Coarse structures and group actions
Tom 111 / 2008
Streszczenie
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 $x_0\in X$.
Theorem 0.2. 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 $\phi_2$.
They generalize the following two basic results of coarse geometry: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 $x_0\in X$.
Theorem 0.4 (Gromov [4, p. 6]). 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.
