双曲群与虚拟自由群的警察与强盗问题
Cops and robbers for hyperbolic and virtually free groups
AI总结：
本文定义强、弱警察数两种群不变量，证明双曲群对应强警察数为1、虚拟自由群对应弱警察数为1，还给出点灯人群等的强警察数计算实例，解答了相关公开问题。
摘要：
Lee、Martínez-Pedroza和Rodríguez-Quinche通过研究有限生成群的Cayley图上特定组合游戏的获胜策略，定义了两种新的群不变量：强警察数sCop和弱警察数wCop。本文证明，有限生成群G是Gromov双曲群当且仅当sCop(G)=1；G是虚拟自由群当且仅当wCop(G)=1，从而回答了Cornect和Martínez-Pedroza提出的问题。本文还证明sCop(ℤ²)=∞，回答了原论文提出的问题。目前是否存在不等于1的有限警察数仍未知，但本文表明CAT(0)群不存在此类情况。本文提供了显式计算强警察数的方法，并通过将其应用于特定的点灯人群、可解Baumslag-Solitar群以及Thompson群F给出了实例。
英文摘要：
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.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
