Homology growth, hyperbolization, and fibering

TL;DR

This paper introduces a new hyperbolic reflection group method to construct closed aspherical manifolds with preserved hyperbolicity and homology growth, and applies it to produce hyperbolic 7-manifolds that do not virtually fiber over a circle.

Contribution

It presents a novel hyperbolic reflection group trick for building aspherical manifolds and demonstrates its application to hyperbolic 7-manifolds with specific fibering properties.

Findings

Constructed closed aspherical manifolds with preserved hyperbolicity.

Developed hyperbolic 7-manifolds that do not virtually fiber over a circle.

Achieved manifolds with controlled $ ext{F}_p$-homology growth.

Abstract

We introduce a hyperbolic reflection group trick which builds closed aspherical manifolds out of compact ones and preserves hyperbolicity, residual finiteness, and -- for almost all primes $p$ -- $\mathbb{F}_p$-homology growth above the middle dimension. We use this trick, embedding theory and manifold topology to construct Gromov hyperbolic $7$-manifolds that do not virtually fiber over a circle out of graph products of large finite groups.