A Note on the Relation between Hartnell’s Firefighter Problem and Growth of Groups

The firefighter game problem on locally finite connected graphs was introduced by Bert Hartnell [6]. The game on a graph G can be described as follows: let fn be a sequence of positive integers; an initial fire starts at a finite set of vertices; at each (integer) time n ≥ 1, fn vertices which are not on fire become protected, and then the fire spreads to all unprotected neighbors of vertices on fire; once a vertex is protected or is on fire, it remains so for all time intervals. The graph G has the fn-containment property if every initial fire admits an strategy that protects fn vertices at time n so that the set of vertices on fire is eventually constant. If the graph G has the containment property for a sequence of the form fn = Cnd, then the graph is said to have polynomial containment. In [5], it is shown that any locally finite graph with polynomial growth has polynomial containment; and it is remarked that the converse does not hold. That article also raised the question of whether the equivalence of polynomial growth and polynomial containment holds for Cayley graphs of finitely generated groups. In this short note, we remark how the equivalence holds for elementary amenable groups and for non-amenable groups from results in the literature. Let G be a connected and locally finite graph, and let d be a non-negative integer. The graph G satisfies a polynomial containment of degree at most d if there exists a constant C > 0 such that any finite subset X0 of vertices of G admits a {Cn}containment strategy, i.e., there is a sequence {Wk : k ≥ 1} of subsets of vertices of G such that (1) for every n ≥ 1, the set Wn has cardinality at most Cn, (2) the sets Xn and Wn+1 are disjoint for n ≥ 0, where Xn for n > 0 is defined as the set of vertices at distance less than or equal to one of Xn−1 and do not belong to W1 ∪ · · · ∪Wn. (3) there is an integer T > 0 such that Xn = XT for every n ≥ T . This definition can be interpreted as the existence of winning strategies for a single player game: a fire starts at an arbitrary finite set of vertices X0, at each time interval the fire spreads to all unprotected vertices that neighbor vertices on fire, and Xn is the set of vertices on fire at time n; the player aims to contain the fire by protecting at time n a set Wn of at most Cn d vertices that are not on fire; once a vertex is protected or is on fire, it remains so for all time intervals. The graph G satisfies polynomial containment of degree at most d if there is C > 0 such that for any initial fire X0 the player has an {Cn}-strategy to contain the fire. The notion of edge-path in a graph G defines a metric on the set of vertices by declaring distG(x, y) to be the length of the shortest path from x to y. The metric dist on the set of vertices of G is called the combinatorial metric on G. The notion 2000 Mathematics Subject Classification. 20F65, 43A07, 57M07, 05C57, 05C10, 05C25.