The Cops and Robber game on geodesic spaces is a pursuit-evasion game with discrete steps which captures the behavior of the game played on graphs, as well as that of continuous pursuit-evasion games. One of the outstanding open problems about the game on graphs is to determine which graphs embeddable in a surface of genus $g$ have largest cop number. It is known that the cop number of genus $g$ graphs is $O(g)$ and that there are examples whose cop number is $\tilde\Omega(\sqrt{g}\,)$. The same phenomenon occurs when the game is played on geodesic surfaces. In this paper we obtain a surprising result about the game on a surface with constant curvature. It is shown that two cops have a strategy to come arbitrarily close to the robber, independently of the genus. We also discuss upper bounds on the number of cops needed to catch the robber. Our results generalize to higher-dimensional hyperbolic manifolds.

翻译：关于测地空间上的警察与小偷博弈是一种具有离散步骤的追逃博弈，它既刻画了图上的博弈行为，也描述了连续追逃博弈的特征。关于图上的博弈，一个突出的未解决问题是确定哪些可嵌入亏格为$g$的曲面上的图具有最大的警察数量。已知亏格$g$的图的警察数量为$O(g)$，且存在警察数量为$\tilde\Omega(\sqrt{g}\,)$的例子。当博弈在测地曲面上进行时，也会出现同样的现象。本文得到了关于恒定曲率曲面上博弈的一个令人惊讶的结果：我们证明，无论亏格如何，两名警察可以采用一种策略无限接近小偷。此外，我们还讨论了抓捕小偷所需警察数量的上界。我们的结论可推广至高维双曲流形。