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 $ {D_{k,A}}: = left{ {eta in {text{Hom}}left( {{F_k},A} right)left| {overline {leftlangle {phi left( {{F_k}} right)} rightrangle } = A} right.} right} 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: = { h ? textHom( F_k,A )| [`(á f( F_k ) ñ )] = A } {D_{k,A}}: = left{ {eta in {text{Hom}}left( {{F_k},A} right)left| {overline {leftlangle {phi left( {{F_k}} right)} rightrangle } = 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.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 k geqslant 4 is even and K is a local field (with char(K) ≠ 2), the action of Aut(F _k ) on D_{k,textPStextL2(K)} {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.