Abstract
Let G ⤻ X 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 ( F N ) , and right–angled Artin groups.
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a callDisclaimer: All third-party content on this website/platform is and will remain the property of their respective owners and is provided on "as is" basis without any warranties, express or implied. Use of third-party content does not indicate any affiliation, sponsorship with or endorsement by them. Any references to third-party content is to identify the corresponding services and shall be considered fair use under The CopyrightLaw.
