Compact metric spaces with infinite cop number

TL;DR

This paper disproves Mohar's conjectures by constructing a compact geodesic metric space, specifically on , with an infinite cop number, challenging previous assumptions about pursuit-evasion games in such spaces.

Contribution

It provides a counterexample to Mohar's conjectures, showing that finitely many cops may not always suffice in compact geodesic metric spaces.

Findings

Constructed a metric on  with infinite cop number

Disproved the conjecture that finitely many cops can always win

Raised new questions about pursuit-evasion in metric spaces

Abstract

Mohar recently adapted the classical game of Cops and Robber from graphs to metric spaces, thereby unifying previously studied pursuit-evasion games. He conjectured that finitely many cops can win on any compact geodesic metric space, and that their number can be upper-bounded in terms of the ranks of the homology groups when the space is a simplicial pseudo-manifold. We disprove these conjectures by constructing a metric on $\mathbb{S}^3$ with infinite cop number. More problems are raised than settled.