  Consider a closed manifold $M$ with two Riemannian metrics: one hyperbolic
metric, and one other metric $g$. What hypotheses on $g$ guarantee that for a
given radius $r$, there are balls of radius $r$ in the universal cover of $(M,
g)$ with greather-than-hyperbolic volumes? We show that this conclusion holds
for all $r \geq 1$ if $(\mathrm{Vol} (M, g))^2$ is less than a small constant
times the hyperbolic volume of $M$. This strengthens a theorem of Sabourau and
is partial progress toward a conjecture of Guth.
