A Zero-one Law for Random Subgroups of Some Totally Disconnected Groups

Let A be a locally compact group topologically generated by d elements and let k > d. Consider the action, by precomposition, of Γ = Aut(Fk) on the set of marked, k-generated, dense subgroups Dk,A := {η ∈ Hom(Fk, A) | 〈φ(Fk)〉 = A}. We prove the ergodicity of this action for the following two families of simple, totally disconnected locally compact groups: • A = PSL2(K) where K is a non-Archimedean local field (of characteristic 6= 2), • A = Aut(Tq+1) the group of orientation preserving automorphisms of a q + 1 regular tree, for q ≥ 2. In contrast, a recent result of Minsky’s shows that the same action is not ergodic when A = PSL2(R) or A = PSL2(C). Therefore if K is a local field (with char(K) 6= 2) the action of Aut(Fk) on Dk,PSL2(K) is ergodic, for every k > 2, if and only if K is nonArchimedean. Ergodicity implies that every “measurable property” either holds or fails to hold for almost every k-generated dense subgroup of A.