Planar boundaries and parabolic subgroups

2020/08/17 by G. Christopher Hruska, Hruska, G. Christopher, Genevieve Walsh +1 · 2 citations
Mathematics · #20F67 (Primary) 20E08 (Secondary) #FOS: Mathematics #Geometric Analysis and Curvature Flows #Geometric Topology (math.GT) #Geometric and Algebraic Topology #Group Theory (math.GR) #Mathematical Dynamics and Fractals

paper · pdf · doi:10.48550/arxiv.2008.07639

openalex publication_date 2020/08/17 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We study the Bowditch boundaries of relatively hyperbolic group pairs, focusing on the case where there are no cut points. We show that if (G,P) is a rigid relatively hyperbolic group pair whose boundary embeds in S², then the action on the boundary extends to a convergence group action on S². More generally, if the boundary is connected and planar with no cut points, we show that every element of P is virtually a surface group. This conclusion is consistent with the conjecture that such a group G is virtually Kleinian. We give numerous examples to show the necessity of our assumptions.

Cited by

• Parabolic cut pairs in boundaries of relatively hyperbolic groups
• Cyclic orders and actions of Leary--Minasyan groups on coarse PD(n) spaces

Related

• Relatively hyperbolic groups with planar boundaries
• Hyperbolic groups with planar boundaries
• Local connectedness of boundaries for relatively hyperbolic groups
• Some groups with planar boundaries
• Parabolic cut pairs in boundaries of relatively hyperbolic groups