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We study the distribution of products of conjugacy classes in finite simplegroups, obtaining various effective uniformity results, which give rise to anapproximation to a conjecture of Thompson. Our results, combined with work ofGowers and Viola, also lead to the solution of recent conjectures they posed oninterleaved products and related complexity lower bounds, extending their workon the groups SL$(2,q)$ to all (nonabelian) finite simple groups. In particularit follows that, if $G$ is a finite simple group, and $A, B \subseteq G^t$ for$t \ge 2$ are subsets of fixed positive densities, then, as $a = (a_1, \ldots ,a_t) \in A$ and $b = (b_1, \ldots , b_t) \in B$ are chosen uniformly, theinterleaved product $a \bullet b := a_1b_1 \cdots a_tb_t$ is almost uniform on$G$ (with quantitative estimates) with respect to the $\ell_{\infty}$-norm. Italso follows that the communication complexity of an old decision problemrelated to interleaved products of $a, b \in G^t$ is at least $\Omega(t \log|G|)$ when $G$ is a finite simple group of Lie type of bounded rank, and atleast $\Omega(t \log \log |G|)$ when $G$ is any finite simple group. Both thesebounds are best possible.

Simple groups, interleaved products, complexity and conjectures of Gowers and Viola