The Ellis group conjecture and variants of definable amenability

TL;DR

This paper investigates the structure of the Ellis group in definable topological dynamics for NIP groups, proving boundedness and model-independence under certain conditions, with applications to o-minimal structures.

Contribution

It provides a description of the Ellis group for NIP groups with specific decompositions, confirming a conjecture and extending results to o-minimal structures.

Findings

Ellis group is of bounded size for these groups.

Under additional assumptions, the Ellis group is model-independent.

Results generalize previous work in o-minimal structures.

Abstract

We consider definable topological dynamics for $NIP$ groups admitting certain decompositions in terms of specific classes of definably amenable groups. For such a group, we find a description of the Ellis group of its universal definable flow. This description shows that the Ellis group is of bounded size. Under additional assumptions, it is shown to be independent of the model, proving a conjecture proposed by Newelski. Finally we apply the results to new classes of groups definable in o-minimal structures, generalizing all of the previous results for this setting.