Kathryn Mann (Cornell University)
Title: Stability for hyperbolic groups acting on their boundaries.
Abstract: A hyperbolic group acts naturally by homeomorphisms on its boundary. The theme of this talk is to say that, in many cases, such an action has very robust dynamics. Jonathan Bowden and I studied a very special case of this, showing if G is the fundamental group of a compact, negatively curved Riemannian manifold, then the action of G on its boundary is topologically stable (small perturbations of it are semi-conjugate, containing all the dynamical information of the original action). In new work with Jason Manning, we get rid of the Riemannian geometry and show that such a result holds for hyperbolic groups with sphere boundary, using purely large-scale geometric techniques. This theme of studying topological dynamics of boundary actions dates back at least as far as work of Sullivan in the 1980's, although we take a very different approach. My talk will give some history and some picture of the large-scale geometry involved in our work.
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