Definably amenable groups in Continuous logic

TL;DR

This paper introduces and analyzes the concepts of definable amenability and extreme definable amenability for groups within continuous logic, establishing their properties, characterizations, and relations to stability, ultracompactness, and f-generic types.

Contribution

It extends classical notions of amenability to continuous structures, providing characterizations, equivalences, and new results on randomizations of definably amenable groups.

Findings

Stable and ultracompact groups are definably amenable.

Definable amenability in dependent theories is equivalent to the existence of an f-generic type.

Randomizations of first-order definably amenable groups are extremely definably amenable.

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.