Coarse geometry of the fire retaining property and group splittings
| Autor: | Eduardo Martínez-Pedroza, Tomasz Prytuła |
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| Jazyk: | angličtina |
| Rok vydání: | 2019 |
| Předmět: | |
| Zdroj: | Geometriae Dedicata |
| Popis: | 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. V2: Version accepted for publication by Geometriae Dedicata |
| Databáze: | OpenAIRE |
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