Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Coarse geometry of the fire retaining property and group splittings   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.

5
The categories of this paper are: gr-qc
21
The categories of this paper are: adap-org
9
The categories of this paper are: nlin
19
The categories of this paper are: eess
16
The categories of this paper are: stat
35
The categories of this paper are: mtrl-th
26
The categories of this paper are: bayes-an
8
The categories of this paper are: hep-ex
4
The categories of this paper are: cond-mat
32
The categories of this paper are: atom-ph
20
The categories of this paper are: acc-phys
34
The categories of this paper are: chem-ph
33
The categories of this paper are: plasm-ph
15
The categories of this paper are: nucl-ex
18
The categories of this paper are: econ
0
The categories of this paper are: hep-ph