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Piecewise visual, linearly connected metrics on boundaries of relatively hyperbolic groups

Matthew Haulmark

https://orcid.org/0000-0002-1603-0453

Department of Mathematics, State University of New York-Binghamton, Binghamton, NY, USA

E-mail Address: haulmark@binghamton.edu

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and

Michael Mihalik

Department of Mathematics, Vanderbilt University, Nashville, TN, USA

E-mail Address: michael.l.mihalik@vanderbilt.edu

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https://doi.org/10.1142/S1793525321500217Cited by:1 (Source: Crossref)

Abstract

Suppose a finitely generated group $G$ $G$ is hyperbolic relative to $𝒫$ a set of proper finitely generated subgroups of $G$ . Established results in the literature imply that a “visual” metric on $\partial (G, 𝒫)$ is “linearly connected” if and only if the boundary $\partial (G, 𝒫)$ has no cut point. Our goal is to produce linearly connected metrics on $\partial (G, 𝒫)$ that are “piecewise” visual when $\partial (G, 𝒫)$ contains cut points. Our main theorem is connected to graph of groups decompositions of relatively hyperbolic groups $(G, 𝒫)$ by work of B. Bowditch. We describe piecewise visual linearly connected metrics on connected boundaries of relatively hyperbolic groups. Our metric on $\partial (G, 𝒫)$ agrees with the visual metric on limit sets of vertex groups and is in this sense piecewise visual.

Keywords:

AMSC: 20F65, 20F67, 20F69

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