Small separations in vertex-transitive graphs

A rough structure theorem for small separations in symmetric graphs is developed. Let G=(V,E) be a vertex transitive graph, let A⊆V be finite with set k=|{v∈V\A:u∼v for some u∈A}|. We show that whenever the diameter of G is at least 312(k+1), either |A|⩽2k3, or G has a (bounded) ring-like structure and A is efficiently contained in an interval. This theorem has applications to the study of product sets and expansion in groups.