Cops and robbers for hyperbolic and virtually free groups

Abstract

Lee, Martínez-Pedroza and Rodríguez-Quinche define two new group invariants, the strong cop number $\mathrm{sCop}$ and the weak cop number $\mathrm{wCop}$, by examining winning strategies for certain combinatorial games played on Cayley graphs of finitely generated groups. We show that a finitely generated group $G$ is Gromov hyperbolic if and only if $\mathrm{sCop}(G)=1$. We show that $G$ is virtually free if and only if $\mathrm{wCop}(G)=1$, answering a question by Cornect and Martínez-Pedroza. We show that $\mathrm{sCop}({\mathbb{Z}}^2)=\infty$, answering a question from the original paper. It is still unknown whether there exist finite cop numbers not equal to $1$, but we show that this is not possible for $\mathrm{CAT}(0)$-groups. We provide machinery to explicitly compute strong cop numbers and give examples by applying it to certain lamplighter groups, the solvable Baumslag–Solitar groups, Grigorchuk’s group and Thompson’s group $F$.