A ZERO-ONE LAW FOR RANDOM SUBGROUPS OF SOME TOTALLY DISCONNECTED GROUPS

作者：Glasner Yair^*

来源：Transformation Groups, 2009, 14(4): 787-800.

DOI：10.1007/s00031-009-9070-0

Let A be a locally compact group topologically generated by d elements and let k > d. Consider the action, by precomposition, of Gamma = Aut(F(k)) on the set of marked, k-generated, dense subgroups D(k,A) := {eta epsilon Hom(F(k,A))| <()over bar> = A}. We prove the ergodicity of this action for the following two families of simple, totally disconnected, locally compact groups:
A = PSL(2)(K) where K is a non-Archimedean local field (of characteristic not equal 2);
A = Aut(0)(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 = PSL(2)(C) and, when k is even, also for A = PSL(2)(R). Therefore, if k >= 4 is even and K is a local field (with char(K) not equal 2), the action of Aut(F(k)) on D(k,PSL2(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.

• 出版日期2009-12