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Stability of hyperbolic groups acting on their boundaries

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arxiv 2206.14914 v2 pith:UNH2NBNX submitted 2022-06-29 math.GR math.GT

Kathrynn Mann , Jason Fox Manning , Theodore Weisman This is my paper · ORCID
classification math.GRmath.GT
keywords boundaryactioncasedynamicalhyperbolicproofstabilityacting
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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.

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  1. Stability for boundary actions of cocompact lattices in Euclidean buildings

    math.DS 2026-07 accept novelty 7.0 of 10

    Cocompact lattice actions on flag boundaries of Euclidean buildings are topologically stable: every sufficiently small perturbation is semi-conjugate to the original action.

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