  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 $\omega = 2$, 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 $\omega=2$ 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 $\omega = 2$
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 $\omega=2$ in this potentially easier setting is a key challenge
that represents an intermediate goal short of actually proving $\omega = 2$. We
give two constructions in the continuous setting, each of which evades one of
our two barriers.
