Coarse structures and group actions

N. Brodskiy¹ ; J. Dydak¹ ; A. Mitra¹

¹ Department of Mathematics University of Tennessee Knoxville, TN 37996, U.S.A.

Colloquium Mathematicum, Tome 111 (2008) no. 1, pp. 149-158

Résumé

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.

Détail

DOI : 10.4064/cm111-1-13

Keywords: main results paper proposition group acting coarsely coarse space mathcal induces coarse equivalence mapsto cdot theorem coarse structures mathcal mathcal set equivalent following conditions satisfied bounded sets mathcal identical bounded sets mathcal there coarse action phi group mathcal coarse action phi group mathcal phi commutes phi generalize following basic results coarse geometry proposition shvarts milnor lemma theorem group acting properly cocompactly via isometries length space finitely generated induces quasi isometry equivalence mapsto cdot theorem gromov finitely generated groups quasi isometric only there locally compact space admitting proper cocompact actions commute