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Counting loxodromics for hyperbolic actions

Introductory Workshop: Geometric Group Theory August 22, 2016 - August 26, 2016

August 23, 2016 (02:30 PM PDT - 03:20 PM PDT)
Speaker(s): Samuel Taylor (Yale University)
Location: MSRI: Simons Auditorium
Tags/Keywords
  • geometric group theory

  • hyperbolic groups

  • hyperbolic metric space

  • loxodromics

  • mapping class groups

  • geodesics

  • Cayley graphs

  • group actions on trees

Primary Mathematics Subject Classification
Secondary Mathematics Subject Classification No Secondary AMS MSC
Video

14593

Abstract

Consider a nonelementary action by isometries of a hyperbolic group G on a hyperbolic metric space X. Besides the action of G on its Cayley graph, some examples to bear in mind are actions of G on trees and quasi-trees, actions on nonelementary hyperbolic quotients of G, or examples arising from naturally associated spaces, like subgroups of the mapping class group acting on the curve graph.

We show that the set of elements of G which act as loxodromic isometries of X (i.e those with sink-source dynamics) is generic. That is, for any finite generating set of G, the proportion of X-loxodromics in the ball of radius n about the identity in G approaches 1 as n goes to infinity. 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 boundary of X. This is joint work with I. Gekhtman and G. Tiozzo

Supplements
26642?type=thumb Taylor Notes 250 KB application/pdf Download
Video/Audio Files

14593

H.264 Video 14593.mp4 297 MB video/mp4 rtsp://videos.msri.org/data/000/026/455/original/14593.mp4 Download
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