Hyperbolic manifolds that fiber algebraically up to dimension 8

Abstract

We construct some cusped finite-volume hyperbolic n-manifolds Mn that fiber algebraically in all the dimensions 5 ≤ n ≤ 8. That is, there is a surjective homomorphism π 1 (Mn) → Z with finitely generated kernel.The kernel is also finitely presented in the dimensions n = 7, 8, and this leads to the first examples of hyperbolic n-manifolds Mn whose fundamental group is finitely presented but not of finite type. These n-manifolds Mn have infinitely many cusps of maximal rank and hence infinite Betti number b n−1 . They cover the finite-volume manifold Mn.We obtain these examples by assigning some appropriate colours and states to a family of right-angled hyperbolic polytopes P 5 , . . . , P 8 , and then applying some arguments of Jankiewicz -Norin -Wise [15] and . We exploit in an essential way the remarkable properties of the Gosset polytopes dual to Pn, and the algebra of integral octonions for the crucial dimensions n = 7, 8.

Hyperbolic groups containing subgroups of type $\mathscr{F}_{3}$ not $\mathscr{F}_{4}$

“…To deduce Theorem 1 from Theorems 3 and 5, we will use a certain hyperbolic manifold M 8 = H 8 /Γ built from the dual of the 8-dimensional Euclidean Gosset polytope studied in [19], together with a continuous map u : M 8 → S 1 . The manifold M 8 and the corresponding map to the circle were constructed in [26]. There the authors prove that the kernel of the homomorphism u * : Γ → Z induced by u is finitely presented.…”

“…Before proving Theorem 5 in Section 3.2, we provide a brief discussion of the relevance of 2 -Betti numbers for the study of finiteness properties in Section 3.1. Their use allows for instance to determine the exact finiteness properties of some of the groups considered in [26]. We emphasize however that all the theorems stated in the introduction are proved without using 2 -Betti numbers.…”

“…We provide the proof for the reader's convenience and we shall comment further on Fisher's work below. In combination with Theorem 24 below, Proposition 14 allows us to determine the precise finiteness properties of the original infinite cyclic coverings considered in [26].…”

“…In [26] Italiano, Martelli and Migliorini construct a 2n-dimensional hyperbolic manifold M 2n together with a homomorphism f n : π 1 (M 2n ) → Z induced by a circlevalued height function for n ∈ {2, 3, 4} from duals of Euclidean Gosset polytopes. As a consequence of Corollary 15 we obtain that ker(f n ) is a subgroup of a relatively hyperbolic group that is not of type FP n (Q) (in particular not F n ).…”

“…In this section we describe the hyperbolic 8-manifold M 8 first studied in [26] and prove Theorem 6. We will use the letter M to denote a general hyperbolic manifold and will write M 8 when refering to the specific example built in [26].…”