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arXiv · 1909.02324

$C^0$ stability of boundary actions and inequivalent Anosov flows

Jonathan Bowden·Kathryn Mann
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Abstract

We give a topological stability result for the action of the fundamental group of a compact manifold of negative curvature on its boundary at infinity: any nearby action of this group by homeomorphisms of the sphere is semi-conjugate to the standard boundary action. Using similar techniques we prove a global rigidity result for the "slithering actions" of 3-manifold groups that come from skew-Anosov flows. As applications, we construct hyperbolic 3-manifolds that admit arbitrarily many topologically inequivalent Anosov flows, answering a question from Kirby's problem list, and also give a more conceptual proof of a theorem of the second author on {\em global} $C^0$--rigidity of geometric surface group actions on the circle.

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Jonathan Bowden, Kathryn Mann. 2019-09-05. $C^0$ stability of boundary actions and inequivalent Anosov flows. https://arxiv.org/abs/1909.02324

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