Cops and robbers for hyperbolic and virtually free groups

TL;DR

This paper introduces new invariants for groups based on combinatorial games, characterizing hyperbolic and virtually free groups, and providing computational tools and examples for these invariants.

Contribution

It establishes characterizations of hyperbolic and virtually free groups via the new cop number invariants and offers methods to compute these invariants for specific groups.

Findings

Gromov-hyperbolic groups have strong cop number 1

Virtually free groups have weak cop number 1

Strong cop number is infinite for Z^2

Abstract

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.