On compact hyperbolic manifolds of Euler characteristic two

Abstract

Abstract. We prove that for n > 4 there is no compact arithmetic hyperbolic n-manifold whose Euler characteristic has absolute value equal to 2. In particular, this shows the nonexistence of arithmetically defined hyperbolic rational homology n-sphere with n even different than 4.1. Main result and discussion 1.1. Smallest hyperbolic manifolds. Let H n be the hyperbolic n-space. By a hyperbolic n-manifold we mean an orientable manifold M = Γ\H n , where Γ is a torsion-free discrete subgroup Γ ⊂ Isom + (H n ). The set of volumes of hyperbolic n-manifolds being well ordered, it is natural to try to determine for each dimension n the hyperbolic manifolds of smallest volume. For n = 3 this problem has recently been solved in [15], the smallest volume being achieved by a unique compact manifold, the Weeks manifold. When n is even the volume is proportional to the Euler characteristic, and this allows to formulate the problem in terms of finding the hyperbolic manifolds M with smallest |χ(M )|. In particular this observation solves the problem in the case of surfaces. For n > 3, noncompact hyperbolic n-manifolds M with |χ(M )| = 1 have been found for n = 4, 6 [14].In the present paper we consider the case of compact manifolds of even dimension. In particular, such manifolds have even Euler characteristic (see [17, Theorem 1.2]). We restrict ourselves to the case of arithmetic manifolds, where Prasad's formula [20] can be used to study volumes. We complete the proof of the following result. The result for n > 10 already follows from the work of Belolipetsky [4, 5], also based on Prasad's volume formula. More precisely, Belolipetsky determined the smallest Euler characteristic |χ(Γ)| for arithmetic orbifold quotients Γ\H n (n even). This smallest value grows fast with the dimension n, and for compact quotients we have |χ(Γ)| > 2 for n > 10. That the result of nonexistence holds for n high enough is already a consequence of Borel-Prasad's general finiteness result [9], which was the first application of Prasad's formula. The proof of Theorem 1 for n = 6, 8, 10 requires a more precise analysis of the Euler characteristic of arithmetic subgroups Γ ⊂ PO(n, 1), and in particular of the special values of Dedekind zeta functions that appear as factors of χ(Γ).For n = 4, the corresponding problem is not solved, but there is the following result [5]. Theorem 2 (Belolipetsky). If M = Γ\H 4 is a compact arithmetic manifold with χ(M ) ≤ 16, then Γ arises as a (torsion-free) subgroup of the following hyperbolic Coxeter group:An arithmetic (orientable) hyperbolic 4-manifold of Euler characteristic 16 has been first constructed by Conder and Maclachlan in [12], using the presentation of W 1 to obtain a torsion-free subgroup with the help of a computer. Further examples with χ(M ) = 16 have been obtained by Long in [18] by considering a homomorphism from W 1 onto the finite simple group PSp 4 (4).1.2. Hyperbolic homology spheres. Our original motivation for Theorem 1 was the problem of existence of hyperbolic homology spheres. A homo...

Infinitely many arithmetic hyperbolic rational homology 3-spheres that bound geometrically

“…There is still no known example of a hyperbolic QHS in dimensions ≥ 4. Moreover, it is known that even such a large class as arithmetic hyperbolic manifolds cannot produce a QHS of even dimension n > 4 [5].…”

Quaternionic hyperbolic lattices of minimal covolume

“…3.7]). A list for is given, for instance, in [10, Table 2], from which we obtain the explicit values for listed in Table 1. We omit for reason of space.…”