## 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. 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 ≠ 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) ≠ 2), the action of Aut(F_{k}) 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 language | English |
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Pages (from-to) | 787-800 |

Number of pages | 14 |

Journal | Transformation Groups |

Volume | 14 |

Issue number | 4 |

DOIs | |

State | Published - 1 Dec 2009 |

## ASJC Scopus subject areas

- Algebra and Number Theory
- Geometry and Topology