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              <text>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, whose strong cop numbers are already determined by known results, we give a direct proof using explicit quasi-retractions onto ℤ². 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 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.&#13;
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