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.
Categories: math.MG math.GT