The frequency space of a free group

2003/11/05 by Ilya Kapovich, Kapovich, Ilya · 1 citation
Mathematics · #20F36 #FOS: Mathematics #Geometric Topology (math.GT) #Geometric and Algebraic Topology #Group Theory (math.GR) #Homotopy and Cohomology in Algebraic Topology #Mathematical Dynamics and Fractals #math.GR #math.GT #msc:20F36

paper · pdf · doi:10.48550/arxiv.math/0311053

Revised version, to appear in IJAC (special issue dedicated to Grigorchuk's 50's birthday)

openalex publication_date 2003/11/05 · arxiv created 2004/08/27 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We analyze the structure of the frequency space Q(F) of a nonabelian free group F=F(a₁,...,a_k) consisting of all shift-invariant Borel probability measures on ∂ F and construct a natural action of Out(F) on Q(F). In particular we prove that for any outer automorphism ϕ of F the conjugacy distortion spectrum of ϕ, consisting of all numbers ||ϕ(w)||/||w||, where w is a nontrivial conjugacy class, is the intersection of \mathbb Q and a closed subinterval of \mathbb R with rational endpoints. We also provide an algorithm for detecting strict hyperbolicity of an automorphism of F.

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• Boundary actions of outer automorphism groups of Thompson-like groups

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