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 | A | ⩽ | V | 2 set k = | { v ∈ V \ A : u ∼ v for some u ∈ A } | . We show that whenever the diameter of G is at least 31 ( k + 1 ) 2 , either | A | ⩽ 2 k 3 , 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.