On the genericity of loxodromic actions

2014/06/26 by Bert Wiest, Wiest, Bert · 3 citations
Mathematics · #20F10 #20F36 #20F65 #57M07 #FOS: Mathematics #Group Theory (math.GR) #math.GR #msc:20F10 #msc:20F36 #msc:20F65 #msc:57M07

paper · pdf · doi:10.48550/arxiv.1406.7041

23 pages, 1 figure. Exposition improved, small errors corrected, added example of hyperbolic groups acting on their Cayley graphs

arxiv created 2015/06/08 · arxiv updated 2015/06/09

Abstract

One way of picking a "generic" element of a finitely generated group is to pick a random element with uniform probability in a large ball centered on 1 in the Cayley graph. If the group acts on a δ-hyperbolic space, with at least one element acting loxodromically, then it is plausible that generic elements should act loxodromically with high probability. In this paper we prove that the probability of acting loxodromically is bounded away from 0, provided the group satisfies a very weak automaticity condition, and provided a certain compatibility condition linking the automatic with the δ-hyperbolic structure is satisfied. We present several applications of this result, including the genericity of pseudo-Anosov braids.

Cited by

• Growth gaps and exponential genericity in acylindrically hyperbolic groups
• A general construction of simultaneously hyperbolic elements
• Acylindrically hyperbolic groups and counting problems

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