Subsets of Cayley Graphs That Induce Many Edges

Let G be a regular graph of degree d and let A subset of V(G). Say that A is eta-closed if the average degree of the subgraph induced by A is at least eta d. This says that if we choose a random vertex x is an element of A and a random neighbour y of x, then the probability that y is an element of A is at least eta. This paper was motivated by an attempt to obtain a qualitative description of closed subsets of the Cayley graph Gamma whose vertex set is F-2(n1 )circle times ... circle times F-2(nd) with two vertices joined by an edge if their difference is of the form u(1) circle times ... circle times u(d) For the matrix case (that is, when d = 2), such a description was obtained by Khot, Minzer and Safra, a breakthrough that completed the proof of the 2-to-2 conjecture. In this paper, we formulate a conjecture for higher dimensions, and prove it in an important special case. Also, we identify a statement about eta-closed sets in Cayley graphs on arbitrary finite Abelian groups that implies the conjecture and can be considered as a "highly asymmetric Balog-Szemeredi-Gowers theorem" when it holds. We conclude the paper by showing that this statement is not true for an arbitrary Cayley graph. It remains to decide whether the statement can be proved for the Cayley graph Gamma.