The Lamplighter groups have infinite weak cop number

The weak-cop number of a graph, a variation of the cop number, is an invariant suitable for infinite graphs and is a quasi-isometric invariant. While for any $m\in\mathbb{Z}_+\cup\{\infty\}$ there exist locally finite infinite graphs with weak-cop number $m$, it is an open question whether there exists locally finite vertex transitive graphs whose weak-cop number is different than $1$ and $\infty$. We test this question on Cayley graphs of wreath products, these are objects known for their exotic geometries. We prove that Cayley graphs of wreath products of nontrivial groups by infinite groups have infinite weak-cop number. The result is proved by defining a new pursuit and evasion game and proving the existence of strategies for the evader. We also include a short argument that Cayley graphs of Thompson's group $F$ have infinite weak cop number.