Infinite Strong Cop Number of the Three-Dimensional Heisenberg Group

Infinite Strong Cop Number of the Three-Dimensional Heisenberg Group

Geometric group theory and coarse cops and robbers on Cayley graphs

September 2026

Dr Jerónimo García Mejía (URSS Supervisor)

This project studies the weak and strong cop numbers of Cayley graphs of finitely generated groups, with particular emphasis on methods for proving that the strong cop number is infinite. After reviewing several known results and techniques, we investigate a number of non-hyperbolic examples. For right-angled Artin groups, we use quasi-retractions onto Z² to determine their strong cop numbers in the non-free case. A novelty of this work is the proof that the discrete three-dimensional Heisenberg group H₃(Z) has infinite strong cop number. To show this, we introduce a weighted coordinate function comparable to the word metric and use the natural Heisenberg dilations to construct a sequence of quasi-homotheties. Motivated by this argument, we study a semidirect product Z³ ⋊ₐ Z and establish an analogous word-metric comparison, while the corresponding quasi-homothety problem is left open.

geometric group theory, cops and robbers, strong cop number, weak cop number, Cayley graphs, Heisenberg group, quasi-homothety, right-angled Artin groups