  The basic idea of quantum complexity geometry is to endow the space of
unitary matrices with a metric, engineered to make complex operators far from
the origin, and simple operators near. By restricting our attention to a finite
subgroup of the unitary group, we observe that this idea can be made rigorous:
the complexity geometry becomes what is known as a Cayley graph. This
connection allows us to translate results from the geometrical group theory
literature into statements about complexity. For example, the notion of
$\delta$-hyperbolicity makes precise the idea that complexity geometry is
negatively curved. We report an exact (in the large N limit) computation of the
average complexity as a function of time in a random circuit model.
