  Given a non-decreasing function $f \colon \mathbb{N} \to \mathbb{N}$ we
define a single player game on (infinite) connected graphs that we call fire
retaining. If a graph $G$ admits a winning strategy for any initial
configuration (initial fire) then we say that $G$ has the $f$-retaining
property; in this case if $f$ is a polynomial of degree $d$, we say that $G$
has the polynomial retaining property of degree $d$.
  We prove that having the polynomial retaining property of degree $d$ is a
quasi-isometry invariant in the class of uniformly locally finite connected
graphs. Henceforth, the retaining property defines a quasi-isometric invariant
of finitely generated groups. We prove that if a finitely generated group $G$
splits over a quasi-isometrically embedded subgroup of polynomial growth of
degree $d$, then $G$ has polynomial retaining property of degree $d-1$. Some
connections to other work on quasi-isometry invariants of finitely generated
groups are discussed and some questions are raised.
