Coarse cops and robber in graphs and groups

We investigate two variants of the classical Cops and robber game in graphs, recently introduced by Lee, Martínez-Pedroza, and Rodríguez-Quinche. The two versions are played in infinite graphs and the goal of the cops is to prevent the robber to visit some ball of finite radius (chosen by the robber) infinitely many times. Moreover the cops and the robber move at a different speed, and the cops can choose a radius of capture before the game starts. Depending on the order in which the parameters are chosen, this naturally defines two games, a weak version and a strong version (in which the cops are more powerful), and thus two variants of the cop number of a graph G: the weak cop number wCop(G) and the strong cop number sCop(G). It turns out that these two parameters are invariant under quasi-isometry and thus we can investigate these parameters in finitely generated groups by considering any of their Cayley graphs; the parameters do not depend on the chosen set of generators. We answer a number of questions raised by Lee, Martínez-Pedroza, and Rodríguez-Quinche, and more recently by Cornect and Martínez-Pedroza. • We show that if some graph G has a quasi-isometric embedding in some graph H, then wCop(G)≤wCop(H) and sCop(G)≤sCop(H). • It was proved by Lee, Martínez-Pedroza, and Rodríguez-Quinche that Gromov-hyperbolic graphs have strong cop number equal to 1. We prove that the converse also holds, so that sCop(G)=1 if and only if G is Gromov-hyperbolic. This gives a new purely game-theoretic definition of hyperbolicity in infinite graphs. • We tie the weak cop number of a graph G to the existence of asymptotic minors of large treewidth in G. We deduce that for any graph G, wCop(G)=1 if and only if G is quasi-isometric to a tree. In particular, for any finitely generated group Γ, wCop(Γ)=1 if and only if Γ is virtually free. We also prove that for any finitely presented group Γ, wCop(Γ)=1or∞. • We prove that sCop(Z²)=∞ (this was only known to hold for the weak version of the game). We have learned very recently that some of our results have been obtained independently by Appenzeller and Klinge, using fairly different arguments.