Cops and Robber on Hyperbolic Manifolds

TL;DR

This work studies pursuit-evasion through the Cops and Robber game on geodesic spaces, with a focus on hyperbolic manifolds. It introduces a covering-space strategy that proves the cop win number $c(M)=2$ for any compact hyperbolic manifold $M$, independent of genus, and extends the approach to higher dimensions while discussing bounds on the strong cop number $c_0$. The results bridge discrete graph pursuit and continuous pursuit-evasion, revealing a genus-insensitive behavior in negative curvature and laying groundwork for generalizations to spaces of constant curvature. Open questions remain about exact strong-capture bounds on constant-curvature surfaces and the tightness of two-cop strategies in higher dimensions.

Abstract

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 $\tildeΩ(\sqrt{g}\,)$. 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.

Figures (10)

• Figure 1: Figure (a) depicts the surface $S(g)$ and Figure (b) its fundamental domain $P\left(4g,\frac{2\pi}{4g}\right)$ inside a partial tessellation of the Poincaré disk.
• Figure 2: A hyperbolic right-angled triangle $ABC$ with the right angle at $C$.
• Figure 3: Copies of the fundamental polygon $P(4g,\frac{2\pi}{4g})$ in $\mathcal{D}$. Depicted is a point and one copy of this point in each of the depicted copies of the fundamental polygon.
• Figure 4: A schematic picture of cop's position $c_1$ and robber's position $r$ at time step $k$.
• Figure 5: The point $\phi(R^{k+1})$ is the reflection point of $R^{k+1}$ over $g^k$.

Theorems & Definitions (49)

• Theorem 1.1: AIGNER19841
• Lemma 1.2: AIGNER19841Mo22
• Conjecture 1.3: mohar2017notesMo22
• Conjecture 1.4: FRANKL1987301
• Conjecture 1.5: Mo22
• Theorem 1.6
• Theorem 2.1: Killing-Hopf Hopf1926Killing1891
• Lemma 2.2
• Lemma 2.3
• Lemma 2.4