Hyperbolic Manifolds of Small Volume

Mikhail Belolipetsky^{1} and Vincent Emery^{2}

Received: December 16, 2013

Communicated by Ursula Hamenstädt

Abstract. We conjecture that for every dimension n ≠ 3 there exists a noncompact hyperbolic n-manifold whose volume is smaller than the volume of any compact hyperbolic n-manifold. For dimensions n ≤ 4 and n = 6 this conjecture follows from the known results. In this paper we show that the conjecture is true for arithmetic hyperbolic n-manifolds of dimension n ≥ 30.

2010 Mathematics Subject Classification: 22E40 (primary); 11E57, 20G30, 51M25 (secondary) Keywords and Phrases: hyperbolic manifold, volume, Euler characteristic, arithmetic group.

1. Introduction

By hyperbolic n-manifold we mean a complete orientable manifold that is locally isometric to the hyperbolic n-space H^{n}. Let M be a complete noncompact hyperbolic 3-manifold of finite volume. By Thurston’s Dehn surgery theorem there exists infinitely many compact hyperbolic 3-manifolds obtained from M by Dehn filling [15, Section 5.8]. These manifolds all have their volume smaller than vol(M) [15, Theorem 6.5.6]. It is known that the Dehn filling procedure is specific to manifolds of dimension n = 3 (cf. [1, Section 2.1]). We believe that so is the minimality of volume of compact manifolds and that for hyperbolic manifolds in other dimensions the following conjecture should hold:

Conjecture 1. Let N be a compact hyperbolic manifold of dimension n ≠ 3. Then there exists a noncompact hyperbolic n-manifold M whose volume is smaller than the volume of N.

1Belolipetsky supported by a CNPq research grant.

2Emery supported by SNSF, project no. PA00P2-139672 and PZ00P2-148100.

The set of volumes of hyperbolic n-manifolds is well-ordered (indeed, discrete if n ≠ 3), thus the conjecture states that the smallest complete hyperbolic n-manifold is noncompact for n ≠ 3. Conjecture 1 is known to be true for dimensions n = 2, 4 and 6: for these n there exist noncompact hyperbolic nmanifolds M with |χ(M)| = 1 [14, 5], whereas it is a general fact that the Euler characteristic of a compact hyperbolic manifold is even (cf. [13, Theorem 4.4]). In this paper we prove the conjecture for arithmetic hyperbolic n-manifolds of sufficiently large dimension:

Theorem 2. Conjecture 1 holds for arithmetic hyperbolic n-manifolds of dimension n ≥ 30.

A folklore conjecture states that the smallest volume is always attained on an arithmetic hyperbolic n-manifold (in both compact and noncompact cases). We will refer to this statement as the minimal volume conjecture. This conjecture is obvious for n = 2, it was proved only recently for n = 3 [6], and there is currently no any potential approach to resolving this conjecture in higher dimensions. If true, the minimal volume conjecture together with Theorem 2 would imply Conjecture 1 for dimensions n ≥ 30. However, Conjecture 1 is weaker than the minimal volume and we think that it might be easier to attack it directly. At this point we would like to mention the related picture for hyperbolic norbifolds. By [2, 3], we know the compact and noncompact minimal volume arithmetic hyperbolic n-orbifolds in all dimensions n ≥ 4. This is complemented by the previous results for n = 2, 3 (see [loc.cit.]). It follows that the smallest arithmetic hyperbolic n-orbifold is compact for n = 2, 3, 4 and noncompact for n ≥ 5. For dimensions n ≤ 9 we do know the smallest volume noncompact hyperbolic n-orbifolds thanks to the result of Hild’s thesis ([8]). It turns out that these orbifolds are arithmetic, supporting the orbifold version of the minimal volume conjecture. Let us now describe the contents of the paper. The proof of the main theorem is based on arithmetic techniques. We start with the minimal volume arithmetic n-orbifold constructed in [2, 3], which is noncompact for n ≥ 5. Then we construct a manifold cover of it and try to show that the volume of this manifold is still smaller than the smallest compact arithmetic n-orbifold, which is also known from our previous work. Constructing a manifold cover of an orbifold Γ\H^{n} is equivalent to finding a torsion-free subgroup of Γ. Our method here is based on Lemma 3, which can be thought of as a variant of Minkowski’s lemma that asserts that the kernel of GL_{N} (ℤ) → GL_{N} (ℤ/m) is torsion-free for m > 2. Minkowski’s lemma is the classical tool to construct hyperbolic manifolds from arithmetic subgroups (see for instance [13, Section 4]). We observe that applying Lemma 3 on two small primes (see Section 2.3) gives better results. In particular we show in Section 3 how our method applies to the problem and proves Theorem 2 for dimensions n ≥ 33. On the other hand, Minkowski’s lemma (with m = 3) cannot be used to obtain a direct proof for n < 50.

The dimension bound in Theorem 2 can be further improved because the compact orbifolds of small volume considered above do have singularities and hence the volumes of the smallest compact manifolds are larger. In order to obtain better estimates for these volumes we use the arithmetic information encoded in the Euler characteristic. This approach works well for even dimensions (see Sections 5.1 and 5.2). In odd dimensions the Euler characteristic is zero, but we can still use a similar method. In order to do so we consider even dimensional totally geodesic suborbifolds of the compact odd dimensional orbifolds of small volume. The details of the argument are explained in Section 5.3 and its application to the case n = 31 is in Section 5.4. It is possible to use similar considerations for smaller dimensions but the number of variants and the computational difficulty increase rapidly for smaller n. The problem becomes very difficult for n < 10 and we expect that some new ideas or significant computational advances are required in order to extend our result to this range.

Acknowledgments. Part of this work was done while the second author was visiting IMPA (Rio de Janeiro, Brazil). He wishes to thank the institute for the hospitality and support.

2. A method of construction

2.1. Let G be a semisimple algebraic group defined over a number field k. We denote by V_{f} the set of finite places of k. For each v ∈ V_{f}, we denote by k_{v}, O_{v} and 𝔣_{v} the completion, local ring and residue field defined by v. We suppose that G is simply connected as an algebraic group. By definition, a principal arithmetic subgroup of G(k) is a subgroup Λ_{P} = G(k) ∩ ∏ _{v∈V_{f}} P_{v}, where P = (P_{v})_{v∈V_{f}} is a coherent collection of parahoric subgroups P_{v} ⊂ G(k_{v}) (see [4]). By Bruhat-Tits theory (see [16]), for each v ∈ V_{f}, there exists a smooth connected O_{v}-group scheme P_{v} with generic fiber G and such that P_{v}(O_{v}) = P_{v}. We denote by M_{v} the 𝔣_{v}-group defined as the maximal reductive quotient of P_{v}/𝔣_{v}. The reduction map P_{v} → M_{v}(𝔣_{v}) is surjective. Moreover, since G is simply connected, we have that P_{v}/𝔣_{v} (and thus M_{v} as well) is connected.

2.2. Let p be the rational prime above v, i.e., p is the characteristic of the residue field 𝔣_{v}. Let us denote by P_{v}^{∗} ⊂ P_{v} the pre-image under the reduction map of a p-Sylow subgroup of M_{v}(𝔣_{v}).

Lemma 3. Each torsion element in P_{v}^{∗} has an order of the form p^{s}.

Proof. Let K_{v} be the kernel of the map P_{v} → P_{v}(𝔣_{v}). Let g ∈ P_{v}^{∗} be an element of finite order q, and denote by \overline{g} its image in P_{v}(𝔣_{v}). It suffices to consider the case where q is prime. The image of P_{v}^{∗} in P_{v}(𝔣_{v}) is, by definition, the extension of a p-group by a unipotent 𝔣_{v}-group. It follows that the image of P_{v}^{∗} is itself a p-group, and thus we have either \overline{g} = 1, or q = p. But in the first case g is contained in K_{v}, which is a pro-p group (see [11, Lemma 3.8]), and the conclusion follows as well. □

2.3. By choosing two primes v and w above two distinct rational primes, and replacing P_{v} (resp. P_{w}) by P_{v}^{∗} (resp. P_{w}^{∗}) in the coherent collection P, we obtain by Lemma 3 a torsion-free subgroup Λ_{P}^{v,w} ⊂ Λ_{P}. By strong approximation of G, the index [Λ_{P} : Λ_{P}^{v,w}] equals [P_{v} : P_{v}^{∗}] · [P_{w} : P_{w}^{∗}]. By construction, the index [P_{v} : P_{v}^{∗}] is equal to the order of M_{v}(𝔣_{v}) divided by its highest p-factor. We will usually work with parahoric subgroups for which M_{v} is semisimple (for instance, P_{v} hyperspecial), and in this case [P_{v} : P_{v}^{∗}] can be easily computed from the type of M_{v} using the list of orders of simple groups over a finite field (see [10]). For example, if G/k_{v} is a split group of type B_{r} and P_{v} ⊂ G(k_{v}) is hyperspecial, then M_{v} is semisimple of the same type and we obtain:

[P_{v} : P_{v}^{∗}]= ∏_{j=1}^{r} (q_{v}^{2j} − 1), (1)

where q_{v} denotes the cardinality of the finite field 𝔣_{v}. The product of two such expressions gives us the index of the torsion-free subgroup Λ_{P}^{v,w} in Λ_{P}. If Λ_{P} acts on H^{n} we can then compute the volume of the quotient manifold Λ_{P}^{v,w} \H^{n} as a product of this index and the covolume of Λ_{P}.

3. Construction of noncompact manifolds

We will construct noncompact hyperbolic manifolds starting from the arithmetic lattices of minimal covolume considered in [2, 3]. They are best understood as normalizers of principal arithmetic subgroups of G = Spin(n,1), the latter Lie group being a two-fold covering of Isom^{+} (H^{n}).

3.1. We showed in [3, Section 2.1] that if a lattice Γ ⊂ G contains the center Z of Spin(n,1), then the hyperbolic volume vol(Γ\H^{n}) is given by 2vol(S^{n})μ(Γ\G), where S^{n} is the n-sphere of constant curvature 1 and μ is the normalization of the Haar measure on G that was used by Prasad in [12], and that the volume is the half of this value otherwise. Suppose now that Γ ⊂ G is torsion-free. Since Z has order 2 in all dimensions, we obtain in this case:

(2)  vol(Γ\H^{n}) = vol(S^{n})μ(Γ\G)

If n is even, then 2μ(Γ\G) = |χ(Γ)| (cf. [4, Section 4.2]) and equation (2) is equivalent to the Gauss-Bonnet formula (note that a torsion-free Γ ⊂ G is isomorphic to – and thus has same Euler characteristic as – its image in Isom^{+} (H^{n})):

(3)  vol(Γ\H^{n}) = \frac{vol(S^{n})}{2}|χ(Γ)|, n even.

We recall that the volume of the n-sphere of curvature 1 is given by the following formula:

vol(S^{n}) = \frac{2π^{\frac{n+1}{2}}}{Γ(\frac{n+1}{2} )} , (4)

where Γ(−) is the Gamma function (recall that for an integer m we have Γ(m) = (m − 1)!).

3.2. Let Λ_{P} ⊂ Spin(n,1) be the (unique) nonuniform principal arithmetic subgroup whose normalizer realizes the smallest covolume. Like all nonuniform arithmetic hyperbolic lattices it is defined over k = ℚ, so that the set V_{f} of finite places corresponds to the rational primes. The structure of the coherent collection P was determined in [2, 3]. We denote by G the algebraic ℚ-group containing Λ_{P}, and we will use the notation introduced in Section 2.

3.3. Let us first deal with the case when n = 2r is even. Then G is of type B_{r} and the coherent collection P determining Λ_{P} has the following structure (see [2]): P_{v} is hyperspecial unless v = 2 and r ≡ 2,3 (4), in which case the associated reductive 𝔽_{2}-group M_{2} is semisimple of type ^{2}D_{r}. By Prasad’s volume formula [12], the Euler characteristic of Λ_{P} is then given by

(5)  |χ(Λ_{P})| = 2λ_{2}(r)C(r) ∏_{j=1}^{r} ζ(2j),

with λ_{q}(r) = 1 if r ≡ 0,1 (4) and λ_{q}(r) = q^{r} − 1 otherwise, and the constant C(r) given by

(6)  C(r) = ∏_{j=1}^{r} \frac{(2j − 1)!}{(2π)^{2j}} .

Using the functional equation for the Riemann zeta function, and expressing its value at negative integers through Bernoulli numbers, we can rewrite (5) as follows:

(7)  |χ(Λ_{P})| = 2λ_{2}(r) ∏_{j=1}^{r} \frac{|B_{2j} |}{4j}.

3.4. We can apply the construction presented in Section 2 to Λ_{P} with v = 2 and w = 3. We find that for all r, we have:

(8)  λ_{2}(r)[P_{2} : P_{2}^{∗}]= ∏_{j=1}^{r} (2^{2j} − 1),

the case of trivial λ_{2}(r) being just the computation in (1). Let Γ be the (isomorphic) image of Λ_{P}^{2,3} in Isom^{+} (H^{n}) and M^{n} = Γ\H^{n} the corresponding quotient manifold, which by the construction is noncompact and arithmetic. From the previous computation we have:

(9)  |χ(M^{2r})| = 2 ∏_{j=1}^{r} (4^{j} − 1)(9^{j} − 1)\frac{|B_{2j} |}{4j}

= 2C(r) ∏_{j=1}^{r} (4^{j} − 1)(9^{j} − 1)ζ(2j). (10)

\begin{matrix}n & |χ(M^{2r})| \\ 4 & 10 \\ 6 & 910 \\ 8 & 3171350 \\ 10 & 725639764850 \\ 12 & 16654568229539490250 \\ 14 & 54376724439679967985482572750 \\ 16 & 33998109351372684068956597092378802073750 \\ 18 & 5272397653068183031816584035192902513000228940543011250 \end{matrix}

Table 1. Euler characteristic of noncompact manifolds M^{n}

3.5. We now construct noncompact hyperbolic manifolds in odd dimensions n > 3. Let r = (n + 1)/2. Then the ℚ-group G containing Λ_{P} has type D_{r}. It is an inner form (type ^{1}D_{r}) unless r is even (i.e., n ≡ 3 (4)), in which case it has type ^{2}D_{r} and becomes inner over ℓ = ℚ(\sqrt{−3} ). The details of the construction are given in [3], here we only briefly recall the relevant parts. Let us first suppose that r is odd. Then P_{v} is hyperspecial unless v = 2 and r ≡ 3 (4). In the latter case the radical of M_{2} is a nonsplit one-dimensional torus, and the semisimple part of M_{2} has type ^{2}D_{r−1} . If we let

λ′_{q}(r) = \frac{(q^{r} − 1)(q^{r−1} − 1)}{q + 1} (11)

for q = 2 with r ≡ 3 (4) and λ′_{q}(r) = 1 otherwise, we find that in all cases with odd r we have

(12)  λ′_{q_{v}}(r)[P_{v} : P_{v}^{∗}]= (q_{v}^{r} − 1) ∏_{j=1}^{r−1} (q_{v}^{2j} − 1).

Moreover, Prasad’s formula together with equations (2) and (4) shows that we have (with C(−) defined in (6)):

vol(Λ_{P}) = vol(S^{n})λ′_{2}(r)\frac{(r − 1)!}{(2π)^{r}} ζ(r) ∏_{j=1}^{r−1} \frac{(2j − 1)!}{(2π)^{2j}} ζ(2j)

= λ′_{2}(r)\frac{C(r − 1)}{2^{r−1}} ζ(r) ∏_{j=1}^{r−1} ζ(2j) (13)

= λ′_{2}(r)ζ(r) ∏_{j=1}^{r−1} \frac{|B_{j} |}{8j}. (14)

Thus in this case we obtain a manifold M^{n} = Λ_{P}^{2,3}\H^{n}, n = 2r−1, r odd, whose volume is given by

(15)  vol(M^{n}) = ζ(r)(2^{r} − 1)(3^{r} − 1) ∏_{j=1}^{r−1} \frac{(4^{j} − 1)(9^{j} − 1)}{8j} |B_{2j} |.

Let us finally consider the remaining case n = 2r − 1 with r even. In this case all parahoric subgroups P_{v} in the collection P are hyperspecial with the only exception of v = 3 (which ramifies in ℓ = ℚ(\sqrt{−3} )), where P_{v} is special. Since 2 does not split in ℓ, the group G is an outer form over ℚ_{2} and the 𝔽_{2}-group M_{2} associated to the hyperspecial parahoric P_{2} is semisimple of type ^{2}D_{r}. Thus we have:

[P_{2} : P_{2}^{∗}]= (2^{r} + 1) ∏_{j=1}^{r−1} (2^{2j} − 1). (16)

For the place v = 3, M_{3} is semisimple of type B_{r−1} , from which we get

[P_{3} : P_{3}^{∗}]= ∏_{j=1}^{r−1} (3^{2j} − 1). (17)

Using Prasad’s formula to compute the covolume of Λ_{P}, we finally deduce that the manifold M^{n} = Λ_{P}^{2,3}\H^{n}, n = 2r − 1, r even, has volume

(18)  vol(M^{2r−1} ) = 3^{r−1/2} (2^{r} + 1)ζ_{ℓ}(r)/ζ(r) ∏_{j=1}^{r−1} \frac{(4^{j} − 1)(9^{j} − 1)}{8j} |B_{2j} |.

3.6. Formulas (15) and (18) allow us to evaluate the volume of the constructed manifolds M^{n} using Pari/GP. Moreover, for n even the volume can be deduced from formula (9) together with (3). We list the values obtained for dimensions n ≤ 20 in Table 2. It shows, in particular, that the volume grows rapidly with the dimension.

4. Proof of Theorem 2 for sufficiently large dimension

4.1. We now compare the volume of M^{n} from the previous section with the volume of the smallest compact arithmetic hyperbolic n-orbifold O^{n} = Γ\H^{n}. In the rest of the paper the Euler characteristic χ(O^{n}) is to be understood in the orbifold sense, that is, χ(O^{n}) = χ(Γ). We begin with even dimensions n = 2r ≥ 4. The orbifold O^{n} is defined over the field k = ℚ(\sqrt{5}) and its Euler characteristic is given by

|χ(O^{n})| = 4 · \overline{λ}_{4}(r) · 5^{r^{2}+r/2} C(r)^{2} ∏_{j=1}^{r} ζ_{k}(2j), (19)

with \overline{λ}_{q}(r) = 1 if r is even and \overline{λ}_{q}(r) = \frac{q^{r} −1}{2} if r is odd [2].

808 Mikhail Belolipetsky, Vincent Emery

n  vol(M^{n})
4  131.594
5  273.467
6  15048.379
7  42504.453
8  47073267.939
9  770938537.303
10  7519493827964.029
11  305396253769850.768
12  98579836734072034892.809
13 36424053767874477954431.531
14  1.555 E29
15  2.059 E32
16  4.074 E40
17  6.691 E44
18  2.335 E54
19  1.797 E59
20  3.734 E70

Table 2. Approximate values of the volume of M^{n}

By (10) and (19), we have:

(20)  \frac{|χ(O^{n})|}{|χ(M^{n})|} = \frac{4 · \overline{λ}_{4}(r) · 5^{r^{2}+r/2} C(r)^{2} ∏ _{j=1}^{r} ζ_{k}(2j)}{2C(r)∏ _{j=1}^{r} (4^{j} − 1)(9^{j} − 1)ζ(2j)}.

Using the basic inequalities ζ_{k}(2j) > 1 and ∏ _{j=1}^{r} ζ(2j) < 2 plus the fact that \overline{λ}_{4}(r) ≥ 1 for all r, we obtain the lower bound

(21)  \frac{|χ(O^{n})|}{|χ(M^{n})|} > \frac{5^{r^{2}+r/2} C(r)}{∏ _{j=1}^{r} (4^{j} − 1)(9^{j} − 1)} > \frac{5^{r^{2}+r/2} C(r)}{36^{r^{2}/2+r/2}}.

By Stirling’s formula,

C(r) = ∏_{j=1}^{r} \frac{(2j − 1)!}{(2π)^{2j}} > ∏_{j=1}^{r} \frac{(2j − 1)^{2j−1}}{2π(2πe)^{2j−1}} .

Applying the Euler–Maclaurin summation formula, we obtain

logC(r) ≳ \frac{1}{4}(2r − 1)^{2} log(2r − 1),

which is ≫ cr^{2} (for any constant c). Hence for r ≫ 0, we have

\frac{|χ(O^{n})|}{|χ(M^{n})|} ≫ 1.

Documenta Mathematica 19 (2014) 801–814

In fact, a computation shows that it is enough to take r ≥ 18 for the ratio in the right hand side of (21) to be > 1. A more precise computation of the expression in (20) using Pari/GP shows that

\frac{|χ(O^{n})|}{|χ(M^{n})|} > 1, for n = 2r ≥ 34.

Since O^{n} has the smallest volume among compact arithmetic hyperbolic norbifolds, this proves Theorem 2 for these values of n.

4.2. Now consider the odd dimensional case n = 2r − 1 ≥ 5. The smallest compact odd dimensional arithmetic orbifold O^{n} is again defined over k = ℚ(\sqrt{5}). Its volume is given by

(22) vol(O^{n}) = \frac{5^{r^{2}−r/2} · 11^{r−1/2} · (r − 1)!}{2^{2r−1} π^{r}}· L_{ℓ_{0}|k}(r) · C(r − 1)^{2} ∏_{j=1}^{r−1} ζ_{k}(2j),

where ℓ_{0} is the quartic field with a defining polynomial x^{4} − x^{3} + 2x − 1 and L_{ℓ_{0}|k} = ζ_{ℓ_{0}}/ζ_{k} [3, Thoerem 1]. Note that for any s ≥ 3 we have L_{ℓ_{0}|k}(s) ≥ 1/ζ_{k}(3) > 0.973. By (15), (18) and (22), we have

(23) \frac{vol(O^{n})}{vol(M^{n})} = \frac{\frac{5^{r^{2}−r/2} ·11^{r−1/2} ·(r−1)!}{2^{2r−1} π^{r}} · L_{ℓ_{0}|k}(r) · C(r − 1)^{2} ∏ _{j=1}^{r−1} ζ_{k}(2j)}{A(r) · \frac{C(r−1)}{2^{r−1}} · ∏ _{j=1}^{r−1} ζ(2j)(4^{j} − 1)(9^{j} − 1)},

where A(r) = ζ(r)(2^{r} − 1)(3^{r} − 1) if r is odd and A(r) = 3^{r−1/2} (2^{r} + 1)ζ_{ℓ}(r)/ζ(r), ℓ = ℚ(\sqrt{−3} ), if r is even, and hence

(24)  A(r) < 6^{r} · 2 for all r.

Therefore,

(25)  \frac{vol(O^{n})}{vol(M^{n})} > \frac{5^{r^{2}−r/2} · 11^{r−1/2} · (r − 1)! · C(r − 1)}{2^{r}π^{r}6^{r} · 4 · ∏ _{j=1}^{r−1} (4^{j} − 1)(9^{j} − 1)}

> \frac{5^{(r−1)(r−1/2)} C(r − 1)}{36^{r^{2}/2−r/2}} · \frac{11^{r−1/2} · (r − 1)!}{5^{1/2−r} · 4 · (12π)^{r}}. (26)

Now the first factor has the same form as the ratio in (21), hence by the previous section it is > 1 for r − 1 ≥ 18. It is an easy exercise to show that for such r, the second factor is > 1 as well. We can improve the bound for r by evaluating (23) using Pari/GP. This gives

\frac{|χ(O^{n})|}{|χ(M^{n})|} > 1, for n = 2r − 1 ≥ 33.

Together with the even dimensional part it proves Theorem 2 for n ≥ 33.

5. Lowering the dimension bound

5.1. Improving the bound for dimension requires more careful analysis. We will begin again with the even dimensional case. The first dimension to consider is n = 32. Here we have a noncompact arithmetic manifold M^{n} with Euler characteristic |χ(M^{n})| = 2.354...· 10^{228} . Ironically, we call it “small”, but our study shows that small manifolds in high dimensions tend to have huge volume. The smallest compact arithmetic norbifold O^{n} has |χ(O^{n})| = 8.777...· 10^{217} , which is less than |χ(M^{n})|. Using Pari/GP we can compute (see remark below) the precise value of |χ(O^{n})|, which is a rational number. Its denominator equals

D = 107887196930872715055177987172922818560000000000000000000,

which is ∼ 10^{56}. A manifold covering of O^{n} has to have an (even) integer Euler characteristic, hence its degree is divisible by D. It follows that the volume of the smallest possible manifold cover of O^{n} is much larger than vol(M^{n}).

Remark 4. A way to compute the exact value of |χ(O^{n})| is to first use the functional equation for ζ_{k} to transform equation (19) into a product of a (known) rational factor and values ζ_{k}(1 − 2j), which are rational. Using Pari/GP the numerical values of ζ_{k}(1 − 2j) can be approximated by rationals, leading to the number D (it is enough to work with precision \p 80). With this method based on approximation, the value obtained for D can only be used as a lower bound for the actual denominator of |χ(O^{n})|, but this is already sufficient for our purpose. However, a cleaner way to proceed is to use the library of special values computed by Alvaro Lozano-Robledo, who obtained them through a procedure under Pari/GP that computes generalized Bernoulli numbers (see [9]). This guarantees the correctness of the value computed for D.

In order to complete the discussion for n = 32 we need to consider other maximal arithmetic subgroups Γ whose covolume is < vol(M^{n}). Recall that any maximal arithmetic subgroup is a normalizer of a principal subgroup. Maximal arithmetic subgroups other than the one defining O^{n} can be either subgroups defined over k = ℚ(\sqrt{5}) or arithmetic subgroups defined over other totally real fields. In the latter case, by [2], we know that the next smallest covolume subgroup has the field of definition ℚ(\sqrt{2}). By the volume formula, the absolute value of its Euler characteristic is > 9.071 · 10^{271} , which is already bigger than |χ(M^{n})|. Therefore, we are only left with the groups defined over k. The covolume of any such subgroup Γ would have an additional lambda factor in comparison with vol(O^{n}). By [2, Sections 2.3 and 3.2], the smallest possible

lambda factor is λ_{v} = \frac{q_{v}^{16} +1}{2} for q_{v} = 4, and it follows that the orbifold corresponding to the principal arithmetic subgroup Λ that Γ normalizes has an Euler characteristic of absolute value 1.884...· 10^{227} with denominator 2D. Hence all manifold covers of this orbifold are much larger than M^{n}. Now, the index [Γ : Λ] can be computed the same way as it is done in [2] and [3] and in particular, since the field k = ℚ(\sqrt{5}) has class number one, this index is a power of 2 (whose exponent can be bounded). Using this the result just obtained immediately extends to the manifold covers of Γ\H^{32}. The same argument excludes the case of orbifolds with a lambda factor λ_{v} corresponding to q_{v} = 5. All other possible lambda factors give rise to orbifolds of volume larger than vol(M^{n}). This shows that Theorem 2 is true for n = 32.

5.2. Dimension n = 30 is treated in a similar way. Here we have a noncompact arithmetic manifold M^{n} with |χ(M^{n})| = 1.252...· 10^{195} . The smallest volume compact arithmetic orbifold O^{n} has |χ(O^{n})| = 8.112...·10^{187} with denominator 5.231...· 10^{48}, hence its manifold covers are larger than M^{n}. The smallest covolume cocompact arithmetic subgroup defined over k ≠ ℚ(\sqrt{5}) has the field of definition ℚ(\sqrt{2}) and |χ| > 3.116 · 10^{235} , which is bigger than |χ(M^{n})|. For the other maximal arithmetic subgroups defined over ℚ(\sqrt{5}) we have a different lambda factor in the volume formula. The values of λ for which |χ| of the principal subgroup is smaller than |χ(M^{n})| are listed below:

λ  |χ|  denominator of χ
\frac{1}{2}(5^{15} − 1) 2.305... · 10^{189} 82391859826240770906019357261824 · 10^{18}
\frac{1}{2}(9^{15} − 1) 1.555... · 10^{193}  41154775137982403049959718912 · 10^{18}
\frac{1}{2}(11^{15} − 1) 3.155... · 10^{194} 32109284800854974718802972372893696 · 10^{16}

As before, considering maximal subgroups instead of principal subgroups does not change the picture and it follows that a manifold cover in each of these cases is larger than the noncompact manifold M^{n}.

5.3. Odd dimensional case presents us a different challenge: here the Euler characteristic is zero and we cannot take advantage of its integral properties in order to bound the degree of the smooth covers. One of the possible ways to proceed is to look at the orders of finite subgroups of π_{1}(O^{n}). This indeed gives a bound for the degree, however, it is much smaller than bounds provided by the denominators of Euler characteristic in the neighboring even dimensions. Considerably stronger results can be obtained based on the following simple observation:

Small volume arithmetic hyperbolic orbifolds tend to contain totally geodesic even dimensional suborbifolds whose Euler characteristic can be used to obtain good bounds on the degrees of the smooth covers.

Indeed, assume that a group Γ has an infinite index subgroup Γ′ and a torsionfree finite index subgroup Γ_{M} with [Γ : Γ_{M} ] = f. Then Γ_{M} ∩Γ′ is a torsion-free subgroup of Γ′ of index d ≤ f:

(27)  Γ  ^{∞}_{f}  Γ′_{d} Γ_{M} _{∞} Γ_{M} ∩ Γ′

This proposition can be easily checked by looking at the cosets Γ/Γ_{M} and Γ′/(Γ_{M} ∩ Γ′). It appears to be very useful for bounding degrees of the smooth covers in odd dimensions.

5.4. We demonstrate the application of the method from 5.3 for dimension n = 31. By [3, Section 3.5], the minimal volume compact orbifold O^{31} is defined by the quadratic form

f_{31}(x_{0},x_{1},...,x_{31}) = (3 − 2\sqrt{5})x_{0}^{2} + x_{1}^{2} + ...x_{31}^{2}.

It has vol(O^{31}) = 2.415...·10^{200} < vol(M^{31}) = 3.113...·10^{202} , so we can not a priori exclude the possibility of some manifold cover of O^{31} being smaller than M^{31}. A restriction f_{30} of the form f_{31} to the first 31 coordinates defines a totally geodesic suborbifold O_{1}^{30} ⊂ O^{31} that has a non-zero Euler characteristic. In order to compute χ(O_{1}^{30}) we need to control the associated integral structure. In what follows we will use the notation from [3] and [7]. Consider the quadratic space (V, \frac{1}{2}f_{31}). We have the discriminant δ(\frac{1}{2}f_{31}) = \frac{1}{2^{32}} (3 − 2\sqrt{5}) and the Hasse-Witt invariant ω(V, \frac{1}{2}f_{31}) = 1 over all places including the dyadic place v_{2} = (2) of k = ℚ(\sqrt{5}). Comparing this data with the local description of the group Γ of O^{31} given in [3, Section 3], we conclude that Γ is isomorphic to the stabilizer of a maximal lattice L in (V, \frac{1}{2}f_{31}) (this can be also confirmed by comparing the covolume of Stab(L) computed using [7, Table 3] with vol(O^{31}) and recalling the uniqueness of O^{31}). The lattice L is defined uniquely up to a conjugation under G but different (conjugate) lattices may have different restriction to the subspace (V′, \frac{1}{2}f_{30}) of V. Geometrically this corresponds to choosing different totally geodesic 30-dimensional suborbifolds of O^{31}. In order to complete the computation we need to fix the lattice L. This can be done as follows: First take a maximal lattice L′ in (V′, \frac{1}{2}f_{30}). Now consider an integral lattice in V generated by L′ and the vector (0,...,0,2). It is contained in some maximal lattice in V and we can choose this lattice to be our lattice L. This construction implies that the restriction of L onto the subspace V′⊂ V is the maximal lattice L′ in (V′, \frac{1}{2}f_{30}). Now we can use [7, Table 4] to compute the covolume of Stab(L′), and hence the Euler characteristic of the corresponding suborbifold:

|χ(O_{1}^{30})| = \frac{4^{15} − 1}{2} · \frac{11^{15} + 1}{2} · \frac{1}{4^{14}} ∏_{i=1}^{15} |ζ_{k}(1 − 2i)| = 1.694...· 10^{202} ,

with denominator

D_{1} = 290623844184270796846629126144000000000000000000.

Hence by (27), the degree f of any smooth cover of O^{31} is ≥ D_{1}, and so its volume

vol(O^{31}) · f > 7.019...· 10^{247} > vol(M^{31}). (28)

It remains to check if there are other compact arithmetic 31-dimensional orbifolds that have volume ≤ vol(M^{31}). Similar to the previous discussion, these can be either defined over a different field k, or have a different splitting field ℓ, or have a non-trivial lambda factor in the volume formula. Note that by (23) the ratio

(29)  \frac{vol(O^{31})}{vol(M^{31})} > 0.007.

If we change the defining field then its discriminant in the volume formula for vol(O^{31}) would contribute a factor of at least (8/5)^{r^{2}−r/2} (1/11)^{r−1/2} > 3.021· 10^{34} (cf. [3]). This immediately brings it out of the range of consideration. Changing the field ℓ gives a factor ≥ ( \frac{400}{275} )^{31/2} = 332.868... (as the field ℓ that corresponds to the minimal volume orbifold has discriminant D_{ℓ} = 275 and the next possible value is D = 400). This factor is already sufficiently large to make the volume > vol(M^{31}). Finally, following [3, equation (5)], the lambda factor is bounded below by \frac{2}{3} ( \frac{3}{4}q_{v})^{r_{v}} with q_{v} ≥ 4, r_{v} = 16 if q_{v} ≠ 11 and r_{v} = 15 for the remaining place v of k that ramifies in ℓ/k. Hence the smallest possible lambda factor is at least 28697814, which again makes the volume of the corresponding orbifolds too large. Similar to the previous sections, this argument based on principal arithmetic subgroups extends to maximal arithmetics in a straightforward way. This finishes the proof of Theorem 2. □

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Mikhail Belolipetsky IMPA Estrada Dona Castorina 110 22460-320 Rio de Janeiro Brazil mbel@impa.br Vincent Emery FSB-MATHGEOM EPF Lausanne Bâtiment MA Station 8 CH-1015 Lausanne Switzerland vincent.emery@gmail.com