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Generators for the level $m$ congruence subgroups of braid groups 辫状群 m$ 级同余子群的生成器
arXiv - MATH - Group Theory Pub Date : 2024-09-15 DOI: arxiv-2409.09612
Ishan Banerjee, Peter Huxford
{"title":"Generators for the level $m$ congruence subgroups of braid groups","authors":"Ishan Banerjee, Peter Huxford","doi":"arxiv-2409.09612","DOIUrl":"https://doi.org/arxiv-2409.09612","url":null,"abstract":"We prove for $mgeq1$ and $ngeq5$ that the level $m$ congruence subgroup\u0000$B_n[m]$ of the braid group $B_n$ associated to the integral Burau\u0000representation $B_ntomathrm{GL}_n(mathbb{Z})$ is generated by $m$th powers\u0000of half-twists and the braid Torelli group. This solves a problem of Margalit,\u0000generalizing work of Assion, Brendle--Margalit, Nakamura, Stylianakis and\u0000Wajnryb.","PeriodicalId":501037,"journal":{"name":"arXiv - MATH - Group Theory","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2024-09-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142268856","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}
引用次数: 0
Finite generation for the group $Fleft(frac32right)$ 组 $Fleft(frac32right)$ 的有限生成
arXiv - MATH - Group Theory Pub Date : 2024-09-13 DOI: arxiv-2409.09195
José Burillo, Marc Felipe
{"title":"Finite generation for the group $Fleft(frac32right)$","authors":"José Burillo, Marc Felipe","doi":"arxiv-2409.09195","DOIUrl":"https://doi.org/arxiv-2409.09195","url":null,"abstract":"In this paper it is proved that the group $Fleft(frac32right)$, a\u0000Thompson-style group with breaks in $mathbb{Z}left[frac16right]$ but whose\u0000slopes are restricted only to powers of $frac32$, is finitely generated, with\u0000a generating set of two elements.","PeriodicalId":501037,"journal":{"name":"arXiv - MATH - Group Theory","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2024-09-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142253759","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}
引用次数: 0
Enumeration of groups in some special varieties of $A$-groups A$ 群的一些特殊品种中的群枚举
arXiv - MATH - Group Theory Pub Date : 2024-09-13 DOI: arxiv-2409.08586
Arushi, Geetha Venkataraman
{"title":"Enumeration of groups in some special varieties of $A$-groups","authors":"Arushi, Geetha Venkataraman","doi":"arxiv-2409.08586","DOIUrl":"https://doi.org/arxiv-2409.08586","url":null,"abstract":"We find an upper bound for the number of groups of order $n$ up to\u0000isomorphism in the variety $G = A_pA_qA_r$, where $p$, $q$ and $r$ are distinct\u0000primes. We also find a bound on the orders and on the number of conjugacy\u0000classes of subgroups that are maximal amongst the subgroups of the general\u0000linear group that are also in the variety $A_qA_r$.","PeriodicalId":501037,"journal":{"name":"arXiv - MATH - Group Theory","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2024-09-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142253760","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}
引用次数: 0
Conjugacy classes of completely reducible cube-free solvable p'-subgroups of GL(2, q) GL(2, q) 的完全还原无立方可解 p'- 子群的共轭类
arXiv - MATH - Group Theory Pub Date : 2024-09-13 DOI: arxiv-2409.08571
Prashun Kumar, Geetha Venkataraman
{"title":"Conjugacy classes of completely reducible cube-free solvable p'-subgroups of GL(2, q)","authors":"Prashun Kumar, Geetha Venkataraman","doi":"arxiv-2409.08571","DOIUrl":"https://doi.org/arxiv-2409.08571","url":null,"abstract":"Let m be a cube-free positive integer and let p be a prime such that p does\u0000not divide m. In this paper we find the number of conjugacy classes of\u0000completely reducible solvable cube-free subgroups in GL(2, q) of order m, where\u0000q is a power of p.","PeriodicalId":501037,"journal":{"name":"arXiv - MATH - Group Theory","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2024-09-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142268858","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}
引用次数: 0
Simplicial maps between spheres and Davis' manifolds with positive simplicial volume 球面与戴维斯流形之间的简易映射，具有正简易体积
arXiv - MATH - Group Theory Pub Date : 2024-09-12 DOI: arxiv-2409.08336
Francesco Milizia
{"title":"Simplicial maps between spheres and Davis' manifolds with positive simplicial volume","authors":"Francesco Milizia","doi":"arxiv-2409.08336","DOIUrl":"https://doi.org/arxiv-2409.08336","url":null,"abstract":"We study the simplicial volume of manifolds obtained from Davis' reflection\u0000group trick, the goal being characterizing those having positive simplicial\u0000volume. In particular, we focus on checking whether manifolds in this class\u0000with nonzero Euler characteristic have positive simplicial volume (Gromov asked\u0000whether this holds in general for aspherical manifolds). This leads to a\u0000combinatorial problem about triangulations of spheres: we define a partial\u0000order on the set of triangulations -- the relation being the existence of a\u0000nonzero-degree simplicial map between two triangulations -- and the problem is\u0000to find the minimal elements of a specific subposet. We solve explicitly the\u0000case of triangulations of the two-dimensional sphere, and then perform an\u0000extensive analysis, with the help of computer searches, of the\u0000three-dimensional case. Moreover, we present a connection of this problem with\u0000the theory of graph minors.","PeriodicalId":501037,"journal":{"name":"arXiv - MATH - Group Theory","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2024-09-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142253762","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}
引用次数: 0
Long relators in groups generated by two parabolic elements 由两个抛物线元素生成的组中的长关系线
arXiv - MATH - Group Theory Pub Date : 2024-09-12 DOI: arxiv-2409.08086
Rotem Yaari
{"title":"Long relators in groups generated by two parabolic elements","authors":"Rotem Yaari","doi":"arxiv-2409.08086","DOIUrl":"https://doi.org/arxiv-2409.08086","url":null,"abstract":"We find a family of groups generated by a pair of parabolic elements in which\u0000every relator must admit a long subword of a specific form. In particular, this\u0000collection contains groups in which the number of syllables of any relator is\u0000arbitrarily large. This suggests that the existing methods for finding non-free\u0000groups with rational parabolic generators may be inadequate in this case, as\u0000they depend on the presence of relators with few syllables. Our results rely on\u0000two variants of the ping-pong lemma that we develop, applicable to groups that\u0000are possibly non-free. These variants aim to isolate the group elements\u0000responsible for the failure of the classical ping-pong lemma.","PeriodicalId":501037,"journal":{"name":"arXiv - MATH - Group Theory","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2024-09-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142206427","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}
引用次数: 0
Certifying Anosov representations 认证阿诺索夫陈述
arXiv - MATH - Group Theory Pub Date : 2024-09-12 DOI: arxiv-2409.08015
J. Maxwell Riestenberg
{"title":"Certifying Anosov representations","authors":"J. Maxwell Riestenberg","doi":"arxiv-2409.08015","DOIUrl":"https://doi.org/arxiv-2409.08015","url":null,"abstract":"By providing new finite criteria which certify that a finitely generated\u0000subgroup of $mathrm{SL}(d,mathbb{R})$ or $mathrm{SL}(d,mathbb{C})$ is\u0000projective Anosov, we obtain a practical algorithm to verify the Anosov\u0000condition. We demonstrate on a surface group of genus 2 in\u0000$mathrm{SL}(3,mathbb{R})$ by verifying the criteria for all words of length\u00008. The previous version required checking all words of length $2$ million.","PeriodicalId":501037,"journal":{"name":"arXiv - MATH - Group Theory","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2024-09-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142206426","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}
引用次数: 0
Regularity of K-finite matrix coefficients of semisimple Lie groups 半简单李群 K 有限矩阵系数的规律性
arXiv - MATH - Group Theory Pub Date : 2024-09-12 DOI: arxiv-2409.07944
Guillaume Dumas
{"title":"Regularity of K-finite matrix coefficients of semisimple Lie groups","authors":"Guillaume Dumas","doi":"arxiv-2409.07944","DOIUrl":"https://doi.org/arxiv-2409.07944","url":null,"abstract":"We consider $G$ a semisimple Lie group with finite center and $K$ a maximal\u0000compact subgroup of $G$. We study the regularity of $K$-finite matrix\u0000coefficients of unitary representations of $G$. More precisely, we find the\u0000optimal value $kappa(G)$ such that all such coefficients are\u0000$kappa(G)$-H\"older continuous. The proof relies on analysis of spherical\u0000functions of the symmetric Gelfand pair $(G,K)$, using stationary phase\u0000estimates from Duistermaat, Kolk and Varadarajan. If $U$ is a compact form of\u0000$G$, then $(U,K)$ is a compact symmetric pair. Using the same tools, we study\u0000the regularity of $K$-finite coefficients of unitary representations of $U$,\u0000improving on previous results obtained by the author.","PeriodicalId":501037,"journal":{"name":"arXiv - MATH - Group Theory","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2024-09-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142206429","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}
引用次数: 0
Types in torsion free Abelian groups 无扭阿贝尔群中的类型
arXiv - MATH - Group Theory Pub Date : 2024-09-12 DOI: arxiv-2409.07728
Elena Bunina
{"title":"Types in torsion free Abelian groups","authors":"Elena Bunina","doi":"arxiv-2409.07728","DOIUrl":"https://doi.org/arxiv-2409.07728","url":null,"abstract":"In this paper we study (logical) types and isotypical equivalence of torsion\u0000free Abelian groups. We describe all possible types of elements and standard\u00002-tuples of elements in these groups and classify separable torsion free\u0000Abelian groups up to isotypicity.","PeriodicalId":501037,"journal":{"name":"arXiv - MATH - Group Theory","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2024-09-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142206430","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}
引用次数: 0
On $σ$-solvable hypergroups and related Hall's Theorem 关于σ$可解超群和相关霍尔定理
arXiv - MATH - Group Theory Pub Date : 2024-09-12 DOI: arxiv-2409.07778
Chi Zhang, Wenbin Guo
{"title":"On $σ$-solvable hypergroups and related Hall's Theorem","authors":"Chi Zhang, Wenbin Guo","doi":"arxiv-2409.07778","DOIUrl":"https://doi.org/arxiv-2409.07778","url":null,"abstract":"In this paper, we establish the theory of $sigma$-solvable hypergroups,\u0000study some properties of $sigma$-solvable hypergroups and give similar results\u0000of Hall's Theorem in $sigma$-solvable hypergroups.","PeriodicalId":501037,"journal":{"name":"arXiv - MATH - Group Theory","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2024-09-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142206428","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}
引用次数: 0
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