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

Coarse geometry of the fire retaining property and group splittings