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Systolic complexes and group presentations 收缩复合物和群体表现
IF 0.6 3区 数学
Groups Geometry and Dynamics Pub Date : 2021-05-04 DOI: 10.4171/ggd/717
Mireille Soergel
{"title":"Systolic complexes and group presentations","authors":"Mireille Soergel","doi":"10.4171/ggd/717","DOIUrl":"https://doi.org/10.4171/ggd/717","url":null,"abstract":"We give conditions on a presentation of a group, which imply that its Cayley complex is simplicial and the flag complex of the Cayley complex is systolic. We then apply this to Garside groups and Artin groups. We give a classification of the Garside groups whose presentation using the simple elements as generators satisfy our conditions. We then also give a dual presentation for Artin groups and identify in which cases the flag complex of the Cayley complex is systolic.","PeriodicalId":55084,"journal":{"name":"Groups Geometry and Dynamics","volume":null,"pages":null},"PeriodicalIF":0.6,"publicationDate":"2021-05-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46675747","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}
引用次数: 3
Property (T) in density-type models of random groups 随机群密度型模型的性质（T）
IF 0.6 3区 数学
Groups Geometry and Dynamics Pub Date : 2021-04-30 DOI: 10.4171/ggd/730
C. Ashcroft
{"title":"Property (T) in density-type models of random groups","authors":"C. Ashcroft","doi":"10.4171/ggd/730","DOIUrl":"https://doi.org/10.4171/ggd/730","url":null,"abstract":"We study Property (T) in the $Gamma(n,k,d)$ model of random groups: as $k$ tends to infinity this gives the Gromov density model, introduced in [Gro93]. We provide bounds for Property (T) in the $k$-angular model of random groups, i.e. the $ Gamma (n,k,d)$ model where $k$ is fixed and $n$ tends to infinity. We also prove that for $d>1slash 3$, a random group in the $Gamma(n,k,d)$ model has Property (T) with probability tending to $1$ as $k$ tends to infinity, strengthening the results of .{Z}uk and Kotowski--Kotowski, who consider only groups in the $Gamma (n,3k,d)$ model.","PeriodicalId":55084,"journal":{"name":"Groups Geometry and Dynamics","volume":null,"pages":null},"PeriodicalIF":0.6,"publicationDate":"2021-04-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46004343","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}
引用次数: 7
Modular orbits on the representation spaces of compact abelian Lie groups 紧阿贝尔李群表示空间上的模轨道
IF 0.6 3区 数学
Groups Geometry and Dynamics Pub Date : 2021-03-27 DOI: 10.4171/ggd/716
Yohann Bouilly, Gianluca Faraco
{"title":"Modular orbits on the representation spaces of compact abelian Lie groups","authors":"Yohann Bouilly, Gianluca Faraco","doi":"10.4171/ggd/716","DOIUrl":"https://doi.org/10.4171/ggd/716","url":null,"abstract":"Let $S$ be a closed surface of genus $g$ greater than zero. In the present paper we study the topological-dynamical action of the mapping class group on the $Bbb T^n$-character variety giving necessary and sufficient conditions for Mod$(S)$-orbits to be dense. As an application, such a characterisation provides a dynamical proof of the Kronecker's Theorem concerning inhomogeneous diophantine approximation.","PeriodicalId":55084,"journal":{"name":"Groups Geometry and Dynamics","volume":null,"pages":null},"PeriodicalIF":0.6,"publicationDate":"2021-03-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"41450593","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}
引用次数: 1
Subgroups of $mathrm{PL}_{+} I$ which do not embed into Thompson’s group $F$ $ mathm {PL}_{+} I$的子群，不嵌入到Thompson的群$F$中
IF 0.6 3区 数学
Groups Geometry and Dynamics Pub Date : 2021-03-27 DOI: 10.4171/ggd/708
J. Hyde, J. Moore
{"title":"Subgroups of $mathrm{PL}_{+} I$ which do not embed into Thompson’s group $F$","authors":"J. Hyde, J. Moore","doi":"10.4171/ggd/708","DOIUrl":"https://doi.org/10.4171/ggd/708","url":null,"abstract":"We will give a general criterion - the existence of an $F$-obstruction - for showing that a subgroup of $mathrm{PL}_+ I$ does not embed into Thompson's group $F$. An immediate consequence is that Cleary's\"golden ratio\"group $F_tau$ does not embed into $F$. Our results also yield a new proof that Stein's groups $F_{p,q}$ do not embed into $F$, a result first established by Lodha using his theory of coherent actions. We develop the basic theory of $F$-obstructions and show that they exhibit certain rigidity phenomena of independent interest. In the course of establishing the main result of the paper, we prove a dichotomy theorem for subgroups of $mathrm{PL}_+ I$. In addition to playing a central role in our proof, it is strong enough to imply both Rubin's Reconstruction Theorem restricted to the class of subgroups of $mathrm{PL}_+ I$ and also Brin's Ubiquity Theorem.","PeriodicalId":55084,"journal":{"name":"Groups Geometry and Dynamics","volume":null,"pages":null},"PeriodicalIF":0.6,"publicationDate":"2021-03-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46591900","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}
引用次数: 0
Erratum to "Generic free subgroups and statistical hyperbolicity" “一般自由子群与统计夸张性”勘误表
IF 0.6 3区 数学
Groups Geometry and Dynamics Pub Date : 2021-03-25 DOI: 10.4171/GGD/594
Suzhen Han, Wen-yuan Yang
{"title":"Erratum to \"Generic free subgroups and statistical hyperbolicity\"","authors":"Suzhen Han, Wen-yuan Yang","doi":"10.4171/GGD/594","DOIUrl":"https://doi.org/10.4171/GGD/594","url":null,"abstract":"","PeriodicalId":55084,"journal":{"name":"Groups Geometry and Dynamics","volume":null,"pages":null},"PeriodicalIF":0.6,"publicationDate":"2021-03-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47254609","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}
引用次数: 0
Finiteness properties for relatives of braided Higman–Thompson groups 编织Higman-Thompson群近亲的有限性质
IF 0.6 3区 数学
Groups Geometry and Dynamics Pub Date : 2021-03-24 DOI: 10.4171/ggd/731
Rachel Skipper, Xiaolei Wu
{"title":"Finiteness properties for relatives of braided Higman–Thompson groups","authors":"Rachel Skipper, Xiaolei Wu","doi":"10.4171/ggd/731","DOIUrl":"https://doi.org/10.4171/ggd/731","url":null,"abstract":"We study the finiteness properties of the braided Higman--Thompson group $bV_{d,r}(H)$ with labels in $Hleq B_d$, and $bF_{d,r}(H)$ and $bT_{d,r}(H)$ with labels in $Hleq PB_d$ where $B_d$ is the braid group with $d$ strings and $PB_d$ is its pure braid subgroup. We show that for all $dgeq 2$ and $rgeq 1$, the group $bV_{d,r}(H)$ (resp. $bT_{d,r}(H)$ or $bF_{d,r}(H)$) is of type $F_n$ if and only if $H$ is. Our result in particular confirms a recent conjecture of Aroca and Cumplido.","PeriodicalId":55084,"journal":{"name":"Groups Geometry and Dynamics","volume":null,"pages":null},"PeriodicalIF":0.6,"publicationDate":"2021-03-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44864634","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}
引用次数: 5
Strongly scale-invariant virtually polycyclic groups 强尺度不变的虚拟多环群
IF 0.6 3区 数学
Groups Geometry and Dynamics Pub Date : 2021-03-10 DOI: 10.4171/ggd/684
Jonas Der'e
{"title":"Strongly scale-invariant virtually polycyclic groups","authors":"Jonas Der'e","doi":"10.4171/ggd/684","DOIUrl":"https://doi.org/10.4171/ggd/684","url":null,"abstract":"A finitely generated group $Gamma$ is called strongly scale-invariant if there exists an injective endomorphism $varphi: Gamma to Gamma$ with the image $varphi(Gamma)$ of finite index in $Gamma$ and the subgroup $displaystyle bigcap_{n>0}varphi^n(Gamma)$ finite. The only known examples of such groups are virtually nilpotent, or equivalently, all examples have polynomial growth. A question by Nekrashevych and Pete asks whether these groups are the only possibilities for such endomorphisms, motivated by the positive answer due to Gromov in the special case of expanding group morphisms. In this paper, we study this question for the class of virtually polycyclic groups, i.e. the virtually solvable groups for which every subgroup is finitely generated. Using the $mathbb{Q}$-algebraic hull, which allows us to extend the injective endomorphisms of certain virtually polycyclic groups to a linear algebraic group, we show that the existence of such an endomorphism implies that the group is virtually nilpotent. Moreover, we fully characterize which virtually nilpotent groups have a morphism satisfying the condition above, related to the existence of a positive grading on the corresponding radicable nilpotent group. As another application of the methods, we generalize a result of Fel'shtyn and Lee about which maps on infra-solvmanifolds can have finite Reidemeister number for all iterates.","PeriodicalId":55084,"journal":{"name":"Groups Geometry and Dynamics","volume":null,"pages":null},"PeriodicalIF":0.6,"publicationDate":"2021-03-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47004503","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}
引用次数: 3
On the Basilica operation 关于大教堂行动
IF 0.6 3区 数学
Groups Geometry and Dynamics Pub Date : 2021-03-09 DOI: 10.4171/ggd/702
Jan Moritz Petschick, Karthik Rajeev
{"title":"On the Basilica operation","authors":"Jan Moritz Petschick, Karthik Rajeev","doi":"10.4171/ggd/702","DOIUrl":"https://doi.org/10.4171/ggd/702","url":null,"abstract":"Inspired by the Basilica group B, we describe a general construction which allows us to associate to any group of automorphisms G ď AutpT q of a rooted tree T a family of Basilica groups BasspGq, s P N`. For the dyadic odometer O2, one has B “ Bas2pO2q. We study which properties of groups acting on rooted trees are preserved under this operation. Introducing some techniques for handling BasspGq, in case G fulfills some branching conditions, we are able to calculate the Hausdorff dimension of the Basilica groups associated to certain GGS-groups and of generalisations of the odometer, O m. Furthermore, we study the structure of groups of type BasspO mq and prove an analogue of the congruence subgroup property in the case m “ p, a prime.","PeriodicalId":55084,"journal":{"name":"Groups Geometry and Dynamics","volume":null,"pages":null},"PeriodicalIF":0.6,"publicationDate":"2021-03-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45161326","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}
引用次数: 4
Virtually free groups are stable in permutations 几乎自由的群在排列中是稳定的
IF 0.6 3区 数学
Groups Geometry and Dynamics Pub Date : 2021-03-09 DOI: 10.4171/ggd/735
Nir Lazarovich, Arie Levit
{"title":"Virtually free groups are stable in permutations","authors":"Nir Lazarovich, Arie Levit","doi":"10.4171/ggd/735","DOIUrl":"https://doi.org/10.4171/ggd/735","url":null,"abstract":"We prove that finitely generated virtually free groups are stable in permutations. As an application, we show that almost-periodic almost-automorphisms of labelled graphs are close to periodic automorphisms.","PeriodicalId":55084,"journal":{"name":"Groups Geometry and Dynamics","volume":null,"pages":null},"PeriodicalIF":0.6,"publicationDate":"2021-03-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42202755","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}
引用次数: 6
Formal conjugacy growth in graph products I 图积的形式共轭增长1
IF 0.6 3区 数学
Groups Geometry and Dynamics Pub Date : 2021-03-08 DOI: 10.4171/ggd/704
L. Ciobanu, S. Hermiller, Valentin Mercier
{"title":"Formal conjugacy growth in graph products I","authors":"L. Ciobanu, S. Hermiller, Valentin Mercier","doi":"10.4171/ggd/704","DOIUrl":"https://doi.org/10.4171/ggd/704","url":null,"abstract":"In this paper we give a recursive formula for the conjugacy growth series of a graph product in terms of the conjugacy growth and standard growth series of subgraph products. We also show that the conjugacy and standard growth rates in a graph product are equal provided that this property holds for each vertex group. All results are obtained for the standard generating set consisting of the union of generating sets of the vertex groups.","PeriodicalId":55084,"journal":{"name":"Groups Geometry and Dynamics","volume":null,"pages":null},"PeriodicalIF":0.6,"publicationDate":"2021-03-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45914791","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}
引用次数: 3
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