Maps between relatively hyperbolic spaces and between their boundaries

John M. Mackay; Alessandro Sisto

doi:10.1090/tran/9063

Maps between relatively hyperbolic spaces and between their boundaries

John M. Mackay, Alessandro Sisto

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Abstract

We study relations between maps between relatively hyperbolic groups/spaces and quasisymmetric embeddings between their boundaries. More specifically, we establish a correspondence between (not necessarily coarsely surjective) quasi-isometric embeddings between relatively hyperbolic groups/spaces that coarsely respect peripherals, and quasisymmetric embeddings between their boundaries satisfying suitable conditions. Further, we establish a similar correspondence regarding maps with at most polynomial distortion.

We use this to characterise groups which are hyperbolic relative to some collection of virtually nilpotent subgroups as exactly those groups which admit an embedding into a truncated real hyperbolic space with at most polynomial distortion, generalising a result of Bonk and Schramm for hyperbolic groups.

Original language	English
Pages (from-to)	1409-1454
Number of pages	46
Journal	Transactions of the American Mathematical Society
Volume	377
Issue number	2
Early online date	19 Oct 2023
DOIs	https://doi.org/10.1090/tran/9063
Publication status	Published - Feb 2024

Keywords

Relatively hyperbolic groups and spaces
boundary at infinity
quasisymmetric map

ASJC Scopus subject areas

Applied Mathematics
General Mathematics

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10.1090/tran/9063

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This article has been accepted for publication and undergone full peer review but has not been through the copyediting, typesetting, pagination and proofreading process which may lead to differences between this version and the Version of Record. Please cite this article as doi: 10.1090/tran/9063
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Maps between relatively hyperbolic spaces and between their boundaries

Abstract

Keywords

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