A zero-one law for random subgroups of some totally disconnected groups

Abstract

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 \( {D_{k,A}}: = \left\{ {\eta \in {\text{Hom}}\left( {{F_k},A} \right)\left| {\overline {\left\langle {\phi \left( {{F_k}} \right)} \right\rangle } = A} \right.} \right\} \) . 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 \(q \geqslant 2.\) 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 \(k \geqslant 4 \) is even and K is a local field (with char(K) ≠ 2), the action of Aut(F k ) on \( {D_{k,{\text{PS}}{{\text{L}}_2}(K)}} \) 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.