Topological stability of relatively hyperbolic groups acting on their boundaries

2024/02/09 by Kathryn Mann, Jason Fox Manning, Mann, Kathryn +3
Mathematics · #FOS: Mathematics #Geometric Topology (math.GT) #Geometric and Algebraic Topology #Group Theory (math.GR) #Mathematical Dynamics and Fractals #advanced mathematical theories

paper · pdf · doi:10.48550/arxiv.2402.06144

openalex publication_date 2024/02/09 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We prove a topological stability result for the actions of hyperbolic groups on their Bowditch boundaries. More precisely, we show that a sufficiently small perturbation of the standard boundary action, if assumed on each parabolic subgroup to be a perturbation by semi-conjugacy, is in fact always globally semi-conjugate to the standard action. This proves a relative version of the main result of arXiv:2206.14914. The assumption of control on the perturbation of parabolics is necessary.

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