Hyperbolic groups with boundary an n-dimensional Sierpinski space

TL;DR

This paper characterizes hyperbolic groups with boundaries homeomorphic to high-dimensional Sierpinski spaces, linking them to fundamental groups of aspherical manifolds with boundary, and provides counterexamples for the converse.

Contribution

It establishes a correspondence between certain hyperbolic groups and aspherical manifolds, and constructs examples showing limitations of this relationship.

Findings

Hyperbolic groups with boundary an (n-2)-dimensional Sierpinski space are fundamental groups of aspherical n-manifolds.

Counterexamples exist of hyperbolic groups with non-Sierpinski boundary, even when the group is fundamental to an aspherical manifold.

The results hold for dimensions greater than 3 and 6, respectively.

Abstract

For n>6, we show that if G is a torsion-free hyperbolic group whose visual boundary is an (n-2)-dimensional Sierpinski space, then G=\pi_1(W) for some aspherical n-manifold W with nonempty boundary. Concerning the converse, we construct, for each n>3, examples of aspherical manifolds with boundary, whose fundamental group G is hyperbolic, but with visual boundary not homeomorphic to an (n-2)-dimensional Sierpinski space.