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Compact Invariant Random Subgroups

Tal Cohen, Helge Glöckner, Gil Goffer, Waltraud Lederle

Abstract

We study ergodic invariant random subgroups that give full measure to the subset of compact subgroups. We show that in real Lie groups, compactly generated $p$-adic Lie groups, locally compact hyperbolic groups and infinitely ended groups they are always contained in a compact normal subgroup. In general $p$-adic Lie groups, we show they are contained in the locally elliptic radical. In totally disconnected locally compact groups, we show they are contained in the intersection of all Levi subgroups of inner automorphisms.

Compact Invariant Random Subgroups

Abstract

We study ergodic invariant random subgroups that give full measure to the subset of compact subgroups. We show that in real Lie groups, compactly generated -adic Lie groups, locally compact hyperbolic groups and infinitely ended groups they are always contained in a compact normal subgroup. In general -adic Lie groups, we show they are contained in the locally elliptic radical. In totally disconnected locally compact groups, we show they are contained in the intersection of all Levi subgroups of inner automorphisms.
Paper Structure (11 sections, 30 theorems, 34 equations)

This paper contains 11 sections, 30 theorems, 34 equations.

Table of Contents

  1. Introduction
  2. Acknowledgements.
  3. Preliminaries
  4. Lie groups and almost connected groups
  5. Totally disconnected groups
  6. p-adic Lie Groups
  7. Algebraic groups
  8. Groups acting on metric spaces
  9. Hyperbolic spaces
  10. Cayley--Abels graphs
  11. Reduction to compactly generated subgroups

Key Result

Theorem 1.1

Let $G$ be a real Lie group, and $\mu$ a compact ergodic IRS of $G$. Then $\overline{\langle \mu \rangle}$ is compact.

Theorems & Definitions (61)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Example 1.4
  • Remark 1.7
  • Theorem 1.8
  • Lemma 2.1
  • proof
  • Lemma 2.2
  • proof
  • ...and 51 more