  We study the fire-retaining problem on groups, a quasi-isometry invariant
introduced by Mart\'inez-Pedroza and Prytula [8], related to the firefighter
problem. We prove that any Cayley graph with degree-$d$ polynomial growth does
not satisfy $\{f(n)\}$-retainment, for any $f(n) = o(n^{d-2})$, matching the
upper bound given for the firefighter problem for these graphs. In the
exponential growth regime we prove general lower bounds for direct products and
wreath products. These bounds are tight, and show that for exponential-growth
groups a wide variety of behaviors is possible. In particular, we construct,
for any $d\geq 1$, groups that satisfy $\{n^{d}\}$-retainment but not
$o(n^d)$-retainment, as well as groups that do not satisfy sub-exponential
retainment.
