On a generalization of Cannon's conjecture for cubulated hyperbolic groups

TL;DR

This paper extends the understanding of cubulated hyperbolic groups by linking their boundary properties to their manifold structures, generalizing previous results related to Cannon's conjecture.

Contribution

It introduces conditions under which cubulated hyperbolic groups with certain boundary dimensions are virtually fundamental groups of aspherical manifolds.

Findings

Hyperbolic groups with 3 or ≥5 dimensional spherical boundary are virtually fundamental groups of aspherical manifolds.

Presence of many quasi-convex, codimension-1 subgroups with locally flat limit sets implies manifold structure.

Generalizes previous work on Cannon's conjecture to higher boundary dimensions.

Abstract

We show that cubulated hyperbolic groups with spherical boundary of dimension 3 or at least 5 are virtually fundamental groups of closed, orientable, aspherical manifolds, provided that there are sufficiently many quasi-convex, codimension-1 subgroups whose limit sets are locally flat subspheres. The proof is based on ideas used by Markovic in his work on Cannon's conjecture for cubulated hyperbolic groups with 2-sphere boundary.