Lee, Mart\'inez-Pedroza and Rodr\'iguez-Quinche define two new group invariants, the strong cop number $\operatorname{ sCop}$ and the weak cop number $\operatorname{ wCop}$, 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 $\operatorname{ sCop(G)} = 1$. We show that $G$ is virtually free if and only if $\operatorname{ wCop(G)}=1$, answering a question by Cornect and Mart\'inez-Pedroza. We show that $\operatorname{ sCop}(\mathbb{ Z}^2) = \infty$, 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 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, and Thompson's group F.
Categories: math.GR