  We prove that any Cayley graph $G$ with degree $d$ polynomial growth does not
satisfy $\{f(n)\}$-containment for any $f=o(n^{d-2})$. This settles the
asymptotic behaviour of the firefighter problem on such graphs as it was known
that $Cn^{d-2}$ firefighters are enough, answering and strengthening a
conjecture of Develin and Hartke. We also prove that intermediate growth Cayley
graphs do not satisfy polynomial containment, and give explicit lower bounds
depending on the growth rate of the group. These bounds can be further improved
when more geometric information is available, such as for Grigorchuk's group.
