{"title":"Subgroups of right-angled Coxeter groups via Stallings-like techniques","authors":"Pallavi Dani, Ivan Levcovitz","doi":"10.4171/jca/54","DOIUrl":null,"url":null,"abstract":"We associate a cube complex to any given finitely generated subgroup of a right-angled Coxeter group, called the completion of the subgroup. A completion characterizes many properties of the subgroup such as whether it is quasiconvex, normal, finite-index or torsion-free. We use completions to show that reflection subgroups are quasiconvex, as are one-ended Coxeter subgroups of a 2-dimensional right-angled Coxeter group. We provide an algorithm that determines whether a given one-ended, 2-dimensional right-angled Coxeter group is isomorphic to some finite-index subgroup of another given right-angled Coxeter group. In addition, we answer several algorithmic questions regarding quasiconvex subgroups. Finally, we give a new proof of Haglund's result that quasiconvex subgroups of right-angled Coxeter groups are separable.","PeriodicalId":48483,"journal":{"name":"Journal of Combinatorial Algebra","volume":null,"pages":null},"PeriodicalIF":0.6000,"publicationDate":"2019-08-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"11","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Journal of Combinatorial Algebra","FirstCategoryId":"100","ListUrlMain":"https://doi.org/10.4171/jca/54","RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q3","JCRName":"MATHEMATICS","Score":null,"Total":0}
引用次数: 11
Abstract
We associate a cube complex to any given finitely generated subgroup of a right-angled Coxeter group, called the completion of the subgroup. A completion characterizes many properties of the subgroup such as whether it is quasiconvex, normal, finite-index or torsion-free. We use completions to show that reflection subgroups are quasiconvex, as are one-ended Coxeter subgroups of a 2-dimensional right-angled Coxeter group. We provide an algorithm that determines whether a given one-ended, 2-dimensional right-angled Coxeter group is isomorphic to some finite-index subgroup of another given right-angled Coxeter group. In addition, we answer several algorithmic questions regarding quasiconvex subgroups. Finally, we give a new proof of Haglund's result that quasiconvex subgroups of right-angled Coxeter groups are separable.