图的反射树作为Coxeter群的边界

Jacek 'Swikatkowski
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引用次数: 1

摘要

对于任何有限图$X$(视为拓扑空间),我们关联一些显式紧度量空间${\cal X}^r(X)$,我们称之为{\it图的反射树$X$}。该空间的拓扑维度为$\le1$,其连接的组件是局部连接的。我们证明,如果$X$被适当三角化(作为一个简单图$\Gamma$,其中$X$是几何实现),那么直角Coxeter系统$(W,S)$与$\Gamma$神经同构的视觉边界$\partial_\infty(W,S)$与${\cal X}^r(X)$是同态的。对于每个$X$,这产生了许多与空间${\cal X}^r(X)$同胚的Gromov边界的词双曲群。
本文章由计算机程序翻译,如有差异,请以英文原文为准。
Reflection trees of graphs as boundaries of Coxeter groups
To any finite graph $X$ (viewed as a topological space) we assosiate some explicit compact metric space ${\cal X}^r(X)$ which we call {\it the reflection tree of graphs $X$}. This space is of topological dimension $\le1$ and its connected components are locally connected. We show that if $X$ is appropriately triangulated (as a simplicial graph $\Gamma$ for which $X$ is the geometric realization) then the visual boundary $\partial_\infty(W,S)$ of the right angled Coxeter system $(W,S)$ with the nerve isomorphic to $\Gamma$ is homeomorphic to ${\cal X}^r(X)$. For each $X$, this yields in particular many word hyperbolic groups with Gromov boundary homeomorphic to the space ${\cal X}^r(X)$.
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