Counting loxodromics for hyperbolic actions
Introductory Workshop: Geometric Group Theory August 22, 2016  August 26, 2016
Location: SLMath: Eisenbud Auditorium
geometric group theory
hyperbolic groups
hyperbolic metric space
loxodromics
mapping class groups
geodesics
Cayley graphs
group actions on trees
20Jxx  Connections of group theory with homological algebra and category theory
00A35  Methodology of mathematics {For mathematics education, see 97XX}
14593
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 quasitrees, 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 sinksource dynamics) is generic. That is, for any finite generating set of G, the proportion of Xloxodromics 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
Taylor Notes

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