In the present paper we prove sandwich classification for the overgroups of the subsystem subgroup $E(Delta,R)$ of the Chevalley group $G(Phi,R)$ for the three types of pair $(Phi,Delta)$ (the root system and its subsystem) such that the group $G(Delta,R)$ is (up to torus) a Levi subgroup of the parabolic subgroup with abelian unipotent radical. Namely we show that for any such an overgroup $H$ there exists a unique pair of ideals $sigma$ of the ring $R$ such that $E(Phi,Delta,R,sigma)le Hle N_{G(Phi,R)}(E(Phi,Delta,R,sigma))$.
{"title":"Overgroups of Levi subgroups I. The case of abelian unipotent radical","authors":"P. Gvozdevsky","doi":"10.1090/spmj/1631","DOIUrl":"https://doi.org/10.1090/spmj/1631","url":null,"abstract":"In the present paper we prove sandwich classification for the overgroups of the subsystem subgroup $E(Delta,R)$ of the Chevalley group $G(Phi,R)$ for the three types of pair $(Phi,Delta)$ (the root system and its subsystem) such that the group $G(Delta,R)$ is (up to torus) a Levi subgroup of the parabolic subgroup with abelian unipotent radical. Namely we show that for any such an overgroup $H$ there exists a unique pair of ideals $sigma$ of the ring $R$ such that $E(Phi,Delta,R,sigma)le Hle N_{G(Phi,R)}(E(Phi,Delta,R,sigma))$.","PeriodicalId":8427,"journal":{"name":"arXiv: Group Theory","volume":"32 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2019-10-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"87614947","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Let G be a torsion--free abelian group of finite rank. The automorphism group Aut(G) acts on the set of maximal independent subsets of G. The orbits of this action are the isomorphism classes of indecomposable decompositions of G. G contains a direct sum of strongly indecomposable groups as a characteristic subgroup of finite index, giving rise to a classification of finite rank strongly indecomposable torsion--free abelian groups.
{"title":"Torsion-free abelian groups revisited","authors":"P. Schultz","doi":"10.4171/rsmup/67","DOIUrl":"https://doi.org/10.4171/rsmup/67","url":null,"abstract":"Let G be a torsion--free abelian group of finite rank. The automorphism group Aut(G) acts on the set of maximal independent subsets of G. The orbits of this action are the isomorphism classes of indecomposable decompositions of G. G contains a direct sum of strongly indecomposable groups as a characteristic subgroup of finite index, giving rise to a classification of finite rank strongly indecomposable torsion--free abelian groups.","PeriodicalId":8427,"journal":{"name":"arXiv: Group Theory","volume":"29 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2019-09-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"75947421","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2019-09-24DOI: 10.2140/INVOLVE.2021.14.53
T. Braun, C. Crotwell, A. Liu, P. Weston, D. Yetter
We consider the problem of when one quandle homomorphism will factor through another, restricting our attention to the case where all quandles involved are connected. We provide a complete solution to the problem for surjective quandle homomorphisms using the structure theorem for connected quandles of Ehrman et al. (2008) and the factorization system for surjective quandle homomorphsims of Bunch et al. (2010) as our primary tools. The paper contains the substantive results obtained by an REU research group consisting of the first four authors under the mentorship of the fifth, and was supported by National Science Foundation, grant DMS-1659123.
我们考虑了一个双核同态何时会因子化另一个双核同态的问题，将我们的注意力限制在所有双核都是连通的情况下。我们使用Ehrman et al.(2008)的连通量子堆的结构定理和Bunch et al.(2010)的满射量子堆同态的分解系统作为我们的主要工具，提供了满射量子堆同态问题的完整解。本文包含了由前4位作者组成的REU课题组在第5位作者的指导下获得的实质性成果，并得到了美国国家科学基金(基金号:DMS-1659123)的支持。
{"title":"Factorizations of surjective maps of connected quandles","authors":"T. Braun, C. Crotwell, A. Liu, P. Weston, D. Yetter","doi":"10.2140/INVOLVE.2021.14.53","DOIUrl":"https://doi.org/10.2140/INVOLVE.2021.14.53","url":null,"abstract":"We consider the problem of when one quandle homomorphism will factor through another, restricting our attention to the case where all quandles involved are connected. We provide a complete solution to the problem for surjective quandle homomorphisms using the structure theorem for connected quandles of Ehrman et al. (2008) and the factorization system for surjective quandle homomorphsims of Bunch et al. (2010) as our primary tools. The paper contains the substantive results obtained by an REU research group consisting of the first four authors under the mentorship of the fifth, and was supported by National Science Foundation, grant DMS-1659123.","PeriodicalId":8427,"journal":{"name":"arXiv: Group Theory","volume":"134 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2019-09-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"79439031","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We study mapping class groups of infinite type surfaces with isolated punctures and their actions on the loop graphs introduced by Bavard-Walker. We classify all of the mapping classes in these actions which are loxodromic with a WWPD action on the corresponding loop graph. The WWPD property is a weakening of Bestvina-Fujiwara's weak proper discontinuity and is useful for constructing non-trivial quasimorphisms. We use this classification to give a sufficient criterion for subgroups of big mapping class groups to have infinite-dimensional second bounded cohomology and use this criterion to give simple proofs that certain natural subgroups of big mapping class groups have infinite-dimensional second bounded cohomology.
{"title":"WWPD elements of big mapping class groups","authors":"Alexander J. Rasmussen","doi":"10.4171/ggd/613","DOIUrl":"https://doi.org/10.4171/ggd/613","url":null,"abstract":"We study mapping class groups of infinite type surfaces with isolated punctures and their actions on the loop graphs introduced by Bavard-Walker. We classify all of the mapping classes in these actions which are loxodromic with a WWPD action on the corresponding loop graph. The WWPD property is a weakening of Bestvina-Fujiwara's weak proper discontinuity and is useful for constructing non-trivial quasimorphisms. We use this classification to give a sufficient criterion for subgroups of big mapping class groups to have infinite-dimensional second bounded cohomology and use this criterion to give simple proofs that certain natural subgroups of big mapping class groups have infinite-dimensional second bounded cohomology.","PeriodicalId":8427,"journal":{"name":"arXiv: Group Theory","volume":"1 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2019-09-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"76247946","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We introduce and study a local combinatorial condition, called the 5/9-condition, on a simplicial complex, implying Gromov hyperbolicity of its universal cover. We hereby give an application of another combinatorial condition, called 8-location, introduced by Damian Osajda. Along the way we prove the minimal filling diagram lemma for 5/9-complexes.
{"title":"Minimal Disc Diagrams of 5 / 9 -Simplicial Complexes","authors":"I. Lazăr","doi":"10.1307/mmj/1585706557","DOIUrl":"https://doi.org/10.1307/mmj/1585706557","url":null,"abstract":"We introduce and study a local combinatorial condition, called the 5/9-condition, on a simplicial complex, implying Gromov hyperbolicity of its universal cover. We hereby give an application of another combinatorial condition, called 8-location, introduced by Damian Osajda. Along the way we prove the minimal filling diagram lemma for 5/9-complexes.","PeriodicalId":8427,"journal":{"name":"arXiv: Group Theory","volume":"26 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2019-09-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"73012747","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2019-08-20DOI: 10.2140/ant.2020.14.2185
M. Larsen
If w is a word in d>1 letters and G is a finite group, evaluation of w on a uniformly randomly chosen d-tuple in G gives a random variable with values in G, which may or may not be uniform. It is known that if G ranges over finite simple groups of given root system and characteristic, a positive proportion of words w give a distribution which approaches uniformity in the limit as |G| goes to infinity. In this paper, we show that the proportion is in fact 1.
{"title":"Most words are geometrically almost uniform","authors":"M. Larsen","doi":"10.2140/ant.2020.14.2185","DOIUrl":"https://doi.org/10.2140/ant.2020.14.2185","url":null,"abstract":"If w is a word in d>1 letters and G is a finite group, evaluation of w on a uniformly randomly chosen d-tuple in G gives a random variable with values in G, which may or may not be uniform. It is known that if G ranges over finite simple groups of given root system and characteristic, a positive proportion of words w give a distribution which approaches uniformity in the limit as |G| goes to infinity. In this paper, we show that the proportion is in fact 1.","PeriodicalId":8427,"journal":{"name":"arXiv: Group Theory","volume":"8 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2019-08-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"75211210","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2019-08-01DOI: 10.22108/IJGT.2021.129815.1708
C. Kumar, S. Pradhan
For an odd prime $p$ and a positive integer $n$, it is well known that there are two types of extra-special $p$-groups of order $p^{2n+1}$, first one is the Heisenberg group which has exponent $p$ and the second one is of exponent $p^2$. In this article, a new way of representing the extra-special $p$-group of exponent $p^2$ is given. These representations facilitate an explicit way of finding formulae for any endomorphism and any automorphism of an extra-special $p$-group $G$ for both the types. Based on these formulae, the endomorphism semigroup $End(G)$ and the automorphism group $Aut(G)$ are described. The endomorphism semigroup image of any element in $G$ is found and the orbits under the action of the automorphism group $Aut(G)$ are determined. As a consequence it is deduced that, under the notion of degeneration of elements in $G$, the endomorphism semigroup $End(G)$ induces a partial order on the automorphism orbits when $G$ is the Heisenberg group and does not induce when $G$ is the extra-special $p$-group of exponent $p^2$. Finally we prove that the cardinality of isotropic subspaces of any fixed dimension in a non-degenerate symplectic space is a polynomial in $p$ with non-negative integer coefficients. Using this fact we compute the cardinality of $End(G)$.
{"title":"On the Endomorphism Semigroups of Extra-special $p$-groups and Automorphism Orbits.","authors":"C. Kumar, S. Pradhan","doi":"10.22108/IJGT.2021.129815.1708","DOIUrl":"https://doi.org/10.22108/IJGT.2021.129815.1708","url":null,"abstract":"For an odd prime $p$ and a positive integer $n$, it is well known that there are two types of extra-special $p$-groups of order $p^{2n+1}$, first one is the Heisenberg group which has exponent $p$ and the second one is of exponent $p^2$. In this article, a new way of representing the extra-special $p$-group of exponent $p^2$ is given. These representations facilitate an explicit way of finding formulae for any endomorphism and any automorphism of an extra-special $p$-group $G$ for both the types. Based on these formulae, the endomorphism semigroup $End(G)$ and the automorphism group $Aut(G)$ are described. The endomorphism semigroup image of any element in $G$ is found and the orbits under the action of the automorphism group $Aut(G)$ are determined. As a consequence it is deduced that, under the notion of degeneration of elements in $G$, the endomorphism semigroup $End(G)$ induces a partial order on the automorphism orbits when $G$ is the Heisenberg group and does not induce when $G$ is the extra-special $p$-group of exponent $p^2$. Finally we prove that the cardinality of isotropic subspaces of any fixed dimension in a non-degenerate symplectic space is a polynomial in $p$ with non-negative integer coefficients. Using this fact we compute the cardinality of $End(G)$.","PeriodicalId":8427,"journal":{"name":"arXiv: Group Theory","volume":"27 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2019-08-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"81368585","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2019-07-25DOI: 10.22108/IJGT.2020.124368.1643
R. Bastos, C. Monetta
Let $n$ be a positive integer and let $G$ be a group. We denote by $nu(G)$ a certain extension of the non-abelian tensor square $G otimes G$ by $G times G$. Set $T_{otimes}(G) = {g otimes h mid g,h in G}$. We prove that if the size of the conjugacy class $left |x^{nu(G)} right| leq n$ for every $x in T_{otimes}(G)$, then the second derived subgroup $nu(G)''$ is finite with $n$-bounded order. Moreover, we obtain a sufficient condition for a group to be a BFC-group.
设$n$为正整数，设$G$为一个组。我们用$nu(G)$表示非阿贝尔张量平方的某个扩展$G otimes G$乘以$G times G$。设置$T_{otimes}(G) = {g otimes h mid g,h in G}$。证明了如果共轭类$left |x^{nu(G)} right| leq n$对于每一个$x in T_{otimes}(G)$的大小，则第二派生子群$nu(G)''$是有限的，阶为$n$ -有界。此外，还得到了群为bfc群的充分条件。
{"title":"Boundedly finite conjugacy classes of tensors.","authors":"R. Bastos, C. Monetta","doi":"10.22108/IJGT.2020.124368.1643","DOIUrl":"https://doi.org/10.22108/IJGT.2020.124368.1643","url":null,"abstract":"Let $n$ be a positive integer and let $G$ be a group. We denote by $nu(G)$ a certain extension of the non-abelian tensor square $G otimes G$ by $G times G$. Set $T_{otimes}(G) = {g otimes h mid g,h in G}$. We prove that if the size of the conjugacy class $left |x^{nu(G)} right| leq n$ for every $x in T_{otimes}(G)$, then the second derived subgroup $nu(G)''$ is finite with $n$-bounded order. Moreover, we obtain a sufficient condition for a group to be a BFC-group.","PeriodicalId":8427,"journal":{"name":"arXiv: Group Theory","volume":"289 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2019-07-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"79440084","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2019-07-14DOI: 10.2140/agt.2020.20.3083
Elizabeth B Field
When 1 -> H -> G -> Q -> 1 is a short exact sequence of three infinite, word-hyperbolic groups, Mahan Mitra (Mj) has shown that the inclusion map from H to G extends continuously to a map between the Gromov boundaries of H and G. This boundary map is known as the Cannon-Thurston map. In this context, Mitra associates to every point z in the Gromov boundary of Q an ''ending lamination'' on H which consists of pairs of distinct points in the boundary of H. We prove that for each such z, the quotient of the Gromov boundary of H by the equivalence relation generated by this ending lamination is a dendrite, that is, a tree-like topological space. This result generalizes the work of Kapovich-Lustig and Dowdall-Kapovich-Taylor, who prove that in the case where H is a free group and Q is a convex cocompact purely atoroidal subgroup of Out(F_N), one can identify the resultant quotient space with a certain $mathbb{R}$-tree in the boundary of Culler-Vogtmann's Outer space.
当1 -> H -> G -> Q -> 1是三个无限双曲群的短精确序列时，Mahan Mitra (Mj)已经证明了从H到G的包含映射连续地延伸到H和G的Gromov边界之间的映射。这个边界映射被称为Cannon-Thurston映射。在这种情况下，Mitra将Q的Gromov边界上的每一个点z与H上的一个由H边界上不同的点对组成的“结束层合”联系起来，证明了对于每一个这样的z，通过该结束层合生成的等价关系，H的Gromov边界的商是一个树形拓扑空间，即树状拓扑空间。这个结果推广了kapoovich - lustig和dowdll - kapoovich - taylor的工作，他们证明了在H是自由群，Q是Out(F_N)的凸紧纯阿托向子群的情况下，可以用Culler-Vogtmann外空间边界上的某$mathbb{R}$-树来识别合成商空间。
{"title":"Trees, dendrites and the Cannon–Thurston\u0000map","authors":"Elizabeth B Field","doi":"10.2140/agt.2020.20.3083","DOIUrl":"https://doi.org/10.2140/agt.2020.20.3083","url":null,"abstract":"When 1 -> H -> G -> Q -> 1 is a short exact sequence of three infinite, word-hyperbolic groups, Mahan Mitra (Mj) has shown that the inclusion map from H to G extends continuously to a map between the Gromov boundaries of H and G. This boundary map is known as the Cannon-Thurston map. In this context, Mitra associates to every point z in the Gromov boundary of Q an ''ending lamination'' on H which consists of pairs of distinct points in the boundary of H. We prove that for each such z, the quotient of the Gromov boundary of H by the equivalence relation generated by this ending lamination is a dendrite, that is, a tree-like topological space. This result generalizes the work of Kapovich-Lustig and Dowdall-Kapovich-Taylor, who prove that in the case where H is a free group and Q is a convex cocompact purely atoroidal subgroup of Out(F_N), one can identify the resultant quotient space with a certain $mathbb{R}$-tree in the boundary of Culler-Vogtmann's Outer space.","PeriodicalId":8427,"journal":{"name":"arXiv: Group Theory","volume":"54 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2019-07-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"86776316","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Strongly bounded groups are those groups for which every action by isometries on a metric space has orbits of finite diameter. Many groups have been shown to have this property, and all the known infinite examples so far have cardinality at least $2^{aleph_0}$. We produce examples of strongly bounded groups of many cardinalities, including $aleph_1$, answering a question of Yves de Cornulier [4]. In fact, any infinite group embeds as a subgroup of a strongly bounded group which is, at most, two cardinalities larger.
强有界群是指在度量空间上的每一个等距作用都具有有限直径轨道的群。许多群已经被证明具有这个性质，并且到目前为止所有已知的无限例子的基数至少为$2^{aleph_0}$。我们给出了许多基数的强有界群的例子，包括$aleph_1$，回答了Yves de Cornulier[4]的问题。事实上，任何无限群都嵌入为强有界群的子群，强有界群最多比它大两个基数。
{"title":"Strongly bounded groups of various cardinalities","authors":"Samuel M. Corson, S. Shelah","doi":"10.1090/proc/14998","DOIUrl":"https://doi.org/10.1090/proc/14998","url":null,"abstract":"Strongly bounded groups are those groups for which every action by isometries on a metric space has orbits of finite diameter. Many groups have been shown to have this property, and all the known infinite examples so far have cardinality at least $2^{aleph_0}$. We produce examples of strongly bounded groups of many cardinalities, including $aleph_1$, answering a question of Yves de Cornulier [4]. In fact, any infinite group embeds as a subgroup of a strongly bounded group which is, at most, two cardinalities larger.","PeriodicalId":8427,"journal":{"name":"arXiv: Group Theory","volume":"63 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2019-06-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"74125278","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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