Matrix multiplication via matrix groups

In 2003, Cohn and Umans proposed a group-theoretic approach to bounding the exponent of matrix multiplication. Previous work within this approach ruled out certain families of groups as a route to obtaining $ω= 2$, while other families of groups remain potentially viable. In this paper we turn our attention to matrix groups, whose usefulness within this framework was relatively unexplored. We first show that groups of Lie type cannot prove $ω=2$ within the group-theoretic approach. This is based on a representation-theoretic argument that identifies the second-smallest dimension of an irreducible representation of a group as a key parameter that determines its viability in this framework. Our proof builds on Gowers' result concerning product-free sets in quasirandom groups. We then give another barrier that rules out certain natural matrix group constructions that make use of subgroups that are far from being self-normalizing. Our barrier results leave open several natural paths to obtain $ω= 2$ via matrix groups. To explore these routes we propose working in the continuous setting of Lie groups, in which we develop an analogous theory. Obtaining the analogue of $ω=2$ in this potentially easier setting is a key challenge that represents an intermediate goal short of actually proving $ω= 2$. We give two constructions in the continuous setting, each of which evades one of our two barriers.