External definability and groups in NIP theories

TL;DR

This paper investigates how key properties of definable groups in NIP theories are maintained under Shelah expansion, advancing the understanding of definable topological dynamics and confirming the Ellis group conjecture in new contexts.

Contribution

It demonstrates the preservation of properties like definable amenability and G^{00} in Shelah expansions and proves the Ellis group conjecture for certain groups in NIP theories.

Findings

Properties like definable amenability are preserved under Shelah expansion.

The Ellis group conjecture is confirmed for definably amenable groups in o-minimal structures.

The study advances the understanding of definable topological dynamics in NIP theories.

Abstract

We prove that many properties and invariants of definable groups in NIP theories, such as definable amenability, G/G^{00}, etc., are preserved when passing to the theory of the Shelah expansion by externally definable sets, M^{ext}, of a model M. In the light of these results we continue the study of the "definable topological dynamics" of groups in NIP theories. In particular we prove the Ellis group conjecture relating the Ellis group to G/G^{00} in some new cases, including definably amenable groups in o-minimal structures.