Coarse structures and group actions

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dc.creatorBrodskiy, N.
dc.creatorDydak, J.
dc.creatorMitra, A.
dc.date2006-07-22
dc.date.accessioned2026-07-07T09:23:25Z
dc.date.available2026-07-07T09:23:25Z
dc.descriptionThe 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 $ϕ_1$ of a group $G_1$ on $(X,\CC_1)$ and a coarse action $ϕ_2$ of a group $G_2$ on $(X,\CC_2)$ such that $ϕ_1$ commutes with $ϕ_2$. \end{enumerate} They generalize the following two basic results of coarse geometry: Proposition: [Švarc-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.
dc.description11 pages
dc.identifierhttps://arxiv.org/abs/math/0607568
dc.identifierhttp://arxiv.org/abs/math/0607568
dc.identifierColloquium Mathematicum 111 (2008), 149-158
dc.identifier.urihttp://salesiana.dossiersoluciones.com/handle/123456789/155725
dc.subjectMetric Geometry
dc.subjectGeometric Topology
dc.titleCoarse structures and group actions
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