕ-conjugacy classes are the usual conjugacy classes in the group G.

If G is a finite group, then the classical Burnside theorem (see e.g. [13, p. 140]) says that the number of classes of irreducible representations is equal to the number of conjugacy classes of elements of G. Let Ĝ be the unitary dual of G, i.e. the set of equivalence classes of unitary irreducible representations of G.

Remark 1.2. If ϕ : G → G is an epimorphism, it induces a map ϕ̂: Ĝ → Ĝ, ϕ̂(ρ) = ρ ∘ ϕ (because a representation is irreducible if and only if the scalar operators in the space of representation are the only ones which commute with all operators of the representation). This is not the case for a general endomorphism ϕ, because ρϕ can be reducible for an irreducible representation ρ, and ϕ̂ can be defined only as a multi-valued map. But nevertheless we can define the set of fixed points Fix ϕ̂ of ϕ̂ on Ĝ.

Therefore, by the Burnside’s theorem, if ϕ is the identity automorphism of any finite group G, then we have R(ϕ) = # Fix(ϕ̂).

To formulate our theorem for the case of a general endomorphism we first need an appropriate definition of the Fix(ϕ̂).

Definition 1.3. Let Rep(G) be the space of equivalence classes of finite dimensional unitary representations of G. Then the corresponding map ϕ̂_{R} : Rep(G) → Rep(G) is defined in the same way as above: ϕ̂_{R}(ρ) = ρ ∘ ϕ.

Let us denote by Fix(ϕ̂) the set of points ρ ∈ Ĝ ⊂ Rep(G) such that ϕ̂_{R}(ρ) = ρ.

Theorem 1.4 (Main Theorem). Let G be a finitely generated discrete group of type I, ϕ one of its endomorphism, R(ϕ) the number of ϕ-conjugacy classes, and S(ϕ) = # Fix(ϕ̂) the number of ϕ-invariant equivalence classes of irreducible unitary representations. If one of R(ϕ) and S(ϕ) is finite, then it is equal to the other.

Let μ(d), d ∈ ℕ, be the Möbius function, i.e.

μ(d) = \{ \begin{matrix}1 & if d = 1, \\ (−1)^{k} & if d is a product of k distinct primes, \\ 0 & if d is not square − free. \end{matrix}

Theorem 1.5 (Congruences for the Reidemeister numbers). Let ϕ : G → G be an endomorphism of a countable discrete group G such that all numbers R(ϕ^{n}) are finite and let H be a subgroup of G with the properties

ϕ(H) ⊂ H
∀x ∈ G ∃n ∈ ℕ such that ϕ^{n}(x) ∈ H.

If the pair (H, ϕ^{n}) satisfies the conditions of Theorem 1.4 for any n ∈ ℕ, then one has for all n,

∑_{d|n} μ(d) · R(ϕ^{n/d}) ≡ 0 mod n.

These theorems were proved previously in a special case of Abelian finitely generated plus finite group [8, 9].

The interest in twisted conjugacy relations has its origins, in particular, in the Nielsen-Reidemeister fixed point theory (see, e.g. [12, 6]), in Selberg theory (see, eg. [14, 1]), and Algebraic Geometry (see, e.g. [11]).

Concerning some topological applications of our main results, they are already obtained in the present paper (Theorem 8.5). The congruences give some necessary conditions for the realization problem for Reidemeister numbers in topological dynamics. The relations with Selberg theory will be presented in a forthcoming paper.

Let us remark that it is known that the Reidemeister number of an endomorphism of a finitely generated Abelian group is finite iff 1 is not in the spectrum of the restriction of this endomorphism to the free part of the group (see, e.g. [12]). The Reidemeister number is infinite for any automorphism of a non-elementary Gromov hyperbolic group [5].

To make the presentation more transparent we start from a new approach (E.T.) for Abelian (Section 2) and compact (Section 3) groups. Only after that we develop this approach and prove the main theorem for finitely generated groups of type I in Section 5. A discussion of some examples leading to conjectures is the subject of Section 6. Then we prove the congruences theorem (Section 7) and describe some topological applications (Section 8).

Acknowledgement. We would like to thank the Max Planck Institute for Mathematics in Bonn for its kind support and hospitality while the most part of this work has been completed. We are also indebted to MPI and organizers of the Workshop on Noncommutative Geometry and Number Theory (Bonn, August 18–22, 2003) where the results of this paper were presented.

The first author thanks the Research Institute in Mathematical Sciences in Kyoto for the possibility of the present research during his visit there.

The authors are grateful to V. Balantsev, B. Bekka, R. Hill, V. Kaimanovich, V. Manuilov, A. Mishchenko, A. Rosenberg, A. Shtern, L. Vainerman, A. Vershik for helpful discussions.

They are also grateful to the referee for valuable remarks and suggestions.

2. Abelian case

Let ϕ be an automorphism of an Abelian group G.

Lemma 2.1. The twisted conjugacy class H of e is a subgroup. The other ones are cosets gH.

Proof. The first statement follows from the equalities

hϕ(h^{−1})gϕ(g^{−1}) = ghϕ((gh)^{−1}, (hϕ(h^{−1}))^{−1} = ϕ(h)h^{−1} = h^{−1}ϕ(h).

For the second statement suppose a ∼ b, i.e. b = haϕ(h^{−1}). Then

gb = ghaϕ(h^{−1}) = h(ga)ϕ(h^{−1}), gb ∼ ga.
□

Lemma 2.2. Suppose, u_{1}, u_{2} ∈ G, χ_{H} is the characteristic function of H as a set. Then

χ_{H}(u_{1}u_{2}^{−1}) = \{ \begin{matrix}1, & if u_{1}, u_{2} are in one coset , \\ 0, & otherwise . \end{matrix}

Proof. Suppose, u_{1} ∈ g_{1}H, u_{2} ∈ g_{2}H, hence, u_{1} = g_{1}h_{1}, u_{2} = g_{2}h_{2}. Then

u_{1}u_{2}^{−1} = g_{1}h_{1}h_{2}^{−1}g_{2}^{−1} ∈ g_{1}g_{2}^{−1}H.

Thus, χ_{H}(u_{1}u_{2}^{−1}) = 1 if and only if g_{1}g_{2}^{−1} ∈ H and u_{1} and u_{2} are in the same class. Otherwise it is 0. □

The following Lemma is well known.

Lemma 2.3. For any subgroup H the function χ_{H} is of positive type.

Proof. Let us take arbitrary elements u_{1}, u_{2}, …, u_{n} of G. Let us reenumerate them in such a way that some first are in g_{1}H, the next ones are in g_{2}H, and so on, till g_{m}H, where g_{j}H are different cosets. By the previous Lemma the matrix ∥p_{it}∥ := ∥χ_{H}(u_{i}u_{t}^{−1})∥ is block-diagonal with square blocks formed by units. These blocks, and consequently the whole matrix are positively semi-defined. □

Lemma 2.4. In the Abelian case characteristic functions of twisted conjugacy classes belong to the Fourier-Stieltjes algebra B(G) = (C^{∗}(G))^{∗}.

Proof. In this case the characteristic functions of twisted conjugacy classes are the shifts of the characteristic function of the class H of e. Indeed, we have the following sequence of equivalent properties:

a ∼ b,  b = haϕ(h^{−1}) for some h,  gb = ghaϕ(h^{−1}) for some h,
gb = hgaϕ(h^{−1}) for some h, ga ∼ gb.

Hence, by Corollary (2.19) of [4], these characteristic functions are in B(G). □

Let us remark that there exists a natural isomorphism (Fourier transform)

u ↦ û,  C^{∗}(G) = C_{r}^{∗}(G) ≅ C(Ĝ),  ĝ(ρ) := ρ(g),

(this is a number because irreducible representations of an Abelian group are 1-dimen-sional). In fact, it is better to look (for what follows) at an algebra C(Ĝ) as an algebra of continuous sections of a bundle of 1-dimensional matrix algebras. over Ĝ.

Our characteristic functions, being in B(G) = (C^{∗}(G))^{∗} in this case, are mapped to the functionals on C(Ĝ) which, by the Riesz-Markov-Kakutani theorem, are measures on Ĝ. Which of these measures are invariant under the induced (twisted) action of G ? Let us remark, that an invariant non-trivial functional gives rise to at least one invariant space – its kernel.

Let us remark, that convolution under the Fourier transform becomes point-wise multiplication. More precisely, the twisted action, for example, is defined as

g[f](ρ) = ρ(g)f(ρ)ρ(ϕ(g^{−1})), ρ ∈ Ĝ, g ∈ G, f ∈ C(Ĝ).

There are 2 possibilities for the twisted action of G on the representation algebra A_{ρ} ≅ ℂ : 1) the linear span of the orbit of 1 ∈ A_{ρ} is equal to all A_{ρ}, 2) and the opposite case (the action is trivial).

The second case means that the space of interviewing operators between A_{ρ} and A_{ϕ̂ρ} equals ℂ, and ρ is a fixed point of the action ϕ̂ : Ĝ → Ĝ. In the first case this is the opposite situation.

If we have a finite number of such fixed points, then the space of twisted invariant measures is just the space of measures concentrated in these points. Indeed, let us describe the action of G on measures in more detail.

Lemma 2.5. For any Borel set E one has g[μ](E) = ∫_{E} g[1] dμ.

Proof. The restriction of measure to any Borel set commutes with the action of G, since the last is point wise on C(Ĝ). For any Borel set E one has

g[μ](E) = ∫_{E} 1 dg[μ] = ∫_{E} g[1] dμ.
□

Hence, if μ is twisted invariant, then for any Borel set E and any g ∈ G one has

∫_{E}(1 − g[1]) dμ = 0.

Lemma 2.6. Suppose, f ∈ C(X), where X is a compact Hausdorff space, and μ is a regular Borel measure on X, i.e. a functional on C(X). Suppose, for any Borel set E ⊂ X one has ∫_{E} f dμ = 0. Then μ(h) = 0 for any h ∈ C(X) such that f(x) = 0 implies h(x) = 0. I.e. μ is concentrated off the interior of supp f.

Proof. Since the functions of the form fh are dense in the space of the refered to above h’s, it is sufficient to verify the statement for fh. Let us choose an arbitrary ε > 0 and a

simple function h′ = ∑_{i=1}^{n} a_{i}χ_{E_{i}} such that |μ(fh′) − μ(fh)| < ε. Then

μ(fh′) = ∑_{i=1}^{n} ∫_{E_{i}} a_{i}f dμ = ∑_{i=1}^{n} a_{i} ∫_{E_{i}} f dμ = 0.

Since ε is an arbitrary one, we are done. □

Applying this lemma to a twisted invariant measure μ and f = 1 − g[1] we obtain that μ is concentrated at our finite number of fixed points of ϕ̂, because outside of them f ≠ 0.

If we have an infinite number of fixed points, then the space is infinite-dimensional (we have an infinite number of measures concentrated in finite number of points, each time different) and Reidemeister number is infinite as well. So, we are done.

3. Compact case

Let G be a compact group, hence Ĝ is a discrete space. Then C^{∗}(G) = ⊕M_{i}, where M_{i} are the matrix algebras of irreducible representations. The infinite sum is in the following sense:

C^{∗}(G) = {f_{i}}, i ∈ {1, 2, 3, ...} = Ĝ, f_{i} ∈ M_{i}, ∥f_{i}∥ → 0(i → ∞).

When G is finite and Ĝ is finite this is exactly Peter-Weyl theorem.

A characteristic function of a twisted class is a functional on C^{∗}(G). For a finite group it is evident, for a general compact group it is necessary to verify only the measurability of the twisted class with the respect to Haar measure, i.e. that twisted class is Borel. For a compact G, the twisted conjugacy classes being orbits of twisted action are compact and hence closed. Then its complement is open, hence Borel, and the class is Borel too.

Under the identification it passes to a sequence {φ_{i}}, where φ_{i} is a functional on M_{i} (the properties of convergence can be formulated, but they play no role at the moment). The conditions of invariance are the following: for each ρ_{i} ∈ Ĝ one has g[φ_{i}] = φ_{i}, i.e. for any a ∈ M_{i} and any g ∈ G one has φ_{i}(ρ_{i}(g)aρ_{i}(ϕ(g^{−1}))) = φ_{i}(a).

Let us recall the following well-known fact.

Lemma 3.1. Each functional on matrix algebra has form a ↦ Tr(ab) for a fixed matrix b.

Proof. One has dim(M(n, ℂ))′ = dim(M(n, ℂ)) = n × n and looking at matrices as at operators in V, dim V = n, with base e_{i}, one can remark that functionals a ↦ ⟨ae_{i}, e_{j}⟩, i, j = 1, …, n, are linearly independent. Hence, any functional takes form

a ↦ ∑_{i,j} b_{j}^{i}⟨ae_{i}, e_{j}⟩ = ∑_{i,j} b_{j}^{i}a_{i}^{j} = Tr(ba), b := ∥b_{j}^{i}∥.

□

Now we can study invariant ones:

Tr(bρ_{i}(g)aρ_{i}(ϕ(g^{−1}))) = Tr(ba), ∀ a, g,
Tr((b − ρ_{i}(ϕ(g^{−1}))bρ_{i}(g))a) = 0, ∀ a, g,

hence,

b − ρ_{i}(ϕ(g^{−1}))bρ_{i}(g) = 0, ∀ g.

Since ρ_{i} is irreducible, the dimension of the space of such b is 1 if ρ_{i} is a fixed point of ϕ̂ and 0 in the opposite case. So, we are done.

Remark 3.2. In fact we are only interested in finite discrete case. Indeed, for a compact G, the twisted conjugacy classes being orbits of twisted action are compact and hence closed. If there is a finite number of them, then are open too. Hence, the situation is more or less reduced to a discrete group: quotient by the component of unity.

4. Extensions and Reidemeister classes

Consider a group extension respecting homomorphism ϕ:

\xymatrix{0 \ar[r] & H \ar[r]^{i} \ar[d]_{ϕ′} & G \ar[r]^{p} \ar[d]^{ϕ} & G/H \ar[d]^{\overline{ϕ}} \ar[r] & 0 \\ 0 \ar[r] & H \ar[r]^{i} & G \ar[r]^{p} & G/H \ar[r] & 0,}

where H is a normal subgroup of G. The following argument has partial intersection with [10].

First of all let us notice that the Reidemeister classes of ϕ in G are mapped epimorphically on classes of \overline{ϕ} in G/H. Indeed,

p(g̃)p(g)\overline{ϕ}(p(g̃^{−1})) = p(g̃gϕ(g̃^{−1}).

Suppose, R(ϕ) < ∞. Then the previous remark implies R(\overline{ϕ}) < ∞. Consider a class K = {h}_{τ_{g}ϕ′}, where τ_{g}(h) := ghg^{−1}, g ∈ G, h ∈ H. The corresponding equivalence relation is

(1)  h ∼ h̃hgϕ′(h̃^{−1})g^{−1}.

Since H is normal, the automorphism τ_{g} : H → H is well defined. We will denote by K the image iK as well. By (1) the shift Kg is a subset of Hg is characterized by

(2)  hg ∼ h̃(hg)ϕ′(h̃^{−1}).

Hence it is a subset of {hg}_{ϕ} ∩ Hg and the partition Hg = ∪({h}_{τ_{g}ϕ′})g is a subpartition of Hg = ∪(Hg ∩ {hg}_{ϕ}).

Lemma 4.1. Suppose, |G/H| = N < ∞. Then R(τ_{g}ϕ′) ≤ NR(ϕ). More precisely, the mentioned subpartition is not more than in N parts.

Proof. Consider the following action of G on itself: x ↦ gxϕ(g^{−1}). Then its orbits are exactly classes {x}_{ϕ}. Moreover it maps classes (2) onto each other. Indeed,

g̃h̃(hg)ϕ′(h̃^{−1})ϕ(g̃^{−1}) = ĥg̃(hg)ϕ(g̃^{−1})ϕ′(ĥ^{−1})

using normality of the H. This map is invertible (g̃ ↔ g̃^{−1}), hence bijection. Moreover, g̃ and g̃ĥ, for any ĥ ∈ H, act in the same way. Or in the other words, H is in the stabilizer of this permutation of classes (2). Hence, the cardinality of any orbit ≤ N. □

Hence, for any finite G/H the number of classes of the form (2) is finite: it is ≤ NR(ϕ).

Lemma 4.2. Suppose, H satisfies the following property: for any automorphism of H with finite Reidemeister number the characteristic functions of Reidemeister classes of ϕ are linear combinations of matrix elements of some finite number of irreducible finite dimensional representations of H. Then the characteristic functions of classes (2) are linear combinations of matrix elements of some finite number of irreducible finite dimensional representations of G.

Proof. Let ρ_{1}, ρ_{2}, …, ρ_{k} be the above irreducible representations of H, ρ its direct sum acting on V, and π the regular (finite dimensional) representation of G/H. Let ρ_{1}^{I}, …, ρ_{k}^{I}, ρ^{I} be the corresponding induced representations of G. Let the characteristic function of K be represented under the form χ_{K}(h) = ⟨ρ(h)ξ, η⟩. Let ξ^{I} ∈ L^{2}(G/H, V) be defined by the formulas ξ^{I}(\overline{e}) = ξ ∈ V, ξ^{I}(\overline{g}) = 0 if \overline{g} ≠ \overline{e}. Define similarly η^{I}. Then for h ∈ iH we have

ρ^{I}(h)ξ^{I}(\overline{g}) = ρ(s(\overline{g})hs(\overline{g}h)^{−1})ξ(\overline{g}h) = ρ(hs(\overline{g})s(\overline{g})^{−1})ξ(\overline{g}) = \begin{cases}ρ(h)ξ, & if \overline{g} = \overline{e}, \\ 0, & otherwise. \end{cases}

Hence, ⟨ρ^{I}(h)ξ^{I}, η^{I}⟩|_{iH} is the characteristic function of iK. Let u, v ∈ L^{2}(G/H) be such vectors that ⟨π(\overline{g})u, v⟩ is the characteristic function of \overline{e}. Then

⟨(ρ^{I} ⊗ π)(ξ^{I} ⊗ u, η^{I} ⊗ v)⟩

is the characteristic function of iK. Other characteristic functions of classes (2) are shifts of this one. Hence matrix elements of the representation ρ^{I} ⊗ π. It is finite dimensional. Hence it can be decomposed in a finite direct sum of irreducible representations. □

Corollary 4.3 (of previous two lemmata). Under the assumptions of the previous lemma, the characteristic functions of Reidemeister classes of ϕ are linear combinations of matrix elements of some finite number of irreducible finite dimensional representations of G.

5. The case of groups of type I

Theorem 5.1. Let G be a discrete group of type I. Then • [3, 3.1.4, 4.1.11] The dual space Ĝ is a T_{1}-topological space.

• [15] Any irreducible representation of G is finite-dimensional.

Remark 5.2. In fact a discrete group G is of type I if and only if it has a normal, Abelian subgroup M of finite index. The dimension of any irreducible representation of G is at most [G : M] [15].

Suppose R = R(ϕ) < ∞, and let F ⊂ L^{∞}(G) be the R-dimensional space of all twisted-invariant functionals on L^{1}(G). Let K ⊂ L^{1}(G) be the intersection of kernels of functionals from F. Then K is a linear subspace of L^{1}(G) of codimension R. For each ρ ∈ Ĝ let us denote by K_{ρ} the image ρ(K). This is a subspace of a (finite-dimensional) full matrix algebra. Let cd_{ρ} be its codimension.

Let us introduce the following set

Ĝ_{F} = {ρ ∈ Ĝ | cd_{ρ} ≠ 0}.

Lemma 5.3. One has cd_{ρ} ≠ 0 if and only if ρ is a fixed point of ϕ̂. In this case cd_{ρ} = 1.

Proof. Suppose, cd_{ρ} ≠ 0 and let us choose a functional φ_{ρ} on the (finite-dimensional full matrix) algebra ρ(L^{1}(G)) such that K_{ρ} ⊂ Ker φ_{ρ}. Then for the corresponding functional φ_{ρ}^{∗} = φ_{ρ} ∘ ρ on L^{1}(G) one has K ⊂ Ker φ_{ρ}^{∗}. Hence, φ_{ρ}^{∗} ∈ F and is twisted-invariant, as well as φ_{ρ}. Then we argue as in the case of compact group (after Lemma 3.1).

Conversely, if ρ is a fixed point of ϕ̂, it gives rise to a (unique up to scaling) non-trivial twisted-invariant functional φ_{ρ}. Let x = ρ(a) be any element in ρ(L^{1}(G)) such that φ_{ρ}(x) ≠ 0. Then x ∉ K_{ρ}, because φ_{ρ}^{∗}(a) = φ_{ρ}(x) ≠ 0, while φ_{ρ}^{∗} is a twisted-invariant functional on L^{1}(G). So, cd_{ρ} ≠ 0.

The uniqueness (up to scaling) of the intertwining operator implies the uniqueness of the corresponding twisted-invariant functional. Hence, cd_{ρ} = 1. □

Hence,

(3)  Ĝ_{F} = Fix(ϕ̂).

¿From the property cd_{ρ} = 1 one obtains for this (unique up to scaling) functional φ_{ρ}:

(4)  Ker φ_{ρ} = K_{ρ}.

Lemma 5.4. R = #Ĝ_{F}, in particular, the set Ĝ_{F} is finite.

Proof. First of all we remark that since G is finitely generated almost Abelian (cf. Remark 5.2) there is a normal Abelian subgroup H of finite index invariant under all ϕ. Hence we can apply Lemma 4.3 to G, H, ϕ. So there is a finite collection of irreducible representations of G such that any twisted-invariant functional is a linear combination of matrix elements of them, i.e. linear combination of functionals on them. If each of them gives a non-trivial contribution, it has to be a twisted-invariant functional on the corresponding matrix algebra. Hence, by the argument above, these representations belong to Ĝ_{F}, and the appropriate functional is unique up to scaling. Hence, R ≤ S.

Then we use T_{1}-separation property. More precisely, suppose some points ρ_{1}, …, ρ_{s} belong to Ĝ_{F}. Let us choose some twisted-invariant functionals φ_{i} = φ_{ρ_{i}} corresponding to these points as it was described (i.e. choose some scaling). Assume that ∥φ_{i}∥ = 1, φ_{i}(x_{i}) = 1, x_{i} ∈ ρ_{i}(L^{1}(G)). If we can find a_{i} ∈ L^{1}(G) such that φ_{i}(ρ_{i}(a_{i})) = φ_{i}^{∗}(a_{i}) is sufficiently large and ρ_{j}(a_{i}), i ≠ j, are sufficiently small (in fact it is sufficient ρ_{j}(a_{i}) to be close enough to K_{j} := K_{ρ_{j}}), then φ_{j}^{∗}(a_{i}) are small for i ≠ j, and φ_{i}^{∗} are linear independent and hence, s < R. This would imply S := #Ĝ_{F} ≤ R is finite. Hence, R = S.

So, the problem is reduced to the search of the above a_{i}. Let d = max_{i=1,...,s} dim ρ_{i}. For

each i let c_{i} := ∥b_{i}∥, where x_{i} is the unitary equivalence of ρ_{i} and ϕ̂ρ_{i} and x_{i} = ρ_{i}(b_{i}).

Let c := max_{i=1,...,s} c_{i} and ε := \frac{1}{2·s^{2}·d·c}.

One can find a positive element a′_{i} ∈ L^{1}(G) such that ∥ρ_{i}(a′_{i})∥ ≥ 1 and ∥ρ_{j}(a′_{i})∥ < ε for j ≠ i. Indeed, since ρ_{i} can be separated from one point, and hence from the finite number of points: ρ_{j}, j ≠ i. Hence, one can find an element v_{i} such that ∥ρ_{i}(v_{i})∥ > 1, ∥ρ_{j}(v_{i})∥ < 1 for j ≠ i [3, Lemma 3.3.3]. The same is true for the positive element u_{i} = v_{i}^{∗}v_{i}. (Due to density we do not distinguish elements of L^{1} and C^{∗}). Now for a sufficiently large n the element a′_{i} := (u_{i})^{n} has the desired properties.

Let us take a_{i} := a′_{i}b_{i}^{∗}. Then

(5) φ_{i}^{∗}(a_{i}) = Tr(x_{i}ρ_{i}(a_{i})) = Tr(x_{i}ρ_{i}(a′_{i})ρ_{i}(b_{i})^{∗}) = Tr(x_{i}ρ_{i}(a′_{i})x_{i}^{∗}) =

= Tr(x_{i}ρ_{i}(a′_{i})(x_{i})^{−1}) = Tr(ρ_{i}(a′_{i})) ≥ \frac{1}{dim ρ_{i}} ≥ \frac{1}{d}.

For j ≠ i one has

(6)  ∥φ_{j}^{∗}(a_{i})∥ = ∥φ_{j}(ρ_{j}(a′_{i}b_{i}^{∗}))∥ ≤ c_{i} · ε.

Then the s×s matrix Φ = φ_{j}^{∗}(a_{i}) can be decomposed into the sum of the diagonal matrix Δ and off-diagonal Σ. By (5) one has Δ ≥ \frac{1}{d}. By (6) one has

∥Σ∥ ≤ s^{2} · c_{i} · ε ≤ s^{2} · c · \frac{1}{2 · s^{2} · d · c} = \frac{1}{2d}.

Hence, Φ is non-degenerate and we are done. □

Lemma 5.4 together with (3) completes the proof of Theorem 1.4 for automorphisms. We need the following additional observations for the proof of Theorem 1.4 for a general

endomorphism (in which (3) is false for infinite-dimensional representations).

Lemma 5.5. (1) If ϕ is an epimorphism, then Ĝ is ϕ̂_{R}-invariant.

(2) For any ϕ the set Rep(G) \ Ĝ is ϕ̂_{R}-invariant.

(3) The dimension of the space of intertwining operators between ρ ∈ Ĝ and ϕ̂_{R}(ρ) is equal to 1 if and only if ρ ∈ Fix(ϕ̂). Otherwise it is 0.

Proof. (1) and (2): This follows from the characterization of irreducible representation as that one for which the centralizer of ρ(G) consists exactly of scalar operators.

(3) Let us decompose ϕ̂_{R}(ρ) into irreducible ones. Since dim H_{ρ} = dim H_{ϕ̂(ρ)} one has only 2 possibilities: ρ does not appear in ϕ̂(ρ) and the intertwining number is 0, otherwise ϕ̂_{R}(ρ) is equivalent to ρ. In this case ρ ∈ Fix(ϕ̂). □

The proof of Theorem 1.4 can be now repeated for the general endomorphism with the new definition of Fix(ϕ̂). The item (3) supplies us with the necessary property.

6. Examples and their discussion

The natural candidate for the dual object to be used instead of Ĝ in the case when the different notions of the dual do not coincide (i.e. for groups more general than type I one groups) is the so-called quasi-dual G^{⌢}, i.e. the set of quasi-equivalence classes of factorrepresentations (see, e.g. [3]). This is a usual object when we need a sort of canonical decomposition for regular representation or group C^{∗}-algebra. More precisely, one needs the support G^{⌢}_{p} of the Plancherel measure.

Unfortunately the following example shows that this is not the case.

Example 6.1. Let G be a non-elementary Gromov hyperbolic group. As it was shown by Fel’shtyn [5] with the help of geometrical methods, for any automorphism ϕ of G the Reidemeister number R(ϕ) is infinite. In particular this is true for free group in two generators F_{2}. But the support (F_{2}^{⌢})_{p} consists of one point (i.e. regular representation is factorial).

The next hope was to exclude from this dual object the II_{1}-points assuming that they always give rise to an infinite number of twisted invariant functionals. But this is also wrong:

Example 6.2. (an idea of Fel’shtyn realized in [10]) Let G = (Z ⊕ Z) ⋊_{θ} Z be the semi-

direct product by a hyperbolic action θ(1) = ( \begin{matrix}2 & 1 \\ 1 & 1 \end{matrix} ). Let ϕ be an automorphism of G

whose restriction to Z is −id and restriction to Z ⊕ Z is ( \begin{matrix}0 & 1 \\ −1 & 0 \end{matrix} ). Then R(ϕ) = 4,

while the space G^{⌢}_{p} consists of a single II_{1}-point once again (cf. [2, p. 94]).

These examples show that powerful methods of the decomposition theory do not work for more general classes of groups.

On the other hand Example 6.2 disproves the old conjecture of Fel’shtyn and Hill [8] who supposed that the Reidemeister numbers of a injective endomorphism for groups of exponential growth are always infinite. More precisely, this group is amenable and of exponential growth. Together with some calculations for concrete groups which are too routine to be included in the present paper, this allow us to formulate the following question. Question. Is the Reidemeister number R(ϕ) infinite for any automorphism ϕ of (nonamenable) finitely generated group G containing F_{2} ?

In this relation the following example seems to be interesting.

Example 6.3. [7] For amenable and non-amenable Baumslag-Solitar groups Reidemeister numbers are always infinite.

For Example 6.2 recently we have found 4 fixed points of ϕ̂ being finite dimensional irreducible representations. They give rise to 4 linear independent twisted invariant functionals. These functionals can also be obtained from the regular factorial representation. There also exist fixed points (at least one) that are infinite dimensional irreducible representations. The corresponding functionals are evidently linear dependent with the first 4. This example will be presented in detail in a forthcoming paper.

7. Congruences for Reidemeister numbers of endomorphisms

Lemma 7.1 ([12]). For any endomorphism ϕ of a group G and any x ∈ G one has ϕ(x) ∈ {x}_{ϕ}.

Proof. ϕ(x) = x^{−1}xϕ(x). □

The following lemma is useful for calculating Reidemeister numbers.

Lemma 7.2. Let ϕ : G → G be any endomorphism of any group G, and let H be a subgroup of G with the properties

ϕ(H) ⊂ H
∀x ∈ G ∃n ∈ ℕ such that ϕ^{n}(x) ∈ H.

Then

R(ϕ) = R(ϕ_{H}),

where ϕ_{H} : H → H is the restriction of ϕ to H.

Proof. Let x ∈ G. Then there is n such that ϕ^{n}(x) ∈ H. By Lemma 7.1 it is known that x is ϕ-conjugate to ϕ^{n}(x). This means that the ϕ-conjugacy class {x}_{ϕ} of x has non-empty intersection with H.

Now suppose that x, y ∈ H are ϕ-conjugate, i.e. there is a g ∈ G such that

gx = yϕ(g).

We shall show that x and y are ϕ_{H}-conjugate, i.e. we can find a g ∈ H with the above property. First let n be large enough that ϕ^{n}(g) ∈ H. Then applying ϕ^{n} to the above equation we obtain

ϕ^{n}(g)ϕ^{n}(x) = ϕ^{n}(y)ϕ^{n+1}(g).

This shows that ϕ^{n}(x) and ϕ^{n}(y) are ϕ_{H}-conjugate. On the other hand, one knows by Lemma 7.1 that x and ϕ^{n}(x) are ϕ_{H}-conjugate, and y and ϕ^{n}(y) are ϕ_{H} conjugate, so x and y must be ϕ_{H}-conjugate.

We have shown that the intersection with H of a ϕ-conjugacy class in G is a ϕ_{H}- conjugacy class in H. Therefore, we have a map

\begin{aligned}Rest : & R(ϕ) & → & R(ϕ_{H}) \\ & \{x\}_{ϕ} & ↦ & \{x\}_{ϕ} ∩ H \end{aligned}

It is evident that it has the two-sided inverse

{x}_{ϕ_{H}} ↦ {x}_{ϕ}.

Therefore Rest is a bijection and R(ϕ) = R(ϕ_{H}). □

Corollary 7.3. Let H = ϕ^{n}(G). Then R(ϕ) = R(ϕ_{H}).

Now we pass to the proof of Theorem 1.5.

Proof. ¿From Theorems 1.4 and Lemma 7.2 it follows immediately that for every n

R(ϕ^{n}) = # Fix [ϕ̂_{H}^{n} : Ĥ → Ĥ] .

Let P_{n} denote the number of periodic points of ϕ̂_{H} of least period n. One obtains immediately

R(ϕ^{n}) = # Fix [ϕ̂_{H}^{n}] = ∑_{d|n} P_{d}.

Applying Möbius’ inversion formula, we have,

P_{n} = ∑_{d|n} μ(d)R(ϕ^{\frac{n}{d}}).

On the other hand, we know that P_{n} is always divisible by n, because P_{n} is exactly n times the number of ϕ̂_{H}-orbits in Ĥ of cardinality n.

Now we pass to the proof of Theorem 1.5 for general endomorphisms.

¿From Theorem 1.4, Lemma 7.2 it follows immediately that for every n

R(ϕ^{n}) = R(ϕ_{H}^{n}) = # Fix({ \atop (ϕ_{H}^{n})}_{R}|Ĥ)

Let P_{n} denote the number of periodic points of (ϕ̂_{H})_{R}|Ĥ of least period n. One obtains by Lemma 5.5 (2)

R(ϕ^{n}) = # Fix({ \atop (ϕ_{H}^{n})}_{R}|Ĥ) = ∑_{d|n} P_{d}.

The proof can be completed as in the case of automorphisms. □

8. Congruences for Reidemeister numbers of a continuous map

Now we pass to the formulation of the topological counterpart of the main theorems. Let X be a connected, compact polyhedron and f : X → X be a continuous map. Let p : X̃ → X be the universal cover of X and f̃: X̃ → X̃ a lifting of f, i.e. p ∘ f̃ = f ∘ p. Two liftings f̃and f̃′ are called conjugate if there is a elementγ in the deck transformation group Γ ≅ π_{1}(X) such that f̃′ = γ ∘ f̃∘ γ^{−1}. The subset p(Fix(f̃)) ⊂ Fix(f) is called the fixed point class of f determined by the lifting class [f̃]. Two fixed points x_{0} and x_{1} of f belong to the same fixed point class iff there is a path c from x_{0} to x_{1} such that c ≅ f ∘ c (homotopy relative to endpoints). This fact can be considered as an equivalent definition of a non-empty fixed point class. Every map f has only finitely many non-empty fixed point classes, each a compact subset of X.

The number of lifting classes of f (and hence the number of fixed point classes, empty or not) is called the Reidemeister number of f, which is denoted by R(f). This is a positive integer or infinity.

Let a specific lifting f̃: X̃ → X̃ be fixed. Then every lifting of f can be written in a unique way as γ ∘ f̃, with γ ∈ Γ. So the elements of Γ serve as ”coordinates” of liftings with respect to the fixed f̃. Now, for every γ ∈ Γ, the composition f̃∘ γ is a lifting of f too; so there is a unique γ′ ∈ Γ such that γ′ ∘ f̃ = f̃∘ γ. This correspondence γ → γ′ is determined by the fixed f̃, and is obviously a homomorphism.

Definition 8.1. The endomorphism f̃_{∗} : Γ → Γ determined by the lifting f̃of f is defined by

f̃_{∗}(γ) ∘ f̃ = f̃∘ γ.

We shall identify π = π_{1}(X, x_{0}) and Γ in the following way. Choose base points x_{0} ∈ X and x̃_{0} ∈ p^{−1}(x_{0}) ⊂ X̃ once and for all. Now points of X̃ are in 1-1 correspondence with homotopy classes of paths in X which start at x_{0}: for x̃ ∈ X̃ take any path in X̃ from x̃_{0} to x̃ and project it onto X; conversely, for a path c starting at x_{0}, lift it to a path in X̃ which starts at x̃_{0}, and then take its endpoint. In this way, we identify a point of X̃ with a path class ⟨c⟩ in X starting from x_{0}. Under this identification, x̃_{0} = ⟨e⟩ is the unit element in π_{1}(X, x_{0}). The action of the loop class α = ⟨a⟩ ∈ π_{1}(X, x_{0}) on X̃ is then given by

α = ⟨a⟩ : ⟨c⟩ → α · c = ⟨a · c⟩.

Now we have the following relationship between f̃_{∗} : π → π and

f_{∗} : π_{1}(X, x_{0}) ⟶ π_{1}(X, f(x_{0})).

Lemma 8.2. Suppose f̃(x̃_{0}) = ⟨w⟩. Then the following diagram commutes:

\xymatrix{π_{1}(X, x_{0}) \ar[r]^{f_{∗}} \ar@{-->}[dr]_{f̃_{∗}} & π_{1}(X, f(x_{0})) \ar[d]^{w_{∗}} \\  & π_{1}(X, x_{0})}

where w_{∗} is the isomorphism induced by the path w.

In other words, for every α = ⟨a⟩ ∈ π_{1}(X, x_{0}), we have

f̃_{∗}(⟨a⟩) = ⟨w(f ∘ a)w^{−1}⟩.

Remark 8.3. In particular, if x_{0} ∈ p(Fix(f̃)) and x̃_{0} ∈ Fix(f̃), then f̃_{∗} = f_{∗}.

Lemma 8.4 (see, e.g. [12]). Lifting classes of f (and hence fixed point classes, empty or not) are in 1-1 correspondence with f̃_{∗}-conjugacy classes in π, the lifting class [γ ∘ f̃] corresponding to the f̃_{∗}-conjugacy class of γ. We therefore have R(f) = R(f̃_{∗}).

We shall say that the fixed point class p(Fix(γ ∘ f̃)), which is labeled with the lifting class [γ ∘ f̃], corresponds to the f̃_{∗}-conjugacy class of γ. Thus f̃_{∗}-conjugacy classes in π serve as ”coordinates” for fixed point classes of f, once a fixed lifting f̃ is chosen.

Using Lemma 8.4 we may apply the Theorem 1.5 to the Reidemeister numbers of continuous maps.

Theorem 8.5. Let f : X → X be a continuous map of a compact polyhedron X such that all numbers R(f^{n}) are finite. Let f_{∗} : π_{1}(X) → π_{1}(X) be an induced endomorphism of the group π_{1}(X) and let H be a subgroup of π_{1}(X) with the properties

(1) f_{∗}(H) ⊂ H,

(2) ∀x ∈ π_{1}(X) ∃n ∈ ℕ such that f_{∗}^{n}(x) ∈ H.

If the couple (H, f_{∗}^{n}) satisfies the conditions of Theorem 1.4 for any n ∈ ℕ, then one has for all n,

∑_{d|n} μ(d) · R(f^{n/d}) ≡ 0 mod n.

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Fachbereich Mathematik, Emmy-Noether-Campus, Universität Siegen, Walter-Flex-Str. 3, D-57068 Siegen, Germany and Instytut Matematyki, Uniwersytet Szczecinski, ul. Wielkopolska 15, 70-451 Szczecin, Poland

E-mail address: felshtyn@math.uni-siegen.de
URL: http://www.math.uni-siegen.de/felshtyn

Dept. of Mech. and Math., Moscow State University, 119992 GSP-2 Moscow, Russia

E-mail address: troitsky@mech.math.msu.su
URL: http://mech.math.msu.su/~troitsky