ON VOLUMES OF ARITHMETIC QUOTIENTS OF SO(1, n).

MIKHAIL BELOLIPETSKY

Abstract. We apply G. Prasad’s volume formula for the arithmetic quotients of semi-simple groups and Bruhat-Tits theory to study the covolumes of arithmetic subgroups of SO(1, n). As a result we prove that for any even dimension n there exists a unique compact arithmetic hyperbolic n-orbifold of the smallest volume. We give a formula for the Euler-Poincaré characteristic of the orbifolds and present an explicit description of their fundamental groups as the stabilizers of certain lattices in quadratic spaces. We also study hyperbolic 4-manifolds defined arithmetically and obtain a number theoretical characterization of the smallest compact arithmetic 4-manifold.

1. Introduction

In this article we consider the problem of determining the smallest hyperbolic manifolds and orbifolds defined arithmetically. This problem has a long history which goes back to Klein and Hurwitz. Its solution for the hyperbolic dimension 2 was known to Hurwitz which allowed him to write down his famous bound for the order of the automorphisms group of a Riemann surface. The first extremal example for the bound is the Klein quartic. Many interesting facts about this classical subject and far reaching generalizations can be found in the book [L]. For the dimension 3 the problem is also completely solved but the results are quite recent [CF], [CFJR]. For the higher dimensions very little is known.

Probably the most interesting case among the dimensions higher than 3 is in dimension 4. Recently it has attracted particular attention due to a possible application in cosmology. Closed orientable hyperbolic 4-manifolds arise as the doubles of the real tunnelling geometries if the cosmological constant is assumed to be negative. In this context the smallest volume orientable hyperbolic 4-manifold is the most probable candidate for the model of the Lorentzian spacetime [Gi]. In view of the known facts and conjectures for the small dimensions it is quite natural to look for the smallest manifold among the arithmetic ones.

In this article we apply G. Prasad’s volume formula for the arithmetic quotients of semi-simple groups and Bruhat-Tits theory to investigate the particular case of the group SO(1, n) whose symmetric space is the hyperbolic n-space. Since our primary interest lies in dimension 4, at some point we restrict our attention to even dimensions and in the final section even more restrictively, we consider only the SO(1, 4)-case. The main results are given in Theorems 4.1 and 5.5.

The first theorem says that the smallest compact arithmetic orbifold in any even dimension greater or equal than 4 is unique and defined over the field Q[\sqrt{5}], it also provides an explicit description for the orbifold and a formula for its volume. Let us remark that the quadratic number field Q[\sqrt{5}] has the interesting property that its fundamental unit ϵ = (1 + \sqrt{5})/2 is the “golden section” unit which was already known to Greek mathematicians. Since for the dimension 2 the situation is different (the field of definition of the smallest hyperbolic 2-orbifold is ℚ[cos(2π/7)]) this result was a little unexpected for us.

The problem of determining the smallest arithmetic manifold is much more delicate than that for the orbifolds. Here we are currently able to present only partial results and only in dimension 4. Still our Theorem 5.5 gives an explicit classification of all the possible candidates and essentially reduces the problem of finding the smallest arithmetic 4-manifold to an extensive computation, which we hope is practically possible.

I would like to thank Professor G. Harder for helpful discussions and encouragement. I would also like to thank Wee Teck Gan for the helpful email correspondence. Finally, I would like to thank Professor S. Lang for reading an early version of this paper and for his suggestions about the presentation. While working on this paper I was visiting Max-Planck-Institut für Mathematik in Bonn, I appreciate the hospitality and financial support from MPIM.

2. Arithmetic subgroups

2.1. Hyperbolic n-space can be obtained as a symmetric space associated to the orthogonal group G of type (1, n):

H^{n} = G/K_{G} = SO(1, n)^{o}/SO(n)

(K_{G} denotes a maximal compact subgroup of G). This way the the connected component of identity of SO(1, n) acts as a group of isometries of H^{n}. A discrete subgroup Γ of G defines a locally symmetric space X = Γ\G/K_{G} which in our case will be an orientable hyperbolic orbifold or manifold if Γ is torsion-free. We will be interested in hyperbolic orbifolds which arise from arithmetic subgroups of G.

Let G be a connected semi-simple Lie group, H/k is a semi-simple algebraic group defined over a number field k and ϕ : H(k ⊗ ℝ) → G is a surjective homomorphism with a compact kernel. We consider H as a k-subgroup of GL(n) for n big enough and define a subgroup Λ of H(k) to be arithmetic if it is commensurable with the subgroup of k-integral points H(k) ∩ GL(n, O_{k}), that is, the intersection Λ ∩ GL(n, O_{k}) is of finite index in both Λ and H(k) ∩ GL(n, O_{k}). The subgroups of G which are commensurable with ϕ(Λ) are called arithmetic subgroups of G defined over the field k. It can be shown that the notion of arithmeticity does not depend on a particular choice of the k-embedding of H into GL(n).

We call an orbifold or manifold X = Γ\G/K_{G} arithmetic if Γ is an arithmetic subgroup of G, and we say that X is defined over k if k is the field of definition of Γ.

Arithmetic subgroups of the orthogonal groups can be constructed as follows. Let now k be a totally real algebraic number field with the ring of integers O and let f be a quadratic form of type (1, n) with the coefficients in k such that for any non-identity embedding σ : k → ℝ the conjugate form f^{σ} is positive definite (such an f is called admissible form). Then given an O-integral lattice L in k^{n+1} the group Γ = G_{L} = {γ ∈ G ≅ SO(f)^{o} | Lγ = L} is an arithmetic subgroup of G defined over k.

It can be shown that for even n this construction gives all arithmetic subgroups of G = SO(1, n)^{o} up to commensurability and conjugation in G. For odd n there is also another construction related to quaternion algebras, and for n = 7 there is a special type of arithmetic subgroups related to the Cayley algebra.

2.2. Looking for the hyperbolic orbifolds and manifolds of the smallest volume we will be interested in the maximal arithmetic subgroups of G. The maximal arithmetic subgroups can be effectively classified in terms of the arithmetic data and the local structure of G. In order to discuss the classification picture we will give some more definitions.

Let G/k be a connected semi-simple algebraic group defined over a number field k, and let V_{f} (resp. V_{∞}) denote the set of finite (resp. infinite) places of k. By [BT] for a local place v ∈ V_{f} the group G(k_{v}) is endowed with the structure of Tits system of affine type (G(k_{v}), B_{v}, N_{v}, Δ_{v}). A subgroup I_{v} ⊂ G(k_{v}) is called Iwahori subgroup if it is conjugate to B_{v}. Subgroup P_{v} ⊂ G(k_{v}) which contains an Iwahori subgroup is called parahoric. A collection P = (P_{v})_{v∈V_{f}} of parahoric subgroups P_{v} is said to be coherent if ∏_{v∈V_{∞}} G(k_{v}) · ∏_{v∈V_{f}} P_{v} is an open subgroup of the adèle group G(𝔸). A coherent collection of parahoric subgroups P = (P_{v})_{v∈V_{f}} defines an arithmetic subgroup Λ = G(k) ⋂ ∏_{v∈V_{f}} P_{v} of G(k) which will be called the principal arithmetic subgroup determined by P. We will also call by the principal arithmetic subgroups corresponding subgroups of the Lie group G.

If the group G/k is simply connected and adjoint then the maximal arithmetic subgroups of G are exactly the principal arithmetic subgroups defined by coherent collections of maximal parahoric subgroups. For the other forms the situation becomes more complicated, but still it is true that any maximal arithmetic subgroup is a normalizer in G of some principal arithmetic subgroup [Pl]. The problem of classification of the principal arithmetic subgroups which give rise to the maximal subgroups was studied in [CR] where, in particular, a criterion for the groups of type B_{r} (this is the type of SO(1, 2r)) is given. However, the criterion of Ryzhkov and Chernousov is a little confusing: it provides explicit conditions on the collections of parahoric subgroups but it does not always guarantee the existence of the global subgroup with the prescribed local properties. Let us consider this more carefully.

2.3. Let Λ be a principal arithmetic subgroup of G/k defined by ∏_{v∈V_{f}} P_{v} ⊂ G(𝔸_{f}) = ∏′_{v∈V_{f}} G(k_{v}) (where Π′ denotes the restricted product with respect to G(O_{v})). For each place v the type of P_{v} depends on the splitting type of G(k_{v}). We claim that there is a natural restriction on the possible splitting types of G(k_{v}) which, in turn, implies a restriction on types of P_{v}.

In [K] to any reductive group G/k Kottwitz assigned an invariant ϵ(G) ∈ {±1}, which can be computed explicitly for G over the completions of k and for which the product formula holds. Thus, for G over a nonarchimedean local field

ϵ(G(k_{v})) = (−1)^{r(G_{qs})−r(G)}

(G_{qs} denotes the quasi-split inner form of G and r(G) is the k_{v}-rank of the derived group of G), and for the archimedean places

ϵ(G(k_{v})) = (−1)^{q(G_{qs})−q(G)}

(q(G) is a half of the dimension of the symmetric space attached to G(k_{v})). From the product formula for ϵ(G) we immediately obtain that the total number of places for which ϵ = −1 is even. This is what we call by the parity condition on the number of non-split places. Let us see what does it mean for our semi-simple groups of type B_{r}.

There are two forms B_{r} and ^{2}B_{r} of type B_{r} over a nonarchimedean local field, the first form is split and the second is a non quasi-split form [T]. In the first case the k_{v}-rank r(G) = r and in the second case r(G) = r − 1, and also always r(G_{qs}) = r. So for v ∈ V_{f} we have:

ϵ(G(k_{v})) = \{ \begin{matrix}1 & if G is split over k_{v}, \\ −1 & otherwise. \end{matrix}

Over the archimedean places of k by the admissibility condition G(k_{v}) is compact for all v ∈ V_{∞} except one place, say v = Id, which implies:

q(G(k_{v})) = 0 for v ≠ Id,
q(G(k_{Id})) = r.

For the quasi-split form we have:

q(G_{qs}) = \frac{1}{2}(dim (G_{qs}) − dim (K_{G_{qs}}))
= \frac{1}{2}(dim (SO(r + 1, r)) − dim (S(O(r + 1) × O(r)))) = (3r^{2} + r)/2.

So for the place over which G is non-compact (v = Id):

ϵ(G(k_{v})) = \{ \begin{matrix}1 if & r ≡ 0, 3 (mod 4), \\ −1 if & r ≡ 1, 2 (mod 4); \end{matrix}

and for all the other infinite places:

ϵ(G(k_{v})) = \{ \begin{matrix}1 if & r ≡ 0, 1 (mod 4), \\ −1 if & r ≡ 2, 3 (mod 4). \end{matrix}

This implies, for example, that over a totally real quadratic field the number of nonarchimedean places over which G does not split is odd for odd r and even for r even. So we have a parity condition on the number of places over which G has type 2B_{r}. As we already remarked, this gives a restriction on the possible types of the collections of parahoric subgroups of G. This kind of restriction can not be seen in [CR], but it appears to be important for the applications.

It can be checked using the stabilizers of lattices at least for the orthogonal groups that this condition is also sufficient for the existence of the prescribed collections of parahoric subgroups.

2.4. From the general theory of arithmetic subgroups of semi-simple Lie groups it follows that any arithmetic subgroup Γ is a discrete subgroup of G and the volume of Γ\G/K_{G} is finite. It is also known that Γ ⊂ SO(1, 2r)^{o} is cocompact if and only if the corresponding quadratic form f does not represent zero non-trivially over k, which for n = 2r ≥ 4 means that Γ is non-cocompact if and only if it is defined over ℚ. We are going to investigate the volumes of Γ\H^{n}.

For the future reference we fix some notations. Throughout this paper k will denote a totally real algebraic number field with the discriminant D_{k}, ring of integers O and adèle ring 𝔸. The set of places V of k is a union of the set V_{∞} of archimedean and V_{f} of finite places. For v ∈ V_{f}, as usually, k_{v} denotes the completion of k at v, O_{v} is the ring of integers of k_{v} with the uniformizer π_{v} and the residue degree #O_{v}/π_{v} = q_{v}.

3. The volume formula

3.1. In a fundamental paper [P] G. Prasad obtained a formula for the volume of a principal arithmetic subgroup of an arbitrary quasi-simple, simply connected group. This is an extensive generalization of the results of Siegel, Tamagawa, Harder and other people who worked in this direction. Gross has extended Prasad’s formula to the arithmetic subgroups of reductive groups [Gr]. From these results we can write down a closed formula for the volume of a principal arithmetic subgroup Λ of G = SO(1, 2r)^{o}.

We use the Euler-Poincaré normalization of the Haar measure on G(𝔸) in the sense of Serre [S]. Namely, for a discrete subgroup Λ with finite covolume, we then have:

|χ(Λ\G)| = μ^{EP} (Λ\G).

For a principal arithmetic subgroup Λ associated to a coherent collection of parahoric subgroups P = (P_{v})_{v∈V_{f}}:

μ^{EP} (Λ\G) = μ(Λ\G) = c_{∞}D_{k}^{\frac{1}{2} dim G} (∏_{i=1}^{r} \frac{m_{i}!}{(2π)^{m_{i}+1}} )^{[k:ℚ]} τ(G) E ∏_{v∈T} λ_{v}, (1)

where

- c_{∞} = 2 is the Euler-Poincaré characteristic of the compact dual of the symmetric space G/SO(n) (see [S], §3 and also [BP], §4 for the discussion);

- dimension dim G and exponents {m_{i}} of our group G of type B_{r} are well known to be dim G = 2r^{2} + r and m_{i} = 2i − 1 (i = 1, …, r) [B];

- the Tamagawa number τ(G) = 2 (see [W]);

- E is an Euler product which in our case is given by E = ζ_{k}(2) · . . . · ζ_{k}(2r) (ζ_{k}(.) is the Dedekind zeta function of k);

- finally, the rational factors λ_{v} ∈ ℚ correspond to the (finite) set T of the finite places of k over which P_{v} ≅ G̸_{qs}^{o}(O_{v}) where G_{qs}^{o} is the identity component of the quasi-split inner form of G.

This formula gives us the (generalized) Euler characteristic of Λ. The hyperbolic volume of Λ\H^{n} can be obtained from χ(Λ\G)/2 by multiplying by the volume of the unit sphere S^{n} in R^{n+1}:

vol (Λ\H^{2r}) = \frac{(2π)^{r}}{1 · 3 · . . . · (2r − 1)} · |χ(Λ)|.

Note, that in odd dimensions the Euler characteristic vanishes but we can still obtain a similar formula for the covolume of an arithmetic subgroup without passing through the Euler-Poincaré measure.

3.2. The λ-factors in (1) are the most subtle matter. Fortunately, we can explicitly compute the factors using the Bruhat-Tits theory. In [GHY] this was done for the parahoric subgroups which arise as the stabilizers of the maximal lattices. We will extend the table from [GHY] for the odd special orthogonal groups to the other maximal parahorics.

Consider orthogonal group G = SO_{2r+1} over a nonarchimedean local field k_{v} whose residue field O/π has order q. By [T] the group G belongs to one of the two possible types: B_{r} or ^{2}B_{r}. The local Dynkin diagrams and relative local index for the nonsplit case are given on Figure 1.

Figure 1

Similarly to [CR], having the local diagrams we can enumerate all the types of the maximal parahoric subgroups P of G, and the type defines a parahoric subgroup up to conjugation in G. However, some parahoric subgroups which are not conjugate in the simply connected group can become conjugate in the adjoint group. This happens exactly when the diagrams defining the types of the parahoric subgroups are symmetric with respect to an automrphism of the entire diagram. So, in our case types Δ_{1}\{α_{0}} and Δ_{1}\{α_{1}} define conjugate subgroups.

For each type, using results of Bruhat and Tits ([BT], the account of what we need can be found in [T], §3), we can determine the type of the maximal reductive quotient \overline{G} of the special fiber \underline{G} of the Bruhat-Tits group scheme associated with P and also the type of the reductive quotient \overline{G}_{qs}^{o} of the smooth affine group scheme \underline{G}_{qs}^{o} which was defined in [Gr], §4 for the quasi-split inner form of G (see also [P]). Now, using the tables of orders of finite groups of Lie type (e.g. [Ono], Table 1) for each of the cases the corresponding λ-factor is readily computed by the formula from [GHY], §2:

λ = λ(P) = \frac{q^{−N(\overline{G}_{qs}^{o})} · #\overline{G}_{qs}^{o}(O/π)}{q^{−N(\overline{G})} · #\overline{G}(O/π)}

(N(\overline{G}) denotes the number of positive roots of \overline{G} over the algebraic closure of the residue field O/π). We list the results in the following table.

maximal type \overline{G}

\overline{G}_{qs}^{o}

λ
Δ_{1}\{α_{0}}  SO_{2r+1}  SO_{2r+1}  1
Δ_{1}\\{α_{0}, α_{1}\}  GL_{1} × SO_{2r−1}  SO_{2r+1}  \frac{q^{2r}−1}{q−1}
Δ_{1}\\{α_{i}\}, i = 2, ..., r − 1 O_{2i} × SO_{2(r−i)+1} SO_{2r+1}  \frac{(q^{i}+1) ∏_{ν=i+1}^{r}(q^{2ν}−1)}{2·∏_{ν=1}^{r−i}(q^{2ν}−1)}
Δ_{1}\\{α_{r}\}  O_{2r}  SO_{2r+1}  \frac{q^{r}+1}{2}
Δ_{2}\\{α_{0}\}  ^{2}O_{2} × SO_{2r−1}  SO_{2r+1}  \frac{q^{2r}−1}{2(q+1)}
Δ_{2}\\{α_{i}\}, i = 1, ..., r − 2 ^{2}O_{2(i+1)} × SO_{2(r−i)−1} SO_{2r+1} \frac{(q^{i+1}−1) ∏_{ν=i+2}^{r}(q^{2ν}−1)}{2·∏_{ν=1}^{r−i−1}(q^{2ν}−1)}
Δ_{2}\\{α_{r−1}\}  ^{2}O_{2r}  SO_{2r+1}  \frac{q^{r}−1}{2}

Table 1 3.3. Proposition. For any rank r ≥ 2 and P_{v} ≅ \underline{G̸}_{qs}^{o}(O_{v}) we have:

1) λ_{v} > 1;

2) λ_{v} > 2 if q_{v} > 2.

Proof. If P_{v} is a maximal parahoric subgroup then the statement reduces to an easy check of the values of λ in Table 1. For an arbitrary parahoric subgroup P_{v} ⊂ G(k_{v}) there exists a maximal parahoric P which containes P_{v}. By the formula for λ we get λ(P_{v}) > λ(P), so the λ-factors of non-maximal parahoric subgroups also satisfy the conditions (1) and (2). □

Remark, that as in the simply connected case ([BP], Section 3.1 and Appendix A) the minimal values of λ correspond to the special parahoric subgroups, i.e. those for which \overline{G} is an irreducible affine reflection group.

3.4. Example. Let us consider the case when f is the unimodular integral quadratic form −x_{0}^{2} + x_{1}^{2} + ⋯ + x_{n}^{2} (n is even) and Γ is the stabilizer of a maximal lattice L on which f takes integral values. Then by [GHY] the group Γ is a principal arithmetic subgroup of SO(f)^{o} = SO(1, n)^{o} and the types of the corresponding parahoric subgroups can be determined from the local invariants of the quadratic form f. We have k = ℚ, O = ℤ, v runs through the primes of ℚ, the determinant δ_{v}(f) = ±1 ∈ ℚ^{×}, the Hasse-Witt invariant w_{v}(f) = ϵ(SO(f; k_{v})^{o}) is 1 for v ≠ 2 and w_{2}(f) = 1 if n ≡ 0, 6 (mod 8), w_{2}(f) = −1 if n ≡ 2, 4 (mod 8). So we take the values for λ_{v} from the Table 4 in [GHY] and immediately obtain:

|χ(Γ)| = 2 ∏_{i=1}^{r} \frac{(2i − 1)!}{(2π)^{2i}} 2 ∏_{i=1}^{r} ζ(2i) = 4 ∏_{i=1}^{r} \frac{|B_{2i}|}{4i} ·\begin{cases}1 if & r ≡ 0, 3 (mod 4), \\ 6^{−1}(2^{2r} − 1) if & r ≡ 1, 2 (mod 4) \end{cases}

(B_{2i} are Bernoulli numbers: B_{2} = 1/6, B_{4} = −1/30, B_{6} = 1/42 …). This can be compared with [RT] where the authors evaluated the Siegel’s limit, but the results will not coincide. The reason is that Ratcliffe and Tschantz consider the arithmetic subgroups SO(1, n; ℤ) of G which are the stabilizers of not maximal but unimodular lattices in the corresponding quadratic spaces. The relation between this two cases is not straightforward, but it appears that it is still possible to use a similar approach to obtain the covolumes of the stabilizers of the unimodular lattices and, in particular, to deduce the results of [RT]. We will explain this in details in [BG].

3.5. We now return to the maximal arithmetic subgroups. As it was already mentioned in Section 2.2 any maximal arithmetic subgroup Γ of G can be obtained as a normalizer in G of a principal arithmetic subgroup. So, in order to have a control over the volumes of Γ = N_{G}(Λ) we need an estimate for the index [Γ : Λ]. Following [BP] such an estimate can be obtained from an exact sequence for the Galois cohomology of k due to Rohlfs [R]. We have ([BP], Section 2.10):

[Γ : Λ] = # ∏_{v∈𝒮} C(k_{v}) · #H^{1}(k, C̃)_{ξ} · ∏_{v∈V \S} #Ξ_{θ_{v}}

where C is the center of G, C̃ is the center of the simply connected inner form of G and all the other notations can be found in [BP]. In our case the center of G is trivial and C̃ ≅ ℤ_{2}, so we get

[Γ : Λ] = #H^{1}(k, C̃)_{ξ} · ∏_{v∈V \S} #Ξ_{θ_{v}}  (2)

By [BP], Proposition 5.1 applied to our group

#H^{1}(k, C̃)_{ξ} ≤ h_{k} · 2^{[k:ℚ]+#T}  (3)

and by [BP], the proof of Proposition 6.1, the class number h_{k} can be estimated as

h_{k} ≤ 10^{2} (\frac{π}{12})^{[k:ℚ]} D_{k}  (4)

(this bound follows from the Brauer-Siegel theorem and Zimmert’s bound for the regulator of k. We refer to [BP] for the details.)

Finally, we obtain (c.f. [BP], Sections 3.6, 4.6):

|μ(Γ\G)|

≤ (#H^{1}(k, C̃))^{−1}c_{∞}D_{k}^{\frac{1}{2} dim G} (∏_{i=1}^{r} \frac{m_{i}!}{(2π)^{m_{i}+1}} )^{[k:ℚ]} τ(G) E ∏_{v∈T} λ_{v}^{s}

≤ \frac{1}{10^{2} · (\frac{π}{6} )^{[k:ℚ]}} · 4 · D_{k}^{\frac{1}{2} dim G−1} (∏_{i=1}^{r} \frac{m_{i}!}{(2π)^{m_{i}+1}} )^{[k:ℚ]} τ(G) E ∏_{v∈T} \frac{λ_{v}^{s}}{2} .  (5)

Here factors λ_{v}^{s} correspond to the collections of parahoric subgroups which are special for each v ∈ V_{f}.

Remark, that it is possible to give a better bound for the case when not all P_{v} are special, but since such subgroups do not contribute to the minimal volume we do not consider it here.

4. Orbifolds

4.1. Theorem. For any n = 2r ≥ 4 there exists a unique compact orientable arithmetic hyperbolic n-orbifold O_{min}^{n} of the smallest volume. It is defined over the field ℚ[\sqrt{5}] and has Euler characteristic

|χ(O_{min}^{n})| = \frac{λ(r)}{N(r)4^{r−1}} ∏_{i=1}^{r} |ζ_{ℚ[\sqrt{5}]}(1 − 2i)|,

where: λ(r) = 1 if r is even and λ(r) = 2^{−1}(4^{r} − 1) if r is odd; N(r) is a positive integer, ≤ 4 if r is even, and ≤ 8 if r is odd.

Proof. 1. We are looking for a maximal arithmetic subgroup of G = SO(1, n)^{o} of the smallest volume which is defined over k ≠ ℚ.

Let k = ℚ[\sqrt{5}] is a totally real quadratic field of the smallest discriminant D_{k} = 5. By (1) the volume is proportional to D_{k}^{dim G/2} and depends exponentially on the degree of the field (for big enough r), so the smallest orbifold O_{0}^{n} = Γ_{0}^{n}\H^{n} defined over k is a good candidate for O_{min}^{n}. Let P = (P_{v})_{v∈V_{f} (k)} be a coherent collection of parahoric subgroups of G(k) such that:

- if r is even P_{v} = G(O_{v}) for all v ∈ V_{f};

- if r is odd P_{v} = G(O_{v}) for all v except one with the residue characteristic 2, for the remaining place v_{2} we choose P_{v_{2}} so that λ_{v_{2}} = (q^{r} − 1)/2.

These collections of parahoric subgroups satisfy the conditions of the maximality criterion [CR] and the parity condition (Section 2.3). Let

Λ_{0}^{n} = G(k) ∩ ∏_{v} P_{v}.

So Λ_{0}^{n} is a principal arithmetic subgroup of G. Let

Γ_{0}^{n} = N_{G}(Λ_{0}^{n}).

Then Γ_{0}^{n} is a maximal arithmetic subgroup. We have:

|μ(Λ_{0}^{n}\G)| = 4 · 5^{r^{2}+r/2}C(r)^{2} ∏_{i=1}^{r} ζ_{k}(2i) λ(r),

|μ(Γ_{0}^{n}\G)| = \frac{μ(Λ_{0}^{n}\G)}{[Γ_{0}^{n} : Λ_{0}^{n}]} = \frac{μ(Λ_{0}^{n}\G)}{N(r)}.

Here C(r) denotes the product ∏_{i=1}^{r}(2i − 1)!/(2π)^{2i}, λ(r) = λ_{v_{2}} is as in the statement of the theorem and N(r) is the order of the group of outer automorphisms of Λ_{0}^{n}. By (2) and (3) N(r) ≤ 4 for even r and N(r) ≤ 8 for odd r. Later on we will state a conjecture about the actual value of N(r).

Consider the other groups defined over quadratic extensions of ℚ. Note, that for odd r the set T (of “bad places”) should contain at least one place due to the parity condition. For Λ_{0}^{n} we have chosen T in such a way that the λ-factor in the volume formula has the smallest possible value. For the other maximal groups Γ^{n} which are defined over quadratic fields k using (5) we get:

|μ(Γ^{n}\G)| ≥ \frac{1}{4h_{k}} · 4D_{k}^{dim G/2}C(r)^{2} ∏_{i=1}^{r} ζ_{k}(2i) ∏_{v∈T} \frac{λ_{v}}{2}

≥ \frac{1}{h_{k}} D_{k}^{r^{2}+r/2}C(r)^{2} ∏_{v∈T} \frac{λ_{v}}{2}

≥ 4 · 5^{r^{2}+r/2}C(r)^{2} ∏_{i=1}^{r} ζ_{ℚ[\sqrt{5}]}(2i) λ(r).

The last line for D_{k} ≠ 8, 12, 28 (k ≠ ℚ[\sqrt{2}], ℚ[\sqrt{3}], ℚ[\sqrt{7}]) is obtained using the bound (4) for the class number h_{k}. For the remaining fields we use the well-known fact that h_{k} = 1. We also consider separately the cases r = 2 and r > 2. For the later we make use of an inequality

∏_{i=1}^{r} ζ_{ℚ[\sqrt{5}]}(2i) ≤ 2 for any r.  (∗)

The product ∏_{i=1}^{∞} ζ_{k}(2i) converges and (∗) can be checked by the computer up to any precision. An analytic proof of this bound is also not difficult and we leave it as an exercise in number theory.

So we are done with the quadratic fields and can proceed to the higher degrees.

Let [k : ℚ] = 3. We have:

\frac{μ(Γ^{n}\G)}{μ(Γ_{0}^{n}\G)} ≥ \frac{\frac{1}{2h_{k}2^{3+#T}} · 4D_{k}^{dim G/2}C(r)^{2} ∏_{i=1}^{r} ζ_{k}(2i) ∏_{v∈T} λ_{v}}{4 · 5^{r^{2}+r/2}C(r)^{2} ∏_{i=1}^{r} ζ_{ℚ[\sqrt{5}]}(2i) λ(r)}

≥ \frac{D_{k}^{r^{2}+r/2}C(r) ∏_{i=1}^{r} ζ_{k}(2i)}{8h_{k}5^{r^{2}+r/2} ∏_{i=1}^{r} ζ_{ℚ[\sqrt{5}]}(2i) λ(r)}.

Our strategy remains the same: for small D_{k} we use the known values of the class numbers and for the remaining fields apply the bound (4), we also consider separately the cases r = 2 and r > 2, making use of (∗) for the later. Thus, for D_{k} = 49, 81, 148, 169:

\frac{μ(Γ^{n}\G)}{μ(Γ_{0}^{n}\G)} ≥ (\frac{49}{5})^{r^{2}+r/2} \frac{C(r) ∏_{i=1}^{r} ζ_{k}(2i)}{8 ∏_{i=1}^{r} ζ_{ℚ[\sqrt{5}]}(2i) λ(r)} ≥ 1;

for D_{k} ≥ 229:

\frac{μ(Γ^{n}\G)}{μ(Γ_{0}^{n}\G)} ≥ (\frac{229}{5})^{r^{2}+r/2} \frac{C(r) ∏_{i=1}^{r} ζ_{k}(2i)}{8h_{k} ∏_{i=1}^{r} ζ_{ℚ[\sqrt{5}]}(2i) λ(r)}

≥ \frac{1}{5} (\frac{229}{5})^{r^{2}+r/2−1} \frac{C(r) ∏_{i=1}^{r} ζ_{k}(2i)}{8 · 100 · (\frac{π}{12})^{3} 2 λ(r)}
≥ 1.

The last inequality becomes obviously true when r is big enough (say, r ≥ 9). For the smaller r it can be easily checked by a direct computation. We allow ourselves to skip the details.

For d = [k : ℚ] ≥ 4 we will make use of the known lower bounds for the discriminants of the totally real number fields (see [Od]):

if d = 4 D_{k} > 5^{d};

if d = 5 D_{k} > 6.5^{d};
if d ≥ 6 D_{k} > 7.9^{d} (and D_{k} > 10^{d} if d ≥ 8).

The cases d = 4, 5 are considered entirely similar to the previous case d = 3 and we omit the details. Let Γ^{n} is defined over a field k of degree d ≥ 6. We have:

\frac{μ(Γ^{n}\G)}{μ(Γ_{0}^{n}\G)} ≥ \frac{1}{5} (\frac{D_{k}}{5} )^{r^{2}+r/2−1} \frac{C(r)^{d−2} ∏_{i=1}^{r} ζ_{k}(2i)}{2^{d} · 100 · (\frac{π}{12})^{d} 2 λ(r)}.

For r ≥ 3 we can estimate D_{k} by 7.9^{d} and then show that χ(Γ^{n})/χ(Γ_{0}^{n}) > 1 for any d. This does not work for r = 2 since for large d the factor C(r)^{d−2} becomes too small. In order to get rid of it we use the second bound D_{k} > 10^{d} for d ≥ 8. Remark, that here we do not need any particular knowledge of the class numbers.

We proved that O_{min}^{n} = O_{0}^{n} = Γ_{0}^{n}\H^{n} has the smallest possible volume for each n. Using the functional equation for the Dedekind zeta function we can write down the formula for μ(Γ_{0}^{n}\G) = χ(O_{min}^{n}) in a compact form which is given in the statement. It remains to show the uniqueness of O_{min}^{n}.

2. Let H_{1}/k and H_{2}/k be two algebraic groups defined over k = ℚ[\sqrt{5}] such that each H_{i}(k ⊗ ℝ) admits a surjective homomorphism onto G with a compact kernel.

Then H_{1} and H_{2} are k-isogeneous and the isogeny takes arithmetic subgroups to arithmetic subgroups. So we can fix an algebraic group H/k and the surjective homomorphism with a compact kernel ϕ : H(k ⊗ ℝ) → G, such that H is of type B_{r} and can be supposed to be the adjoint group since G is centerless. Let Λ_{1} and Λ_{2} be two arithmetic subgroups of H(k) of the same maximal type P = (P_{v}). We want to prove that N_{G}(ϕ(Λ_{1})) is conjugate in G to N_{G}(ϕ(Λ_{2})).

For each finite place v ∈ V_{f} there exists g_{v} such that Λ_{1,v} = g_{v}Λ_{2,v}g_{v}^{−1} (see Section 3.2). The set of places

S = {v ∈ V_{f} | Λ_{1,v} ≠ H(O_{v}) and Λ_{2,v} ≠ H(O_{v})}

is finite. This is a known fact. To prove it one can first show the finiteness of such a set of places for the simply connected inner form of H using the strong approximation property (see e.g. [CR]), and then transfer it to H itself by an inner twist. So the set

U = {g ∈ H(𝔸_{f}) | ∏_{v∈V_{f}} Λ_{1,v} = g( ∏_{v∈V_{f}} Λ_{2,v})g^{−1}}

is not empty. Moreover, for each g = (g_{v}) ∈ U we have an open subset

∏_{v∈S} g_{v}Λ_{2,v} ∏_{v∈V_{f} \S} Λ_{2,v} ⊂ U.

If S = ∅ then Λ_{1} = Λ_{2} and there is nothing to prove. Suppose S is non empty. We consider H(k) diagonaly embedded into H(𝔸_{f}) and in ∏_{v∈S} H(k_{v}). By the week approximation property for H (H is an adjoint group so it has week approximation [H]) H(k) ∩ ∏_{v∈S} g_{v}Λ_{2,v} is dense in ∏_{v∈S} g_{v}Λ_{2,v} with respect to the product topology, so there exists a non empty open subset X ⊂ H(k) such that for any x ∈ X and any v ∈ S:

Λ_{1,v} = xΛ_{2,v}x^{−1}.

Let p_{1}, …, p_{n} be the set of prime ideals in O which define the places from S. Consider the ring R = O[\frac{1}{p_{1}} , …, \frac{1}{p_{n}}]. Since S ≠ ∅ it is a dense subset of k, and so H(R) is dense in H(k). Consequently there exists r ∈ X ∩ H(R). We have:

rΛ_{2}r^{−1} = r(H(k) ∩ ∏_{v∈V_{f}} Λ_{2,v})r^{−1} = H(k) ∩ r( ∏_{v∈V_{f}} Λ_{2,v})r^{−1} =

H(k) ∩ ∏_{v∈S} rΛ_{2,v}r^{−1} ∏_{v∈V_{f} \S} rH(O_{v})r^{−1} = H(k) ∩ ∏_{v∈V_{f}} Λ_{1,v} = Λ_{1};

ϕ(r)ϕ(Λ_{2})ϕ(r)^{−1} = ϕ(rΛ_{2}r^{−1} = ϕ(Λ_{1});

ϕ(r)N_{G}(ϕ(Λ_{2}))ϕ(r)^{−1} = N_{G}(ϕ(Λ_{1})).
□

We proceed with some remarks concerning the Theorem.

4.2. The value of N(r) is the order of the outer automorphisms group of the principal arithmetic subgroup Λ_{0}^{n}. Since we are in a very extremal situation we suppose that, in fact, Λ_{0}^{n} has no non-trivial symmetries. This can be checked for small n for which the group Λ_{0}^{n} is reflective (see [V]). By now I do not know how to prove this observation for the higher dimensions, but still we would like to have it as a conjecture.

Conjecture. For all r ≥ 2 N(r) = 1 and so

|χ(O_{min}^{n})| = \frac{λ(r)}{4^{r−1}} ∏_{i=1}^{r} |ζ_{ℚ[\sqrt{5}]}(2i − 1)|.

4.3. We can describe groups Γ_{0}^{n} of the smallest orbifolds as the stabilizers of lattices in quadratic spaces. Let

f = −\frac{1 + \sqrt{5}}{2}x_{0}^{2} + x_{1}^{2} + … + x_{n}^{2}.

and (V, f) is the corresponding (n + 1)-dimensional quadratic space. By [GHY] for even r = n/2 the coherent collection of parahoric subgroups defining the principal arithmetic subgroups Λ_{0}^{n} has the same type as the one that gives the stabilizer of the maximal lattice in (V, f). So by the uniqueness argument from the proof of the Theorem Λ_{0}^{n} is the stabilizer of the maximal lattice in (V, f). Similarly, by [BG] for r odd Λ_{0}^{n} is the stabilizer of the odd unimodular lattice in (V, f) or, equivalently, it is the stabilizer of the maximal lattice in (V, 2f). Consequently, the groups Γ_{0}^{n} are obtained as the normalizers in G of the stabilizers of the lattices.

4.4. For completeness, let us also consider the non-compact case which is easy because the only possible field of definition is k = ℚ. Similarly to the previous constructions (see also Example 3.4) we obtain:

For any n ≥ 4 there is an unique non-compact orientable arithmetic hyperbolic n-orbifold O′_{min}^{n} of the smallest volume. It has Euler characteristic

|χ(O′_{min}^{n})| = \frac{λ′(r)}{N′(r)2^{r−2}} ∏_{i=1}^{r} |ζ(1 − 2i)|.

where: λ(r) = 1 if r ≡ 0, 3 (mod 4), λ(r) = 2^{−1}(2^{r} − 1) if r ≡ 1, 2 (mod 4); N(r) is a positive integer, ≤ 2 if r ≡ 0, 3 (mod 4), and ≤ 4 if r ≡ 1, 2 (mod 4).

For r ≡ 0, 3 (mod 4) the group of O′_{min}^{n} is the normalizer of the stabilizer of the maximal lattice in quadratic space (V, f) defined by

f = −x_{0}^{2} + x_{1}^{2} + … + x_{n}^{2};

and for r ≡ 1, 2 (mod 4) it is the normalizer of the stabilizer of the odd unimodular lattice in (V, f).

Conjecture 4.2 also applies to O′_{min}^{n} and says that N′(r) = 1 for all r.

4.5. We will now compute the Euler characteristic of the smallest orbifolds for small n. We will give the values for the principal arithmetic subgroups and then either the Conjecture 4.2 is true or one should divide by the actual value of N(r) in order to obtain the Euler characteristic of the smallest orbifolds. In any case, since N(r) is bounded and always smaller then 8, this will not change the general picture. We have:

n = 2r ≥ 4 4  6  8  10  12
|χ(Λ_{min}^{n})| \frac{1}{7200} \frac{67}{576000} \frac{24187}{8709120000}  \frac{169449613}{1907405430}  \frac{8595242581}{4077148563}  …
|χ(Λ′_{min}^{n})| \frac{1}{960}  \frac{1}{725760}  \frac{1}{348364800}  \frac{31}{183936614400} \frac{691}{191294078976000}
14  16  18
… \frac{4198086141344681742549054686934}{51305855012682724567} 3.3481... · 10^{16}  1.7455... · 10^{34}
\frac{691}{144618323705856000}  \frac{4143}{3912297344326361317} \frac{24331699}{58893302680351508172}

Table 2

The smallest non-compact orbifold for all n (which is also the smallest among all the arithmetic orbifolds) has dimension n = 16 and χ = 4143/3912297344326361317 = 1.0589... · 10^{−15}. The smallest compact orbifold has dimension n = 8 and χ = 24187/8709120000 = 0.08883775... . For n = 4 the volume of the smallest compact orbifold is less then that of the non-compact one and for all bigger n the noncompact orbifolds are smaller. After n = 10 in the compact case and n = 18 for noncompact (and so for all the arithmetic hyperbolic orbifolds) the minimal volumes start to increase and then grow exponentially with respect to the dimension.

4.6. The previous remark can be considered in a wider context of [BP] where the discreteness of the set of covolumes of arithmetic subgroups was proved in a very general setting. In particular, it follows from [BP] that there exist “absolutely smallest” among all the S-arithmetic subgroups of G over k when G runs through the algebraic k-groups of absolute rank ≥ 2, the global field k can be either a number or a function field, S is any finite subset of places of k containing all the archimedean places and the Haar measures are chosen in a consistent way. Our results imply that for the adjoint groups G of type B_{r} and real rank 1 over the totally real number fields the smallest arithmetic subgroup is the unique smallest arithmetic non-cocompact subrgoup of the group G of rank r = 8. It is not hard to generalize this results (except the uniqueness) to the S-arithmetic subgroups and to the other forms of type B_{r} over totally real fields. Extending this to the other types and other fields of definition would require much work. The geometric or any other meaning of the absolutely smallest groups is completely mysterious. The only thing we know is that such a group or groups exist.

5. 4-Manifolds

5.1. Let us now consider the problem of determining the smallest arithmetic manifolds. This is a much more difficult task. From the arithmetic point of view the first difficulty is that we can not just estimate the Euler products but rather we have to deal with their rational structure. So, obtaining the precise values of the Euler characteristic for the groups of interest will be the first step. After this one needs to study the low index subgroups lattice of the distinguished groups and find the torsion-free subgroups. We will restrict our attention to the hyperbolic dimension 4.

5.2. The smallest known example of a compact orientable hyperbolic 4-manifold was constructed by Davis [D]. It can be shown that the Davis manifold M_{D} is arithmetic and defined over the field ℚ[\sqrt{5}] [EM]. The Euler characteristic χ(M_{D}) is equal to 26. We will be looking for smaller examples, so we are interested in the manifolds with χ < 26. It is well-known that the Euler characteristic of a compact orientable 4-manifold is an even integer which gives us one more natural restriction on χ.

5.3. We start with refining some estimates from the proof of Theorem 4.1 for the case r = 2 and a larger bound for χ.

Lemma. If an orientable compact arithmetic hyperbolic 4-manifold defined over a field k has χ ≤ 24 then one of the following possibilities hold:

(1) d = 2, D_{k} ≤ 362;
(2) d = 3, D_{k} ≤ 3104;
(3) d = 4, D_{k} ≤ 26574;
(4) d = 5, D_{k} ≤ 227481;
(5) d = 6, D_{k} ≤ 1947276

where d = [k : ℚ] is the degree of k and D_{k} is its discriminant.

Proof. The group of an arithmetic manifold is a (torsion-free) subgroup of a maximal arithmetic subgroup Γ, so we have χ(Γ) ≤ 24. From the other side

χ(Γ) ≥ \frac{1}{2^{d}h_{k}} · 4D_{k}^{5} (\frac{6}{2^{6}π^{6}} )^{d} ζ_{k}(2)ζ_{k}(4)

≥ \frac{1}{2^{d} · 10^{2} · (\frac{π}{12})^{d}D_{k}} · 4D_{k}^{5} (\frac{6}{2^{6}π^{6}} )^{d}

≥ D_{k}^{4} · \frac{1}{25} · (\frac{6^{2}}{2^{6}π^{7}} )^{d} .

And thus we get

D_{k} ≤ (24 · 25 · (\frac{6^{2}}{2^{6}π^{7}} )^{d})^{1/4} .

For d ≥ 8 this upper bound becomes smaller then the lower bound 10^{d} for the discriminant of a totally real field of degree d from [Od]. For d = 7 the precise smallest value of D_{k} is known to be (11.051...)^{7} [Od], and it again appears to be bigger then our upper bound for D_{k}. So we are left with the 5 remaining values of d and for each of them we compute the corresponding upper bound for D_{k} from the above inequality. □

5.4. Using the tables of number fields of low degree [BFPOD] we can perform a more careful analysis of the groups over the fields which satisfy the conditions of Lemma 5.3. There are many fields which fit the conditions and we used a simple program for GP/PARI to perform the calculations. We obtain that among 109 + 98 + 182 + 45 + 32 = 466 totally real fields which have discriminant in one of the 5 ranges only 21 + 12 + 12 + 2 = 47 can actually admit the groups with χ ≤ 24 if we use the precise values of the class numbers in the volume estimate. For the remaining fields we compute the numerator ν of the Euler characteristic χ of the smallest arithmetic group Γ defined over the field (Γ’s correspond to the principal arithmetic subgroups for which all the λ-factors in the volume formula are equal to 1). Since when passing to a finite index subgroup Γ′ of Γ the number ν still divides the numerator of the Euler characteristic χ(Γ′), we can discard all the groups with ν > 24 or ν is odd and > 12. There are also several maximal groups with nontrivial λ-factors that fit into our range and these need to be checked in an entirely similar way. Finally, we are left with only two groups Γ_{1} and Γ_{2} which are defined over ℚ[\sqrt{5}], ℚ[\sqrt{2}] and have χ = 1/7200, 11/5760, respectively. The group Γ_{1} is the groups of the smallest arithmetic 4-orbifold (see Section 4), Γ_{2} is the smallest group defined over ℚ[\sqrt{2}] (the same argument as in the proof of Theorem 4.1 can be used to show that Γ_{2} is defined uniquely up to conjugations in G).

Let us summarize the results.

5.5. Theorem. If there exists a compact orientable arithmetic hyperbolic 4-manifold M having χ(M) ≤ 24 then it satisfies one of the following conditions:

1) M is defined over ℚ[\sqrt{5}] and has the form Γ_{M}\H^{4} with Γ_{M} is a torsionfree subgroup of index 7200χ(M) of the group Γ_{1} of the smallest arithmetic 4-orbifold;

2) M has Euler characteristic 22, is defined over ℚ[\sqrt{2}], and its group is a torsion-free subgroup of index 11520 of Γ_{2} which is the smallest arithmetic subgroup of SO(1, 4)^{o} defined over ℚ[\sqrt{2}].

5.6. This result reduces the problem of finding the smallest compact arithmetic 4-manifold to a computational problem: we need to search for the “low” index torsion-free subgroups of the groups Γ_{1} and Γ_{2} defined above. The first step to implement this in practice is to find good presentations for the maximal groups. For the group Γ_{1} this can be done by identifying it with the orientation-preserving subgroup of a Coxeter group Γ′_{1} which has the Coxeter diagram given on Figure 2.

Figure 2

(It is easy to check that Γ′_{1} is an arithmetic subgroup of O(1, 4) defined over ℚ[\sqrt{5}] and χ(Γ′_{1}) = 1/14400, so its orientation-preserving subgroup is Γ_{1} by the uniqueness of the smallest arithmetic orbifold.)

Now we can search for the torsion-free subgroups of Γ′_{1} of index 14400χ(M). This, in principal, can be done by using the computer programs like GAP. The indexes of the subgroups we are interested in are quite large, but, as it can be checked, the Coxeter group Γ′_{1} has not many subgroups of low index, and so the computation looks more or less realistic.

We do not know whether or not the group Γ_{2} is also reflective in a sense of [V], and we suppose that it is not. Since here we are dealing with a stabilizer of not a modular but a maximal lattice the Vinberg’s algorithm [V] can not be directly applied to check this. It is possible to write down an explicit matrix representation for the generators of Γ_{2} in SO(1, 4), but we will not do it here. In any case this group can provide only an example with χ = 22 which is already quite large.

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Sobolev Institute of Mathematics, Koptyuga 4, 630090 Novosibirsk, Russia Max Planck Institute of Mathematics, Vivatsgasse 7, 53111 Bonn, Germany

E-mail address: mbel@math.nsc.ru