Local connectedness of boundaries for relatively hyperbolic groups

Pub Date : 2024-06-10 DOI:10.1112/topo.12347

Ashani Dasgupta, G. Christopher Hruska

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Abstract

Let $(Γ, P)$ be a relatively hyperbolic group pair that is relatively one ended. Then, the Bowditch boundary of $(Γ, P)$ is locally connected. Bowditch previously established this conclusion under the additional assumption that all peripheral subgroups are finitely presented, either one or two ended, and contain no infinite torsion subgroups. We remove these restrictions; we make no restriction on the cardinality of $Γ$ and no restriction on the peripheral subgroups $P \in P$ .

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相对双曲群边界的局部连通性

让 ( Γ , P ) $(\Gamma,\mathbb {P})$ 是相对一端的相对双曲群对。那么，( Γ , P ) $(\Gamma,\mathbb {P})$ 的鲍迪奇边界是局部连通的。鲍迪奇之前是在所有外围子群都是有限呈现、一端或两端、不包含无限扭转子群的额外假设下得出这个结论的。我们取消了这些限制；我们不限制Γ $\Gamma$ 的心性，也不限制外围子群 P ∈ P $P \ in \mathbb {P}$ 。

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