  The weak-cop number of a graph, introduced by Lee et al (2023), is a
quasi-isometric invariant of graphs and hence of finitely generated groups via
their Cayley graphs. While for any $m\in\mathbb{Z}_+\cup\{\infty\}$ there exist
graphs with weak-cop number $m$, it is an open question whether there exists
finitely generated groups whose weak-cop number is different than $1$ and
$\infty$. We prove that wreath products of nontrivial groups by infinite groups
have infinite weak-cop number. We also prove that Thompson's group $F$ has
infinite weak-cop number. The results are proved by defining two new pursuit
and evasion games and proving the existence of strategies for the evader. In
the case of Thompson's group $F$, we also present an alternative and more
algebraic argument proving that it has infinite weak-cop number.
