Maps between relatively hyperbolic spaces and between their boundaries

John M. Mackay, Alessandro Sisto

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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 languageEnglish
Pages (from-to)1409-1454
Number of pages46
JournalTransactions of the American Mathematical Society
Issue number2
Early online date19 Oct 2023
Publication statusPublished - Feb 2024


  • Relatively hyperbolic groups and spaces
  • boundary at infinity
  • quasisymmetric map

ASJC Scopus subject areas

  • Applied Mathematics
  • Mathematics(all)


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