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On dissociated infinite permutation groups

Rémi Barritault, Colin Jahel, Matthieu Joseph

TL;DR

This paper develops and exploits the notion of dissociation for closed permutation groups to unify independence and orthogonality phenomena across unitary representations and measure-preserving actions. It provides a complete classification of irreducible unitary representations for dissociated groups, establishes Property (T) and Howe-Moore properties in broad settings, and delivers strong ergodic-theoretic rigidity results such as stabilizer rigidity and a de Finetti-type description of invariant processes. Two general strategies for constructing dissociated groups are introduced: approximating sequences and strong cofinite amalgamation over countable subsets (σ-SAP), with ω-categoricity playing a key role in enabling the latter. The framework yields numerous new examples, including isometry groups of metrically homogeneous graphs and automorphism groups of diversities, many of which lie outside classical Roelcke precompact regimes, thereby broadening the landscape of dissociated groups and their analytic properties. Overall, the work provides a cohesive methodology and a suite of concrete constructions that link abstract representation-theoretic rigidity to concrete automorphism groups of rich combinatorial and metric structures.

Abstract

The goal of this paper is threefold. First, we describe the notion of dissociation for closed subgroups of the group of permutations on a countably infinite set and explain its numerous consequences on unitary representations (classification of unitary representations, Property (T), Howe-Moore property, etc.) and on ergodic actions (non-existence of type III non-singular actions, Stabilizer rigidity, etc.). Some of the results presented here are new, others were proved in different contexts. Second, we introduce a new method to prove dissociation. It is based on a reinforcement of the classical notion of strong amalgamation, where we allow to amalgamate over countable sets. Third, we apply this technique of amalgamation to provide new examples of dissociated closed permutation groups, including isometry groups of some metrically homogeneous graphs, automorphism groups of diversities, and more.

On dissociated infinite permutation groups

TL;DR

This paper develops and exploits the notion of dissociation for closed permutation groups to unify independence and orthogonality phenomena across unitary representations and measure-preserving actions. It provides a complete classification of irreducible unitary representations for dissociated groups, establishes Property (T) and Howe-Moore properties in broad settings, and delivers strong ergodic-theoretic rigidity results such as stabilizer rigidity and a de Finetti-type description of invariant processes. Two general strategies for constructing dissociated groups are introduced: approximating sequences and strong cofinite amalgamation over countable subsets (σ-SAP), with ω-categoricity playing a key role in enabling the latter. The framework yields numerous new examples, including isometry groups of metrically homogeneous graphs and automorphism groups of diversities, many of which lie outside classical Roelcke precompact regimes, thereby broadening the landscape of dissociated groups and their analytic properties. Overall, the work provides a cohesive methodology and a suite of concrete constructions that link abstract representation-theoretic rigidity to concrete automorphism groups of rich combinatorial and metric structures.

Abstract

The goal of this paper is threefold. First, we describe the notion of dissociation for closed subgroups of the group of permutations on a countably infinite set and explain its numerous consequences on unitary representations (classification of unitary representations, Property (T), Howe-Moore property, etc.) and on ergodic actions (non-existence of type III non-singular actions, Stabilizer rigidity, etc.). Some of the results presented here are new, others were proved in different contexts. Second, we introduce a new method to prove dissociation. It is based on a reinforcement of the classical notion of strong amalgamation, where we allow to amalgamate over countable sets. Third, we apply this technique of amalgamation to provide new examples of dissociated closed permutation groups, including isometry groups of some metrically homogeneous graphs, automorphism groups of diversities, and more.

Paper Structure

This paper contains 30 sections, 59 theorems, 40 equations.

Table of Contents

  1. Dissociation for unitary representations.
  2. Dissociation for Boolean p.m.p. actions
  3. Dissociated groups.
  4. Compendium on dissociated groups.
  5. How to obtain dissociation?
  6. New examples of dissociated groups.
  7. Acknowledgments.
  8. About the definition of dissociation
  9. Dissociation: unitary representations and Boolean p.m.p. actions
  10. Lattice of open subgroups and topological simplicity
  11. Unitary representations of dissociated groups
  12. Classification of unitary representations
  13. Property (T)
  14. Howe-Moore property
  15. Ergodic theory of dissociated groups
  16. ...and 15 more sections

Key Result

Theorem 1

Let $G$ be a closed permutation group on $\Omega$. If $G$ is dissociated, then the following statements hold.

Theorems & Definitions (107)

  • Theorem
  • Lemma 1.1
  • proof
  • Lemma 1.2
  • Lemma 1.3
  • proof
  • Theorem 1.4
  • Theorem 1.5
  • proof
  • Lemma 1.6
  • ...and 97 more