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arXiv math/0607568math.MGmath.GT
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粗结构与群作用

Coarse structures and group actions

N. Brodskiy, J. Dydak, A. Mitra

备注：
11 pages
发表信息：
Colloquium Mathematicum 111 (2008), 149-158
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摘要：

本文的主要结果是：\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}定理：\label{GenGromovThm}同一集合上的两个粗结构$\CC_1$和$\CC_2$是等价的，如果满足下列条件：\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}它们推广了粗几何的下列两个基本结果：命题：[Švarc-Milnor引理{ \cite [Theorem 1.18]{Roe lectures}}] \label{Svarc-Milnor}在长度空间$X$上通过等距变换适当地和共紧地作用的群$G$被随机生成，并且对于任何$x_0\in X$，诱导出从$G$到$X$的准等距等价$g\to g\cdot x_0$。定理：[Gromov { \cite [page 6]{Gro asym invar}}] \label{GromovThm}两个双生成群$G$和$H$是拟等距的当且仅当存在一个局部紧空间$X$，它允许$G$和$H$的适当的和余紧的作用是可交换的。

英文摘要：

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 $ϕ_1$ 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.

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