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(F_k) on the set of marked, k-generated, dense subgroups. 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 ≠ 2); • A = Aut⁰(T_q+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 fails to be ergodic for A = PSL₂(C) and, when k is even, also for A = PSL₂(R). Therefore, if k ≥ 4 is even and K is a local field (with char(K) ≠ 2), the action of Aut(F_k) on is ergodic if and only if K is non-Archimedean. Ergodicity implies that every "measurable property" either holds or fails to hold for almost every k-generated dense subgroup of A.