Cops and robbers for hyperbolic and virtually free groups

Lee, Martínez-Pedroza and Rodríguez-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 Martínez-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 \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 .