Most actions on regular trees are almost free Academic Article uri icon

abstract

  • Let T be a k-regular tree (k ≥ 3) and let Aut(T) be its automor- phism group. The topology of pointwise convergence turns Aut(T) into a locally compact, totally disconnected, unimodular topological group. Let µ be a Haar measure on Aut(T) normalized so that vertex stabi- lizers have measure 1. Since Aut(T) is not compact, µ is an infinite measure … Definition. A subgroup Γ ≤ Aut(T) acts almost freely, if every γ ∈ Γ (γ = 1) has finitely many fixed points on T … Free actions on T are completely understood (see Serre's book [Ser80]). As the following theorem and its corollary show, almost free actions have a much richer structure … Theorem. (Main theorem) Let Γ < Aut(T) be a countable subgroup acting almost freely. Then for µ-almost all elements γ ∈ Aut(T), the group 〈Γ,γ〉 acts almost freely and is isomorphic to the free product Γ ∗ Z … Corollary. Let a1,...,an be independent Haar-random elements of Aut(T) and let Γ …

publication date

  • January 1, 2008