On compactifications and the topological dynamics of definable groups

TL;DR

This paper explores the model-theoretic framework of definable compactifications and topological dynamics of groups, establishing universal objects and properties like amenability within this definable setting.

Contribution

It introduces the notions of definable compactifications and flows, characterizes universal objects, and connects topological dynamics with model theory for definable groups.

Findings

Universal definable compactification as G*/G*00_M

Existence and uniqueness of universal minimal definable G-flows

Characterization of amenability and extreme amenability in the definable context

Abstract

We discuss definable compactifications and topological dynamics. For G a group definable in some structure M, we define notions of "definable" compactification of G and "definable" action of G on a compact space X (definable G-flow), where the latter is under a definability of types assumption on M. We describe the universal definable compactification of G as G*/G*00_M and the universal definable G-ambit as the type space S_{G}(M). We also prove existence and uniqueness of "universal minimal definable G-flows", and discuss issues of amenability and extreme amenability in this definable category, with a characterization of the latter. For the sake of completeness we also describe the universal (Bohr) compactification and universal G-ambit in model-theoretic terms, when G is a topological group (although it is essentially well-known).