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The firefighter game problem on locally finite connected graphs wasintroduced by Bert Hartnell. The game on a graph $G$ can be described asfollows: let $f_n$ be a sequence of positive integers; an initial fire startsat a finite set of vertices; at each (integer) time $n\geq 1$, $f_n$ verticeswhich are not on fire become protected, and then the fire spreads to allunprotected neighbors of vertices on fire; once a vertex is protected or is onfire, it remains so for all time intervals. The graph $G$ has the\emph{$f_n$-containment property} if every initial fire admits an strategy thatprotects $f_n$ vertices at time $n$ so that the set of vertices on fire iseventually constant. If the graph $G$ has the containment property for asequence of the form $f_n=Cn^d$, then the graph is said to have\emph{polynomial containment}. In [5], it is shown that any locally finitegraph with polynomial growth has polynomial containment; and it is remarkedthat the converse does not hold. That article also raised the question ofwhether the equivalence of polynomial growth and polynomial containment holdsfor Cayley graphs of finitely generated groups. In this short note, we remarkhow the equivalence holds for elementary amenable groups and for non-amenablegroups from results in the literature.

A note on the relation between Hartnell's firefighter problem and growth of groups