Sharp hypercontractivity for symmetric groups and its applications

A recently fertile strand of research in Group Theory is developing non-abelian analogues of classical combinatorial results for arithmetic Cayley graphs, describing properties such as growth, expansion, mixing, diameter, etc. We consider these problems for the symmetric and alternating groups. The case of normal Cayley graphs (those generated by unions of conjugacy classes) has seen significant progress via character theory (whereby Larsen and Shalev resolved several open problems), but the general case still remains poorly understood. In this paper we generalise the background assumption from being normal to being global (a pseudorandomness condition), replacing character bounds by spectral estimates for convolution operators of global functions, thus obtaining qualitative generalisations of several results on normal Cayley graphs. Furthermore, our theory in the pseudorandom setting can be applied (via density increment arguments) to several results for general sets that are not too sparse, including analogues of Polynomial Freiman-Ruzsa, Bogolyubov's lemma, Roth's theorem, the Waring problem and essentially sharp estimates for the diameter problem of Cayley graphs whose density is at least exponential in -n. Our main tool is a sharp new hypercontractive inequality for global functions on the symmetric group.