The Lamplighter groups and Thompson's group F have infinite weak-cop number

The 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₊∪∞ 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 ∞. 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.