Stability of hyperbolic groups acting on their boundaries

Abstract

A hyperbolic group $\Gamma$ acts by homeomorphisms on its Gromov boundary $\partial \Gamma$. 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 $\partial \Gamma$ is a topological sphere. Our proof here is independent and gives additional information about the semi-conjugacy in that case. Our techniques also give a new proof of global stability when $\partial \Gamma =S^1$.