# Stability in a group

## TL;DR

This paper extends local stable group theory from topological dynamics to model-theoretic stability, analyzing subsets of groups that avoid infinite half-graphs using advanced stability characterizations.

## Contribution

It introduces a new framework connecting topological dynamics and model-theoretic stability for groups, generalizing existing results to the setting of stability in models.

## Key findings

- Characterization of stable sets via Grothendieck's double-limit theorem
- Extension of stable group theory to model-theoretic context
- Analysis of subsets avoiding infinite half-graphs in groups

## Abstract

We develop local stable group theory directly from topological dynamics, and extend the main results in this subject to the setting of stability "in a model". Specifically, given a group $G$, we analyze the structure of sets $A\subseteq G$ such that the bipartite relation $xy\in A$ omits infinite half-graphs. Our proofs rely on the characterization of stability via Grothendieck's "double-limit" theorem (as shown by Ben Yaacov), and the work of Ellis and Nerurkar on weakly almost periodic $G$-flows.