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.
{"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}
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.
{"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}
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.
{"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}
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.
{"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}
{"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}
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.
{"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}
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.
如果在$Gamma$中存在一个具有有限索引像$varphi(Gamma)$的内射自同态$varphi: Gamma to Gamma$，且子群$displaystyle bigcap_{n>0}varphi^n(Gamma)$有限，则称有限生成群$Gamma$为强尺度不变群。这些群的唯一已知的例子实际上是幂零的，或者等价地说，所有的例子都是多项式增长的。Nekrashevych和Pete提出的一个问题是，这些群是否是这种自同态的唯一可能性，其动机是Gromov在扩展群态射的特殊情况下给出的正答案。本文研究了一类虚多环群，即每一子群都有限生成的虚可解群的这一问题。利用$mathbb{Q}$ -代数壳，我们将某些虚多环群的内射自同态推广到一个线性代数群上，证明了这种自同态的存在意味着这个群是虚幂零的。此外，我们充分刻画了哪些虚幂零群具有满足上述条件的态射，这与相应的可根幂零群上存在一个正分级有关。作为该方法的另一个应用，我们推广了Fel'shtyn和Lee关于次可解流形上的映射对所有迭代都具有有限Reidemeister数的结论。
{"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}
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.
受Basilica群B的启发，我们描述了一个一般的结构，它允许我们将一个Basilica群族BasspGq, s P N '与根树T的任意自同构群G G G T T关联起来。对于二元里程计O2，有B " Bas2pO2q。我们研究了作用于根树的群在这种操作下保留了哪些性质。引入一些处理BasspGq的技术，在G满足某些分支条件的情况下，我们能够计算出与某些ggs群相关的Basilica群的Hausdorff维数，以及omom的推广。此外，我们研究了BasspGq型群的结构，并证明了在m ' p, a素数的情况下的同余子群性质的类似。
{"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}
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.
{"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}
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.
{"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}
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