Stability in a group

Abstract

We develop local stable group theory directly from topological dynamics, and extend the main tools 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 model-theoretic 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.