Coboundary expansion and Gromov hyperbolicity

TL;DR

This paper establishes a connection between coboundary expansion properties in high-dimensional covers of manifolds and the hyperbolicity of their fundamental groups, providing new insights into geometric group theory.

Contribution

It proves that coboundary expansion in residual covers implies Gromov hyperbolicity of the fundamental group of the manifold.

Findings

Residual covers with coboundary expansion lead to hyperbolic fundamental groups.

Non-hyperbolic 3-manifolds do not admit residual coboundary expanders.

The result links topological expansion properties to geometric group properties.

Abstract

We prove that if a compact $n$-manifold admits a sequence of residual covers that form a coboundary expander in dimension $n-2$, then the manifold has Gromov-hyperbolic fundamental group. In particular, residual sequences of covers of non-hyperbolic compact connected irreducible 3-manifolds are not 1-coboundary expanders.