  In this paper, we prove uniform lower bounds on the volume growth of balls in
the universal covers of Riemannian surfaces and graphs. More precisely, there
exists a constant $\delta>0$ such that if $(M,hyp)$ is a closed hyperbolic
surface and $h$ another metric on $M$ with $\area(M,h)\leq \delta \area(M,hyp)$
then for every radius $R\geq 1$ the universal cover of $(M,h)$ contains an
$R$-ball with area at least the area of an $R$-ball in the hyperbolic plane.
This positively answers a question of L. Guth for surfaces. We also prove an
analog theorem for graphs.
