Definable Topological Dynamics of $SL_2(\mathbb{C}((t))$

TL;DR

This paper explores the topological dynamics of the special linear group over the field of formal Laurent series, revealing new structural insights and answering a longstanding question about metastability's role in model theory.

Contribution

It provides the first detailed analysis of definable topological dynamics for $SL_2$ over $ ext{C}((t))$, including explicit descriptions of minimal flows and the Ellis group.

Findings

$SL_2( ext{C}((t)))$ is not definably amenable.

Explicit minimal flow and Ellis group are described.

The Ellis group differs from $G/G^{00}$, challenging previous conjectures.

Abstract

We initiate a study of definable topological dynamics for groups definable in metastable theories. Specifically, we consider the special linear group $G = SL_2$ with entries from $M = \mathbb{C}((t))$; the field of formal Laurent series with complex coefficients. We prove such a group is not definably amenable, find a suitable group decomposition, and describe the minimal flows of the additive and multiplicative groups of $\mathbb{C}((t))$. The main result is an explicit description of the minimal flow and Ellis Group of $(G(M),S_G(M))$ and we observe that this is not isomorphic to $G/G^{00}$, answering a question as to whether metastability is a suitable weakening of a conjecture of Newelski.