A Zero-One Law for Random Subgroups of some Totally Disconnected Groups

TL;DR

This paper proves the ergodicity of the action of automorphism groups on dense subgroups in certain totally disconnected groups, highlighting a stark contrast with the behavior in real or complex cases.

Contribution

It establishes a zero-one law for random subgroups in specific totally disconnected groups, showing ergodicity in non-Archimedean cases and contrasting with Archimedean groups.

Findings

Ergodicity holds for PSL(2,K) with non-Archimedean K.

Ergodicity holds for automorphisms of regular trees.

Non-ergodic behavior in PSL(2,R) and PSL(2,C).

Abstract

Let A be a locally compact group topologically generated by d elements and let k>d. Consider the action, by pre-composition, of Aut(F_k) on the set of marked, k-generated, dense subgroups D_{k,A} := {h:F_k --> A | h(F_k) is dense in A}. We prove the ergodicity of this action for two families of simple, totally disconnected locally compact groups. (i) A = PSL(2,K) where K is a non-Archimedean local field (of characteristic not equal to 2), (ii) A = Aut^{0}(T) - the group of orientation preserving automorphisms of a (q+1)-regular tree, for q > 1. In contrast, a recent result of Minsky's shows that the same action is not ergodic when A = PSL(2,R) or A = PSL(2,C). Therefore if K is a local field (with characteristic different than 2) the action of Aut(F_k) on D_{k,PSL(2,K)} is ergodic, for every k>2, 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.