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Relative hyperbolicity, thickness, and the hierarchically hyperbolic boundary

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arxiv 2305.16906 v1 pith:SWZUTG45 submitted 2023-05-26 math.GR math.GT

Carolyn Abbott , Jason Behrstock , Jacob Russell This is my paper · ORCID
classification math.GRmath.GT
keywords boundaryhyperbolichhgsstructurehierarchicallyhyperbolicityrelativerelatively
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We study the boundaries of relatively hyperbolic HHGs. Using the simplicial structure on the hierarchically hyperbolic boundary, we characterize both relative hyperbolicity and being thick of order 1 among HHGs. In the case of relatively hyperbolic HHGs, we show that the Bowditch boundary of the group is the quotient of the HHS boundary obtained by collapsing the limit sets of the peripheral subgroups to a point. In establishing this, we give a construction that allows one to modify an HHG structure by including a collection of hyperbolically embedded subgroups into the HHG structure.

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Cited by 1 Pith paper

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  1. On the connectedness of the boundary of hierarchically hyperbolic spaces

    math.GR 2025-08 conditional novelty 7.0 of 10

    For hierarchically hyperbolic groups, the boundary is connected if and only if the group is one-ended, and free product boundaries are characterized by factor boundaries.

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