Hyperbolic manifolds that fiber algebraically up to dimension 8

TL;DR

This paper constructs new hyperbolic n-manifolds for dimensions 5 to 8 that fiber algebraically, providing the first examples with finitely presented but not finite type fundamental groups, using polytopes, colorings, and octonions.

Contribution

It introduces the first hyperbolic n-manifolds with finitely presented but not finite type fundamental groups for dimensions 7 and 8.

Findings

Constructed hyperbolic manifolds fibered algebraically in dimensions 5 to 8.

First examples of hyperbolic n-manifolds with finitely presented but not finite type groups.

Manifolds have infinitely many cusps and infinite Betti number.

Abstract

We construct some cusped finite-volume hyperbolic $n$-manifolds $M_n$ that fiber algebraically in all the dimensions $5\leq n \leq 8$. That is, there is a surjective homomorphism $\pi_1(M_n) \to \mathbb Z$ 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 $\widetilde M_n$ whose fundamental group is finitely presented but not of finite type. These $n$-manifolds $\widetilde M_n$ have infinitely many cusps of maximal rank and hence infinite Betti number $b_{n-1}$. They cover the finite-volume manifold $M_n$. We obtain these examples by assigning some appropriate colours and states to a family of right-angled hyperbolic polytopes $P_5, \ldots, P_8$, and then applying some arguments of Jankiewicz, Norin, Wise and Bestvina, Brady. We exploit in an essential way the remarkable properties of the Gosset polytopes dual to $P_n$, and the algebra of integral octonions for the crucial dimensions $n=7,8$.