Stability for hyperbolic groups acting on boundary spheres

TL;DR

This paper proves that hyperbolic groups acting on boundary spheres exhibit topological stability, meaning small perturbations of the action are semi-conjugate to the original boundary action, highlighting robustness in their dynamical behavior.

Contribution

It establishes topological stability for hyperbolic groups acting on boundary spheres, a novel result linking geometric group theory and dynamical systems.

Findings

Hyperbolic groups with boundary as an n-sphere have topologically stable actions.

Nearby actions are semi-conjugate to the standard boundary action.

The result connects boundary topology with dynamical stability properties.

Abstract

A hyperbolic group acts by homeomorphisms on its Gromov boundary. We show that if this boundary is a topological n-sphere the action is topologically stable in the dynamical sense: any nearby action is semi-conjugate to the standard boundary action.