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Definably amenable groups in Continuous logic

Juan Felipe Carmona, Alf Onshuus

TL;DR

This work generalizes amenability notions to groups in continuous logic, defining definable amenability (DA) via invariant means on $M$-definable predicates and extreme definable amenability (EDA) via invariant global types. It establishes fixed-point characterizations: a definable group is DA/EDA exactly when every weak definable action on a compact convex space has a fixed point, with stronger definability yielding fixed points for definable actions. The authors show DA holds for stable and pseudocompact groups, and in dependent theories, DA is equivalent to the existence of an $f$-generic type. Finally, they prove that randomizations of classical definably amenable groups are extremely definably amenable, enriching the landscape with new EDA examples and linking model-theoretic amenability to continuous dynamics.

Abstract

We introduce the notions of definable amenability and extreme definable amenability for groups in continuous structures and conduct an extensive analysis of them, drawing parallels with the classical first-order case. We characterize both notions using fixed-point properties. We show that stable and ultracompact groups are definably amenable and prove that, for groups definable in dependent theories, definable amenability is equivalent to the existence of an f-generic type. Finally, we show the randomizations of first-order definably amenable groups are extremely definably amenable.

Definably amenable groups in Continuous logic

TL;DR

This work generalizes amenability notions to groups in continuous logic, defining definable amenability (DA) via invariant means on -definable predicates and extreme definable amenability (EDA) via invariant global types. It establishes fixed-point characterizations: a definable group is DA/EDA exactly when every weak definable action on a compact convex space has a fixed point, with stronger definability yielding fixed points for definable actions. The authors show DA holds for stable and pseudocompact groups, and in dependent theories, DA is equivalent to the existence of an -generic type. Finally, they prove that randomizations of classical definably amenable groups are extremely definably amenable, enriching the landscape with new EDA examples and linking model-theoretic amenability to continuous dynamics.

Abstract

We introduce the notions of definable amenability and extreme definable amenability for groups in continuous structures and conduct an extensive analysis of them, drawing parallels with the classical first-order case. We characterize both notions using fixed-point properties. We show that stable and ultracompact groups are definably amenable and prove that, for groups definable in dependent theories, definable amenability is equivalent to the existence of an f-generic type. Finally, we show the randomizations of first-order definably amenable groups are extremely definably amenable.
Paper Structure (13 sections, 32 theorems, 64 equations)

This paper contains 13 sections, 32 theorems, 64 equations.

Table of Contents

  1. Introduction
  2. Definable Amenability and Fixed-Point Properties
  3. Preliminaries
  4. Definable amenability and extreme definable amenability
  5. Fixed-point theorems
  6. Stable Groups
  7. Generic Types, and S1 Ideals in dependent groups
  8. Stabilizers and amenability in dependent theories
  9. S1-ideals
  10. Randomizations
  11. Continuous Logic
  12. Functional Analysis
  13. C*-algebras

Key Result

Theorem 2.1

The space of types $S_A^{\Bar{x}}$ is compact with the Logic Topology.

Theorems & Definitions (85)

  • Definition 1.1
  • Definition 1.2
  • Theorem 2.1: Compactness yaacov2008model Proposition 8.6
  • Theorem 2.2: yaacov2008model Theorem 9.9
  • Theorem 2.3
  • proof
  • Definition 2.4
  • Definition 2.5
  • Remark 2.6
  • Definition 2.7
  • ...and 75 more