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<p>We prove a topological stability result for the actions of relatively 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 Mann et al. (Stability of hyperbolic groups acting on their boundaries. Groups Geom. Dyn., to appear). The assumption of control on the perturbation of parabolics is necessary.</p>

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Topological stability of relatively hyperbolic groups acting on their boundaries

  • 期刊论文

  • 发表日期：

    2025年10月22日

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  • Kathryn Mann,
  • Jason Fox Manning,
  • Theodore Weisman

摘要

We prove a topological stability result for the actions of relatively 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 Mann et al. (Stability of hyperbolic groups acting on their boundaries. Groups Geom. Dyn., to appear). The assumption of control on the perturbation of parabolics is necessary.