{"title":"图的反射树作为Coxeter群的边界","authors":"Jacek 'Swikatkowski","doi":"10.2140/AGT.2021.21.351","DOIUrl":null,"url":null,"abstract":"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 ${\\cal X}^r(X)$.","PeriodicalId":8427,"journal":{"name":"arXiv: Group Theory","volume":null,"pages":null},"PeriodicalIF":0.0000,"publicationDate":"2019-05-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"1","resultStr":"{\"title\":\"Reflection trees of graphs as boundaries of Coxeter groups\",\"authors\":\"Jacek 'Swikatkowski\",\"doi\":\"10.2140/AGT.2021.21.351\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"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 ${\\\\cal X}^r(X)$.\",\"PeriodicalId\":8427,\"journal\":{\"name\":\"arXiv: Group Theory\",\"volume\":null,\"pages\":null},\"PeriodicalIF\":0.0000,\"publicationDate\":\"2019-05-18\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"1\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"arXiv: Group Theory\",\"FirstCategoryId\":\"1085\",\"ListUrlMain\":\"https://doi.org/10.2140/AGT.2021.21.351\",\"RegionNum\":0,\"RegionCategory\":null,\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"\",\"JCRName\":\"\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"arXiv: Group Theory","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/10.2140/AGT.2021.21.351","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"","JCRName":"","Score":null,"Total":0}
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 ${\cal X}^r(X)$.