Generic groups acting on regular trees

Abstract

Let T T be a k k -regular tree ( k ≥ 3 k\geq 3 ) and A = A u t ( T ) A=\mathrm {Aut}(T) its automorphism group. We analyze a generic finitely generated subgroup Γ \Gamma of A A . We show that Γ \Gamma is free and establish a trichotomy on the closure Γ ¯ \overline {\Gamma } of Γ \Gamma in A . A. It turns out that Γ ¯ \overline {\Gamma } is either discrete, compact or has index at most 2 2 in A A .