Abstract. In this article, we study Hartnell's Firefighter Problem through the group theoretic notions of growth and quasi-isometry. A graph has the ncontainment property if for every finite initial fire, there is a strategy to contain the fire by protecting n vertices at each turn. A graph has the constant containment property if there is an integer n such that it has the n-containment property. Our first result is that any locally finite connected graph with quadratic growth has the constant containment property; the converse does not hold. A second result is that in the class of graphs with bounded degree, having the constant containment property is closed under quasi-isometry. We prove analogous results for the {fn}-containment property, where fn is an integer sequence corresponding to the number of vertices protected at time n. In particular, we positively answer a conjecture by Develin and Hartke by proving that the d-dimensional square grid L d does not satisfy the cn d−3 -containment property for any constant c.
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