Sphere boundaries of hyperbolic groups

TL;DR

This paper characterizes hyperbolic groups with certain surface subgroups as having a boundary homeomorphic to a 2-sphere, linking group properties to 3-manifold topology.

Contribution

It provides a new criterion for identifying hyperbolic groups that are virtually fundamental groups of hyperbolic 3-manifolds based on boundary and subgroup structure.

Findings

Hyperbolic groups with specific surface subgroups have boundary homeomorphic to a sphere.

The result offers a new characterization of hyperbolic 3-manifold groups.

Connections established between group boundaries and 3-manifold topology.

Abstract

We show that a one-ended simply connected at infinity hyperbolic group $G$ with enough codimension-1 surface subgroups has $\partial G \cong \mathbb{S}^2$. Combined with a result of Markovic, our result gives a new characterization of virtually fundamental groups of hyperbolic 3-manifolds. Keywords: CAT(0) cube complexes, Hyperbolic groups, hyperbolic 3-manifolds.