## Coarse structures and group actions

### Volume 111 / 2008

#### Abstract

The main results of the paper are:

Proposition 0.1.
*A group $G$ acting coarsely on a coarse space $(X,{\mathcal{C}})$ induces a
coarse equivalence $g\mapsto g\cdot x_0$ from $G$ to $X$ for any
$x_0\in X$.*

*Two coarse structures ${\mathcal{C}}_1$ and ${\mathcal{C}}_2$ on the same set $X$ are equivalent if the following conditions are satisfied:*

(1) *Bounded sets in ${\mathcal{C}}_1$ are identical with bounded sets in ${\mathcal{C}}_2$.*

(2) *There is a coarse action $\phi_1$ of a group $G_1$
on $(X,{\mathcal{C}}_1)$ and a coarse action $\phi_2$ of a group $G_2$ on $(X,{\mathcal{C}}_2)$
such that $\phi_1$ commutes with $\phi_2$.*

Proposition 0.3 (Shvarts–Milnor lemma [5, Theorem 1.18]).
*A group $G$ acting properly and cocompactly via isometries on a
length space $X$ is finitely generated and induces a
quasi-isometry equivalence $g\mapsto g\cdot x_0$ from $G$ to $X$ for
any $x_0\in X$.*

*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.*