Groups with no coarse embeddings into hyperbolic groups

David Hume, Alessandro Sisto

Research output: Contribution to journalArticlepeer-review

7 Citations (Scopus)


We introduce an obstruction to the existence of a coarse embedding of a given group or space into a hyperbolic group, or more generally into a hyperbolic graph of bounded degree. The condition we consider is “admitting exponentially many fat bigons”, and it is preserved by a coarse embedding between graphs with bounded degree. Groups with exponential growth and linear divergence (such as direct products of two groups one of which has exponential growth, solvable groups that are not virtually nilpotent, and uniform higher-rank lattices) have this property and hyperbolic graphs do not, so the former cannot be coarsely embedded into the latter. Other examples include certain lacunary hyperbolic and certain small cancellation groups.

Original languageEnglish
Pages (from-to)1657-1670
Number of pages14
JournalNew York Journal of Mathematics
Publication statusPublished - 14 Nov 2017


  • Coarse embeddings
  • Divergence
  • Hyperbolic group
  • Subgroups

ASJC Scopus subject areas

  • Mathematics(all)


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