层次双曲群中的无界域

IF 0.6 3区数学 Q3 MATHEMATICS

Groups Geometry and Dynamics Pub Date : 2020-07-24 DOI:10.4171/ggd/706

H. Petyt, Davide Spriano

{"title":"层次双曲群中的无界域","authors":"H. Petyt, Davide Spriano","doi":"10.4171/ggd/706","DOIUrl":null,"url":null,"abstract":"We investigate unbounded domains in hierarchically hyperbolic groups and obtain constraints on the possible hierarchical structures. Using these insights, we characterise the structures of virtually abelian HHGs and show that the class of HHGs is not closed under finite extensions. This provides a strong answer to the question of whether being an HHG is invariant under quasiisometries. Along the way, we show that infinite torsion groups are not HHGs. By ruling out pathological behaviours, we are able to give simpler, direct proofs of the rank-rigidity and omnibus subgroup theorems for HHGs. This involves extending our techniques so that they apply to all subgroups of HHGs.","PeriodicalId":55084,"journal":{"name":"Groups Geometry and Dynamics","volume":null,"pages":null},"PeriodicalIF":0.6000,"publicationDate":"2020-07-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"17","resultStr":"{\"title\":\"Unbounded domains in hierarchically hyperbolic groups\",\"authors\":\"H. Petyt, Davide Spriano\",\"doi\":\"10.4171/ggd/706\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"We investigate unbounded domains in hierarchically hyperbolic groups and obtain constraints on the possible hierarchical structures. Using these insights, we characterise the structures of virtually abelian HHGs and show that the class of HHGs is not closed under finite extensions. This provides a strong answer to the question of whether being an HHG is invariant under quasiisometries. Along the way, we show that infinite torsion groups are not HHGs. By ruling out pathological behaviours, we are able to give simpler, direct proofs of the rank-rigidity and omnibus subgroup theorems for HHGs. This involves extending our techniques so that they apply to all subgroups of HHGs.\",\"PeriodicalId\":55084,\"journal\":{\"name\":\"Groups Geometry and Dynamics\",\"volume\":null,\"pages\":null},\"PeriodicalIF\":0.6000,\"publicationDate\":\"2020-07-24\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"17\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Groups Geometry and Dynamics\",\"FirstCategoryId\":\"100\",\"ListUrlMain\":\"https://doi.org/10.4171/ggd/706\",\"RegionNum\":3,\"RegionCategory\":\"数学\",\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"Q3\",\"JCRName\":\"MATHEMATICS\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"Groups Geometry and Dynamics","FirstCategoryId":"100","ListUrlMain":"https://doi.org/10.4171/ggd/706","RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q3","JCRName":"MATHEMATICS","Score":null,"Total":0}

引用次数: 17

摘要

我们研究了层次双曲群中的无界域，并得到了可能的层次结构的约束。利用这些见解，我们刻画了几乎阿贝尔HHG的结构，并证明了HHG类在有限扩展下是不闭合的。这为作为HHG在拟等距下是否不变的问题提供了一个强有力的答案。在此过程中，我们证明了无限扭群不是HHG。通过排除病理行为，我们能够给出HHG的秩刚性和综合子群定理的更简单、直接的证明。这涉及到扩展我们的技术，使其适用于HHG的所有亚组。

本文章由计算机程序翻译，如有差异，请以英文原文为准。

查看原文本刊更多论文

Unbounded domains in hierarchically hyperbolic groups

We investigate unbounded domains in hierarchically hyperbolic groups and obtain constraints on the possible hierarchical structures. Using these insights, we characterise the structures of virtually abelian HHGs and show that the class of HHGs is not closed under finite extensions. This provides a strong answer to the question of whether being an HHG is invariant under quasiisometries. Along the way, we show that infinite torsion groups are not HHGs. By ruling out pathological behaviours, we are able to give simpler, direct proofs of the rank-rigidity and omnibus subgroup theorems for HHGs. This involves extending our techniques so that they apply to all subgroups of HHGs.

求助全文

通过发布文献求助，成功后即可免费获取论文全文。去求助

来源期刊

Groups Geometry and Dynamics 数学-数学

CiteScore

1.10

自引率

0.00%

发文量

审稿时长

>12 weeks

期刊介绍： Groups, Geometry, and Dynamics is devoted to publication of research articles that focus on groups or group actions as well as articles in other areas of mathematics in which groups or group actions are used as a main tool. The journal covers all topics of modern group theory with preference given to geometric, asymptotic and combinatorial group theory, dynamics of group actions, probabilistic and analytical methods, interaction with ergodic theory and operator algebras, and other related fields. Topics covered include: geometric group theory; asymptotic group theory; combinatorial group theory; probabilities on groups; computational aspects and complexity; harmonic and functional analysis on groups, free probability; ergodic theory of group actions; cohomology of groups and exotic cohomologies; groups and low-dimensional topology; group actions on trees, buildings, rooted trees.