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

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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(Fk) on the set of marked, k-generated, dense subgroups. 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(Tq+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 = PSL2(C) and, when k is even, also for A = PSL2(R). Therefore, if k ≥ 4 is even and K is a local field (with char(K) ≠ 2), the action of Aut(Fk) on 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.

Original languageEnglish
Pages (from-to)787-800
Number of pages14
JournalTransformation Groups
Volume14
Issue number4
DOIs
StatePublished - 1 Dec 2009

ASJC Scopus subject areas

  • Algebra and Number Theory
  • Geometry and Topology

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