Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Coarse structures and group actions   The main results of the paper are: \begin{Prop}\label{GenSvarc-Milnor} A
group $G$ acting coarsely on a coarse space $(X,\CC)$ induces a coarse
equivalence $g\to g\cdot x_0$ from $G$ to $X$ for any $x_0\in X$. \end{Prop}
  Theorem: \label{GenGromovThm} Two coarse structures $\CC_1$ and $\CC_2$ on
the same set $X$ are equivalent if the following conditions are satisfied:
\begin{enumerate} \item Bounded sets in $\CC_1$ are identical with bounded sets
in $\CC_2$, \item There is a coarse action $\phi_1$ of a group $G_1$ on
$(X,\CC_1)$ and a coarse action $\phi_2$ of a group $G_2$ on $(X,\CC_2)$ such
that $\phi_1$ commutes with $\phi_2$. \end{enumerate}
  They generalize the following two basic results of coarse geometry:
  Proposition: [\v{S}varc-Milnor Lemma {\cite[Theorem 1.18]{Roe lectures}}]
\label{Svarc-Milnor} A group $G$ acting properly and cocompactly via isometries
on a length space $X$ is finitely generated and induces a quasi-isometry
equivalence $g\to g\cdot x_0$ from $G$ to $X$ for any $x_0\in X$.
  Theorem: [Gromov {\cite[page 6]{Gro asym invar}}] \label{GromovThm} Two
finitely generated groups $G$ and $H$ are quasi-isometric if and only if there
is a locally compact space $X$ admitting proper and cocompact actions of both
$G$ and $H$ that commute.

4
The categories of this paper are: cond-mat
9
The categories of this paper are: nlin
22
The categories of this paper are: chao-dyn
16
The categories of this paper are: stat
30
The categories of this paper are: q-alg
37
The categories of this paper are: supr-con
8
The categories of this paper are: hep-ex
35
The categories of this paper are: mtrl-th
20
The categories of this paper are: acc-phys
32
The categories of this paper are: atom-ph
18
The categories of this paper are: econ
11
The categories of this paper are: hep-lat
6
The categories of this paper are: astro-ph
13
The categories of this paper are: quant-ph
17
The categories of this paper are: q-fin
14
The categories of this paper are: math-ph