Cayley graphs of quasirandom groups

A finite group $G$ is $\varepsilon$-quasirandom if all its nontrivial irreducible complex representations have degree at least $|G|^\varepsilon$. Building on recent work of Golsefidy-Srinivas, we prove that expansion in a quasirandom group is controlled by expansion in its simple quotients. As a consequence, we remove the product theorem from the hypotheses of the Bourgain-Gamburd expansion machine. Moreover, we combine this result with crown theory to deduce that $1 + \lfloor \varepsilon^{-1} \rfloor$ random elements give an expander Cayley graph with high probability. Finally, generalizing results of Breuillard-Green-Tao and Pyber-Szab\'o, we prove that the diameter of any connected Cayley graph of a quasirandom group is polylogarithmic.