{"title":"A property of closed geodesics on hyperbolic surfaces","authors":"Max Neumann-Coto, Peter Scott","doi":"10.4171/ggd/699","DOIUrl":"https://doi.org/10.4171/ggd/699","url":null,"abstract":"","PeriodicalId":55084,"journal":{"name":"Groups Geometry and Dynamics","volume":" ","pages":""},"PeriodicalIF":0.6,"publicationDate":"2023-01-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47895019","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Dynamics of actions of automorphisms of discrete groups $G$ on $mathrm{Sub}_G$ and applications to lattices in Lie groups","authors":"Rajdip Palit, Manoj B. Prajapati, R. Shah","doi":"10.4171/ggd/672","DOIUrl":"https://doi.org/10.4171/ggd/672","url":null,"abstract":"","PeriodicalId":55084,"journal":{"name":"Groups Geometry and Dynamics","volume":" ","pages":""},"PeriodicalIF":0.6,"publicationDate":"2023-01-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48120906","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"On the Bieri–Neumann–Strebel–Renz invariants of the weak commutativity construction $mathfrak{X}(G)$","authors":"D. Kochloukova","doi":"10.4171/ggd/680","DOIUrl":"https://doi.org/10.4171/ggd/680","url":null,"abstract":"","PeriodicalId":55084,"journal":{"name":"Groups Geometry and Dynamics","volume":" ","pages":""},"PeriodicalIF":0.6,"publicationDate":"2022-12-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42955993","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"A description of $operatorname{Aut}(dV_n)$ and $operatorname{Out}(dV_n)$ using transducers","authors":"L. Elliott","doi":"10.4171/ggd/697","DOIUrl":"https://doi.org/10.4171/ggd/697","url":null,"abstract":"","PeriodicalId":55084,"journal":{"name":"Groups Geometry and Dynamics","volume":"1 1","pages":""},"PeriodicalIF":0.6,"publicationDate":"2022-12-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"41727836","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Ergodicity of the mapping class group action on Deroin–Tholozan representations","authors":"Arnaud Maret","doi":"10.4171/ggd/695","DOIUrl":"https://doi.org/10.4171/ggd/695","url":null,"abstract":"","PeriodicalId":55084,"journal":{"name":"Groups Geometry and Dynamics","volume":" ","pages":""},"PeriodicalIF":0.6,"publicationDate":"2022-11-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44979353","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Two generalisations of Leighton’s theorem (with an appendix by Giles Gardam and Daniel J. Woodhouse)","authors":"Sam Shepherd","doi":"10.4171/ggd/682","DOIUrl":"https://doi.org/10.4171/ggd/682","url":null,"abstract":"","PeriodicalId":55084,"journal":{"name":"Groups Geometry and Dynamics","volume":" ","pages":""},"PeriodicalIF":0.6,"publicationDate":"2022-10-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42590669","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Hyperbolic groups with almost finitely presented subgroups (with an appendix by Robert Kropholler and Federico Vigolo)","authors":"P. Kropholler","doi":"10.4171/ggd/644","DOIUrl":"https://doi.org/10.4171/ggd/644","url":null,"abstract":"","PeriodicalId":55084,"journal":{"name":"Groups Geometry and Dynamics","volume":" ","pages":""},"PeriodicalIF":0.6,"publicationDate":"2022-05-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48465052","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Cubulation of some triangle-free Artin groups","authors":"T. Haettel","doi":"10.4171/ggd/648","DOIUrl":"https://doi.org/10.4171/ggd/648","url":null,"abstract":"","PeriodicalId":55084,"journal":{"name":"Groups Geometry and Dynamics","volume":" ","pages":""},"PeriodicalIF":0.6,"publicationDate":"2022-03-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47934106","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
A group $Gamma$ is said to be “finitely non-co-Hopfian,” or “renormalizable,” if there exists a self-embedding $varphi colon Gamma to Gamma$ whose image is a proper subgroup of finite index. Such a proper self-embedding is called a “renormalization for $Gamma$.” In this work, we associate a dynamical system to a renormalization $varphi$ of $Gamma$. The discriminant invariant ${mathcal D}_{varphi}$ of the associated Cantor dynamical system is a profinite group which is a measure of the asymmetries of the dynamical system. If ${mathcal D}_{varphi}$ is a finite group for some renormalization, we show that $Gamma/C_{varphi}$ is virtually nilpotent, where $C_{varphi}$ is the kernel of the action map. We introduce the notion of a (virtually) renormalizable Cantor action, and show that the action associated to a renormalizable group is virtually renormalizable. We study the properties of virtually renormalizable Cantor actions, and show that virtual renormalizability is an invariant of continuous orbit equivalence. Moreover, the discriminant invariant of a renormalizable Cantor action is an invariant of continuous orbit equivalence. Finally, the notion of a renormalizable Cantor action is related to the notion of a self-replicating group of automorphisms of a rooted tree.
如果存在一个自嵌入的$varphi colon Gamma to Gamma$，其像是有限索引的适当子群，则群$Gamma$被称为“有限非共hopfian”或“可重整的”。这种适当的自嵌入被称为“$Gamma$的重整化”。在这项工作中，我们将动力系统与$Gamma$的重整化$varphi$联系起来。关联康托动力系统的判别不变量${mathcal D}_{varphi}$是一个无限群，它是动力系统不对称性的度量。如果${mathcal D}_{varphi}$是某种重整化的有限群，我们证明$Gamma/C_{varphi}$实际上是幂零的，其中$C_{varphi}$是动作图的核。我们引入了(虚拟)可重整康托动作的概念，并证明了与可重整群相关的动作是虚拟可重整的。研究了虚可重整康托作用的性质，证明了虚可重整是连续轨道等价的不变量。此外，可重整康托作用的判别不变量是连续轨道等价的不变量。最后，可重整康托动作的概念与根树的自同构的自复制群的概念相关。
{"title":"Cantor dynamics of renormalizable groups","authors":"Steven Hurder, Olga Lukina, Wouter van Limbeek","doi":"10.4171/ggd/636","DOIUrl":"https://doi.org/10.4171/ggd/636","url":null,"abstract":"A group $Gamma$ is said to be “finitely non-co-Hopfian,” or “renormalizable,” if there exists a self-embedding $varphi colon Gamma to Gamma$ whose image is a proper subgroup of finite index. Such a proper self-embedding is called a “renormalization for $Gamma$.” In this work, we associate a dynamical system to a renormalization $varphi$ of $Gamma$. The discriminant invariant ${mathcal D}_{varphi}$ of the associated Cantor dynamical system is a profinite group which is a measure of the asymmetries of the dynamical system. If ${mathcal D}_{varphi}$ is a finite group for some renormalization, we show that $Gamma/C_{varphi}$ is virtually nilpotent, where $C_{varphi}$ is the kernel of the action map. We introduce the notion of a (virtually) renormalizable Cantor action, and show that the action associated to a renormalizable group is virtually renormalizable. We study the properties of virtually renormalizable Cantor actions, and show that virtual renormalizability is an invariant of continuous orbit equivalence. Moreover, the discriminant invariant of a renormalizable Cantor action is an invariant of continuous orbit equivalence. Finally, the notion of a renormalizable Cantor action is related to the notion of a self-replicating group of automorphisms of a rooted tree.","PeriodicalId":55084,"journal":{"name":"Groups Geometry and Dynamics","volume":"54 11-12","pages":""},"PeriodicalIF":0.6,"publicationDate":"2021-12-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138512885","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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