  We establish a lower bound for the cop number of graphs of high girth in
terms of the minimum degree, and more generally, in terms of a certain growth
condition. We show, in particular, that the cop number of any graph with girth
$g$ and minimum degree $\delta$ is at least $\tfrac{1}{g}(\delta - 1)^{\lfloor
\frac{g-1}{4}\rfloor}$. We establish similar results for directed graphs. While
exposing several reasons for conjecturing that the exponent $\tfrac{1}{4}g$ in
this lower bound cannot be improved to $(\tfrac{1}{4}+\varepsilon)g$, we are
also able to prove that it cannot be increased beyond $\frac{3}{8}g$. This is
established by considering a certain family of Ramanujan graphs. In our proof
of this bound, we also show that the "weak" Meyniel's conjecture holds for
expander graph families of bounded degree.
