# Coarse geometry of the fire retaining property and group splittings

## TL;DR

This paper introduces a new geometric property called the polynomial retaining property for graphs and groups, showing it is a quasi-isometry invariant and relating group splittings to this property.

## Contribution

It establishes the polynomial retaining property as a quasi-isometry invariant and links group splittings over polynomial growth subgroups to this property.

## Key findings

- Polynomial retaining property is a quasi-isometry invariant.
- Groups splitting over polynomial growth subgroups have related retaining properties.
- Connections to quasi-isometry invariants of finitely generated groups are discussed.

## 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.