The frequency space of a free group
Ilya Kapovich
Abstract
We analyze the structure of the frequency space Q(F) of a nonabelian free group F=F(a1,...,ak) 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 Q and a closed subinterval of R with rational endpoints. We also provide an algorithm for detecting strict hyperbolicity of an automorphism of F.
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