Stability of hyperbolic groups acting on their boundaries

TL;DR

This paper proves that hyperbolic groups acting on their boundaries exhibit topological stability, meaning small perturbations of the action are semi-conjugate to the original, extending known results to more general boundary types.

Contribution

It introduces a new dynamical coding method to establish topological stability for hyperbolic group actions on their boundaries, generalizing previous sphere boundary results.

Findings

Actions are topologically stable under perturbations

Provides a new proof of stability when boundary is a circle

Extends stability results to more general boundary types

Abstract

A hyperbolic group acts by homeomorphisms on its Gromov boundary. We use a dynamical coding of boundary points to show that such actions are topologically stable in the dynamical sense: any nearby action is semi-conjugate to (and an extension of) the standard boundary action. This result was previously known in the special case that the boundary is a topological sphere. Our proof here is independent and gives additional information about the semiconjugacy in that case. Our techniques also give a new proof of global stability when the boundary is a circle.