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On the genericity of loxodromic actions

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Abstract

Suppose that a finitely generated group G acts by isometries on a δ-hyperbolic space, with at least one element acting loxodromically. Suppose that the elements of G have a normal form such that the language of normal forms can be recognized by a finite state automaton. Suppose also that a certain compatibility condition linking the automatic and the δ-hyperbolic structures is satisfied. Then we prove that in the “ball” consisting of elements of G whose normal form is of length at most l, the proportion of elements which act loxodromically is bounded away from zero, as l tends to infinity. We present several applications of this result, including the genericity of pseudo-Anosov braids.

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  1. UFR Mathématiques, Université de Rennes 1, 35042, Rennes Cedex, France

    Bert Wiest

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  1. Bert Wiest
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Wiest, B. On the genericity of loxodromic actions. Isr. J. Math. 220, 559–582 (2017). https://doi.org/10.1007/s11856-017-1540-9

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  • DOI: https://doi.org/10.1007/s11856-017-1540-9

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