We prove for $mgeq1$ and $ngeq5$ that the level $m$ congruence subgroup $B_n[m]$ of the braid group $B_n$ associated to the integral Burau representation $B_ntomathrm{GL}_n(mathbb{Z})$ is generated by $m$th powers of half-twists and the braid Torelli group. This solves a problem of Margalit, generalizing work of Assion, Brendle--Margalit, Nakamura, Stylianakis and Wajnryb.
{"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}
In this paper it is proved that the group $Fleft(frac32right)$, a Thompson-style group with breaks in $mathbb{Z}left[frac16right]$ but whose slopes are restricted only to powers of $frac32$, is finitely generated, with a generating set of two elements.
{"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}
Let m be a cube-free positive integer and let p be a prime such that p does not divide m. In this paper we find the number of conjugacy classes of completely reducible solvable cube-free subgroups in GL(2, q) of order m, where q is a power of p.
让 m 是一个无立方的正整数，让 p 是一个素数，使得 p 不能整除 m。在本文中，我们要找出阶数为 m 的 GL(2, q) 中完全可还原可解的无立方子群的共轭类的数量，其中 q 是 p 的幂。
{"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}
We find an upper bound for the number of groups of order $n$ up to isomorphism in the variety $G = A_pA_qA_r$, where $p$, $q$ and $r$ are distinct primes. We also find a bound on the orders and on the number of conjugacy classes of subgroups that are maximal amongst the subgroups of the general linear group that are also in the variety $A_qA_r$.
{"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}
We study the simplicial volume of manifolds obtained from Davis' reflection group trick, the goal being characterizing those having positive simplicial volume. In particular, we focus on checking whether manifolds in this class with nonzero Euler characteristic have positive simplicial volume (Gromov asked whether this holds in general for aspherical manifolds). This leads to a combinatorial problem about triangulations of spheres: we define a partial order on the set of triangulations -- the relation being the existence of a nonzero-degree simplicial map between two triangulations -- and the problem is to find the minimal elements of a specific subposet. We solve explicitly the case of triangulations of the two-dimensional sphere, and then perform an extensive analysis, with the help of computer searches, of the three-dimensional case. Moreover, we present a connection of this problem with the theory of graph minors.
{"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}
We find a family of groups generated by a pair of parabolic elements in which every relator must admit a long subword of a specific form. In particular, this collection contains groups in which the number of syllables of any relator is arbitrarily large. This suggests that the existing methods for finding non-free groups with rational parabolic generators may be inadequate in this case, as they depend on the presence of relators with few syllables. Our results rely on two variants of the ping-pong lemma that we develop, applicable to groups that are possibly non-free. These variants aim to isolate the group elements responsible for the failure of the classical ping-pong lemma.
{"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}
By providing new finite criteria which certify that a finitely generated subgroup of $mathrm{SL}(d,mathbb{R})$ or $mathrm{SL}(d,mathbb{C})$ is projective Anosov, we obtain a practical algorithm to verify the Anosov condition. We demonstrate on a surface group of genus 2 in $mathrm{SL}(3,mathbb{R})$ by verifying the criteria for all words of length 8. The previous version required checking all words of length $2$ million.
{"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}
We consider $G$ a semisimple Lie group with finite center and $K$ a maximal compact subgroup of $G$. We study the regularity of $K$-finite matrix coefficients of unitary representations of $G$. More precisely, we find the optimal value $kappa(G)$ such that all such coefficients are $kappa(G)$-H"older continuous. The proof relies on analysis of spherical functions of the symmetric Gelfand pair $(G,K)$, using stationary phase estimates from Duistermaat, Kolk and Varadarajan. If $U$ is a compact form of $G$, then $(U,K)$ is a compact symmetric pair. Using the same tools, we study the regularity of $K$-finite coefficients of unitary representations of $U$, improving on previous results obtained by the author.
{"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}
In this paper we study (logical) types and isotypical equivalence of torsion free Abelian groups. We describe all possible types of elements and standard 2-tuples of elements in these groups and classify separable torsion free Abelian groups up to isotypicity.
{"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}
In this paper, we establish the theory of $sigma$-solvable hypergroups, study some properties of $sigma$-solvable hypergroups and give similar results of Hall's Theorem in $sigma$-solvable hypergroups.
{"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}
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