The Lamplighter groups and Thompson's group $F$ have infinite weak-cop numberThe weak-cop number of a graph, introduced by Lee et al (2023), is a quasi-isometry invariant of graphs and hence of finitely generated groups. While for any $m\in\mathbb{Z}_+\cup\infty$ there exists 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, as well as Thompson's group $F$, have infinite weak-cop number. The argument for Thompson's group relies on the representation of its elements by Belk and Brown Forest diagrams.
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