# Stability of hyperbolic groups acting on their boundaries

Kathrynn Mann, Jason Fox Manning, Theodore Weisman

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.
Stability of boundary actions
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17 days ago
Eliane ▴ 30

In theorem 1.1 the authors prove that the standard action of a hyperbolic group on its Gromov boundary is stable up to a semi conjugacy.

They show that the result is not true if hyperbolic is replaced by relatively hyperbolic and the Gromov boundary is replaced by the Bowditch boundary. For hyperbolic groups if we replace the Gromov boundary with another boundary such as the Poisson boundary is the action stable and if so can it be proved using the dynamical coding techniques in this paper?

boundary theory • 39 views