  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.

Group Theory;Combinatorics;Number Theory