Topological dynamics and NIP fields

TL;DR

This paper investigates the topological dynamics of algebraic group actions over NIP fields, revealing conditions under which the Ellis group is non-trivial and providing counterexamples to the Ellis group conjecture.

Contribution

It demonstrates that the Ellis group of $ ext{SL}_2(K)$ can be non-trivial over NIP fields when the multiplicative group is not type-definably connected, offering new counterexamples.

Findings

Ellis group of $ ext{SL}_2(K)$ is non-trivial under certain conditions.

Counterexamples to the Ellis group conjecture are identified in dp-minimal fields.

Structure theory of algebraic groups with definable f-generics over NIP fields is developed.

Abstract

We study definable topological dynamics of some algebraic group actions over an arbitrary NIP field $K$. We show that the Ellis group of the universal definable flow of $\mathrm{SL}_2(K)$ is non-trivial if the multiplicative group of $K$ is not type-definably connected, providing a way to find multiple counterexamples to the Ellis group conjecture, particularly in the case of dp-minimal fields. We also study some structure theory of algebraic groups over $K$ with definable f-generics.