Hyperbolic manifolds that fiber algebraically up to dimension 8

Abstract

We construct some cusped finite-volume hyperbolic $n$-manifolds $M_n$ that fiber algebraically in all the dimensions $5\le n\le 8$. That is, there is a surjective homomorphism $\ddot{I}{€}_1(M_n)\to \mathbb{Z}$ 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 ${\tilde{M}}_n$ whose fundamental group is finitely presented but not of finite type. These $n$-manifolds ${\tilde{M}}_n$ 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 $P_5,\dots ,P_8$, 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 $n=7,8$.

40 pages, 21 figures, final version