Quasi-hyperbolic planes in relatively hyperbolic groups

John M. Mackay, Alessandro Sisto

Research output: Contribution to journalArticlepeer-review

5 Citations (Scopus)
30 Downloads (Pure)


We show that any group that is hyperbolic relative to virtually nilpotent subgroups, and does not admit peripheral splittings, contains a quasi-isometrically embedded copy of the hyperbolic plane. In natural situations, the specific embeddings we find remain quasi-isometric embeddings when composed with the inclusion map from the Cayley graph to the coned-off graph, as well as when composed with the quotient map to "almost every" peripheral (Dehn) filling. We apply our theorem to study the same question for fundamental groups of 3-manifolds. The key idea is to study quantitative geometric properties of the boundaries of relatively hyperbolic groups, such as linear connectedness. In particular, we prove a new existence result for quasi-arcs that avoid obstacles.

Original languageEnglish
Pages (from-to)139-174
Number of pages36
JournalAnnales Academiæ Scientiarum Fennicæ. Mathematica
Issue number1
Publication statusPublished - 1 Jan 2020


  • Hyperbolic plane
  • Quasi-isometric embedding
  • Quasiarcs
  • Relatively hyperbolic group

ASJC Scopus subject areas

  • General Mathematics


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