  We introduce two variations of the cops and robber game on graphs. These
games yield two invariants in $\mathbb{Z}_+\cup\{\infty\}$ for any connected
graph $\Gamma$, the {weak cop number $\mathsf{wcop}(\Gamma)$} and the {strong
cop number $\mathsf{scop}(\Gamma)$}. These invariants satisfy that
$\mathsf{scop}(\Gamma)\leq\mathsf{wcop}(\Gamma)$. Any graph that is finite or a
tree has strong cop number one. These new invariants are preserved under small
local perturbations of the graph, specifically, both the weak and strong cop
numbers are quasi-isometric invariants of connected graphs. More generally, we
prove that if $\Delta$ is a quasi-retract of $\Gamma$ then
$\mathsf{wcop}(\Delta)\leq\mathsf{wcop}(\Gamma)$ and
$\mathsf{scop}(\Delta)\leq\mathsf{scop}(\Gamma)$. We exhibit families of
examples of graphs with arbitrary weak cop number (resp. strong cop number). We
prove that hyperbolic graphs have strong cop number one. We also prove that
one-ended non-amenable locally-finite vertex-transitive graphs have infinite
weak cop number. We raise the question of whether there exists a connected
vertex transitive graph with finite weak (resp. strong) cop number different
than one.
