2016/01/05 by Aner Shalev, Shalev, Aner
Computer Science · Mathematics · #03D15 #20D06 #20P05 #FOS: Mathematics #Finite Group Theory Research #Group Theory (math.GR) #Limits and Structures in Graph Theory #semigroups and automata theory
paper · pdf · doi:10.48550/arxiv.1601.00795
openalex publication_date 2016/01/05 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We study the distribution of products of conjugacy classes in finite simple groups, obtaining various effective uniformity results, which give rise to an approximation to a conjecture of Thompson. Our results, combined with work of Gowers and Viola, also lead to the solution of recent conjectures they posed on interleaved products and related complexity lower bounds, extending their work on the groups SL(2,q) to all (nonabelian) finite simple groups. In particular it follows that, if G is a finite simple group, and A, B ⊆ Gt for t ≥ 2 are subsets of fixed positive densities, then, as a = (a1, … , at) ∈ A and b = (b1, … , bt) ∈ B are chosen uniformly, the interleaved product a \bullet b := a1b1 ⋯ atbt is almost uniform on G (with quantitative estimates) with respect to the ℓ∞-norm. It also follows that the communication complexity of an old decision problem related to interleaved products of a, b ∈ Gt is at least Ω(t log |G|) when G is a finite simple group of Lie type of bounded rank, and at least Ω(t log log |G|) when G is any finite simple group. Both these bounds are best possible.