Coarse geometry of the fire retaining property and group splittings

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