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

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### 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.

### publication date

• January 1, 2009