Abstract

I will present two recent joint works with Michael Larsen and Pham Tiep. The first shows that short products of words (in disjoint variables) which are not an identity of any (non-abelian) finite simple group are surjective on ALL finite simple groups. This is carried out in a more general framework of characteristic collections of groups, and may be regarded as a generalization of work by Guralnick, Liebeck, O’Brien, Tiep and me (2018), which in turn extends the Odd Order Theorem. The second part of the talk will focus on representations of finite simple groups, exploring related growth phenomena. It is motivated by the theory of approximate subgroups, which establishes 3-step growth phenomena for simple groups of Lie type of bounded rank. We replace products of subsets by tensor products of representations, and establish stronger 2-step growth results also in unbounded rank.