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| Main Authors: | , |
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| Format: | Preprint |
| Published: |
2025
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2502.04540 |
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Table of Contents:
- 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 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.