Counting loxodromics for hyperbolic actions

Ilya Gekhtman, Samuel J. Taylor, Giulio Tiozzo

Research output: Contribution to journalArticlepeer-review


Let (Formula presented.) be a non-elementary action by isometries of a hyperbolic group G on a (not necessarily proper) hyperbolic metric space X. We show that the set of elements of G which act as loxodromic isometries of X has density one in the word metric on G. That is, for any finite generating set of G, the proportion of elements in G of word length at most n, which are X–loxodromics, approaches 1 as n → ∞. We also establish several results about the behavior in X of the images of typical geodesic rays in G; for example, we prove that they make linear progress in X and converge to the Gromov boundary ∂X. We discuss various applications, in particular to mapping class groups, Out(FN), and right–angled Artin groups.

Original languageEnglish
Pages (from-to)379-419
Number of pages41
JournalJournal of Topology
Issue number2
StatePublished - 2018
Externally publishedYes


  • 20H15 (secondary)
  • 20P05 (primary)

All Science Journal Classification (ASJC) codes

  • Geometry and Topology


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