Small separations in vertex transitive graphs

Let $k$ be an integer. We prove a rough structure theorem for separations of order at most $k$ in finite and infinite vertex transitive graphs. Let $G = (V,E)$ be a vertex transitive graph, let $A \subseteq V$ be a finite vertex-set with $|A| \le |V|/2$ and |\{v \in V \setminus A : {u \sim v$for some$u \in A} \}|\le k. We show that whenever the diameter of $G$ is at least $31(k+1)^2$, either $|A| \le 2k^3+k^2$, or $G$ has a ring-like structure (with bounded parameters), and $A$ is efficiently contained in an interval. This theorem may be viewed as a rough characterization, generalizing an earlier result of Tindell, and has applications to the study of product sets and expansion in groups.