Coarse geometry of the fire retaining property and group splittings

Eduardo Martínez-Pedroza; Tomasz Prytuła

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arXiv · 1904.04658

Coarse geometry of the fire retaining property and group splittings

Eduardo Martínez-Pedroza·Tomasz Prytuła

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Abstract

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.

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Eduardo Martínez-Pedroza, Tomasz Prytuła. 2019-04-09. Coarse geometry of the fire retaining property and group splittings. https://arxiv.org/abs/1904.04658

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