On hyperbolic groups with spheres as boundary

TL;DR

This paper proves that torsion-free hyperbolic groups with spherical boundary are fundamental groups of closed aspherical manifolds, extending understanding of the geometric structures associated with such groups.

Contribution

It establishes a new characterization of hyperbolic groups with spherical boundary as fundamental groups of closed aspherical manifolds.

Findings

Hyperbolic groups with spherical boundary are fundamental groups of closed aspherical manifolds.

The boundary homeomorphic to a sphere implies a specific geometric structure.

Extension of the boundary characterization to higher dimensions (n > 5).

Abstract

Let G be a torsion-free hyperbolic group and let n > 5 be an integer. We prove that G is the fundamental group of a closed aspherical manifold if the boundary of G is homeomorphic to an (n-1)-dimensional sphere.