Multi-way expanders and imprimitive group actions on graphs

For n at least 2, the concept of n-way expanders was defined by various researchers. Bigger n gives a weaker notion in general, and 2-way expanders coincide with expanders in usual sense. Koji Fujiwara asked whether these concepts are equivalent to that of ordinary expanders for all n for a sequence of Cayley graphs. In this paper, we answer his question in the affirmative. Furthermore, we obtain universal inequalities on multi-way isoperimetric constants on any finite connected vertex-transitive graph, and show that gaps between these constants imply the imprimitivity of the group action on the graph.Comment: Accepted in Int. Math. Res. Notices. 18 pages, rearrange all of the arguments in the proof of Main Theorem (Theorem A) in a much accessible way (v4); 14 pages, appendix splitted into a forthcoming preprint (v3); 17 pages, appendix on noncommutative L_p spaces added (v2); 12 pages, no figure