Coarse cops and robber in graphs and groups

TL;DR

The paper develops a coarse, infinite-graph variant of Cops and Robber with weak and strong cop numbers $\mathsf{wCop}$ and $\mathsf{sCop}$, proving these parameters are invariant under quasi-isometry. It establishes a sharp dichotomy: $\mathsf{sCop}(G)=1$ iff $G$ is hyperbolic, and $\mathsf{wCop}(G)=1$ iff $G$ is quasi-isometric to a tree, with wide consequences for finitely generated groups (e.g., finitely presented groups have $\mathsf{wCop}\in\{1,\infty\}$). The grid $\mathbb{Z}^2$ is shown to have $\mathsf{sCop}=\infty$, while $\mathsf{wCop}$ is linked to asymptotic minors and large treewidth, yielding structural and group-theoretic corollaries (including results on virtually free groups). The work also introduces divergence games and primed variants of the cop numbers, clarifying the scope and limits of the game-theoretic approach in coarse geometry and suggesting several open directions.

Abstract

(abstract shortened to meet arxiv's length requirements) 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 and the strong cop number. 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. This includes a proof that the weak and strong cop numbers are monotone under quasi-isometric embedding, characterizations of graphs of weak cop number 1 and graphs of strong cop number 1, and a proof that the grid has infinite strong cop number. Moreover we tie the weak cop number of a graph $G$ to the existence of asymptotic minors of large tree-width in $G$, and use the result to prove that any finitely presented group has weak cop number 1 or $\infty$. We have learned very recently that some of our results have been obtained independently by Appenzeller and Klinge, using fairly different arguments.

Figures (3)

• Figure 1: The paths $P$ and $Q$ in the proof of Claim \ref{['C:move']}.
• Figure 2: The graph described in Proposition \ref{['ex: tw2-wcop-infty']}. Blue lines represent subdivided edges.
• Figure 3: The $t+1$ paths between any two rooms $R$ and $R'$.

Theorems & Definitions (49)

• Theorem 1.1
• Theorem 1.2
• Theorem 1.3
• Theorem 1.4
• Theorem 1.5
• Theorem 1.6
• Theorem 3.1
• proof
• Claim 3.2
• Claim 3.3