The Lamplighter groups have infinite weak cop number

TL;DR

This paper investigates the weak-cop number in infinite, vertex transitive graphs, proving that certain Cayley graphs, including those of wreath products and Thompson's group F, have infinite weak-cop number.

Contribution

It establishes that Cayley graphs of wreath products and Thompson's group F possess infinite weak-cop number, advancing understanding of pursuit-evasion dynamics in exotic infinite graphs.

Findings

Cayley graphs of wreath products have infinite weak-cop number

Thompson's group F Cayley graphs also have infinite weak-cop number

Introduces a new pursuit and evasion game for analysis

Abstract

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.