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The algebraic cheap rebuilding property 代数廉价重建特性
arXiv - MATH - Group Theory Pub Date : 2024-09-09 DOI: arxiv-2409.05774
Kevin Li, Clara Loeh, Marco Moraschini, Roman Sauer, Matthias Uschold
{"title":"The algebraic cheap rebuilding property","authors":"Kevin Li, Clara Loeh, Marco Moraschini, Roman Sauer, Matthias Uschold","doi":"arxiv-2409.05774","DOIUrl":"https://doi.org/arxiv-2409.05774","url":null,"abstract":"We present an axiomatic approach to combination theorems for various\u0000homological properties of groups and, more generally, of chain complexes.\u0000Examples of such properties include algebraic finiteness properties,\u0000$ell^2$-invisibility, $ell^2$-acyclicity, lower bounds for Novikov--Shubin\u0000invariants, and vanishing of homology growth. We introduce an algebraic version\u0000of Ab'ert--Bergeron--Frk{a}czyk--Gaboriau's cheap rebuilding property that\u0000implies vanishing of torsion homology growth and admits a combination theorem.\u0000As an application, we show that certain graphs of groups with amenable vertex\u0000groups and elementary amenable edge groups have vanishing torsion homology\u0000growth.","PeriodicalId":501037,"journal":{"name":"arXiv - MATH - Group Theory","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2024-09-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142206441","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
Positive entropy actions by higher-rank lattices 高阶网格的正熵作用
arXiv - MATH - Group Theory Pub Date : 2024-09-09 DOI: arxiv-2409.05991
Aaron Brown, Homin Lee
{"title":"Positive entropy actions by higher-rank lattices","authors":"Aaron Brown, Homin Lee","doi":"arxiv-2409.05991","DOIUrl":"https://doi.org/arxiv-2409.05991","url":null,"abstract":"We study smooth actions by lattices $Gamma$ in higher-rank simple Lie groups\u0000$G$ assuming one element of the action acts with positive topological entropy\u0000and prove a number of new rigidity results. For lattices $Gamma$ in\u0000$mathrm{SL}(n,mathbb{R})$ acting on $n$-manifolds, if the action has positive\u0000topological entropy we show the lattice must be commensurable with\u0000$mathrm{SL}(n,mathbb{Z})$. Moreover, such actions admit an absolutely\u0000continuous probability measure with positive metric entropy; adapting arguments\u0000by Katok and Rodriguez Hertz, we show such actions are measurably conjugate to\u0000affine actions on (infra)tori. In our main technical arguments, we study families of probability measures\u0000invariant under sub-actions of the induced $G$-action on an associated fiber\u0000bundle. To control entropy properties of such measures, in the appendix we\u0000establish certain upper semicontinuity of entropy under weak-$*$ convergence,\u0000adapting classical results of Yomdin and Newhouse.","PeriodicalId":501037,"journal":{"name":"arXiv - MATH - Group Theory","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2024-09-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142206440","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
Non-planar ends are continuously unforgettable 非平面末端持续令人难忘
arXiv - MATH - Group Theory Pub Date : 2024-09-09 DOI: arxiv-2409.05502
Javier Aramayona, Rodrigo De Pool, Rachel Skipper, Jing Tao, Nicholas G. Vlamis, Xiaolei Wu
{"title":"Non-planar ends are continuously unforgettable","authors":"Javier Aramayona, Rodrigo De Pool, Rachel Skipper, Jing Tao, Nicholas G. Vlamis, Xiaolei Wu","doi":"arxiv-2409.05502","DOIUrl":"https://doi.org/arxiv-2409.05502","url":null,"abstract":"We show that continuous epimorphisms between a class of subgroups of mapping\u0000class groups of orientable infinite-genus 2-manifolds with no planar ends are\u0000always induced by homeomorphisms. This class of subgroups includes the pure\u0000mapping class group, the closure of the compactly supported mapping classes,\u0000and the full mapping class group in the case that the underlying manifold has a\u0000finite number of ends or is perfectly self-similar. As a corollary, these\u0000groups are Hopfian topological groups.","PeriodicalId":501037,"journal":{"name":"arXiv - MATH - Group Theory","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2024-09-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142206444","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
Affine groups as flag-transitive and point-primitive automorphism groups of symmetric designs 仿射群作为对称设计的旗变换群和点原初自变群
arXiv - MATH - Group Theory Pub Date : 2024-09-07 DOI: arxiv-2409.04790
Seyed Hassan Alavi, Mohsen Bayat, Ashraf Daneshkhah, Alessandro Montinaro
{"title":"Affine groups as flag-transitive and point-primitive automorphism groups of symmetric designs","authors":"Seyed Hassan Alavi, Mohsen Bayat, Ashraf Daneshkhah, Alessandro Montinaro","doi":"arxiv-2409.04790","DOIUrl":"https://doi.org/arxiv-2409.04790","url":null,"abstract":"In this article, we investigate symmetric designs admitting a flag-transitive\u0000and point-primitive affine automorphism group. We prove that if an automorphism\u0000group $G$ of a symmetric $(v,k,lambda)$ design with $lambda$ prime is\u0000point-primitive of affine type, then $G=2^{6}{:}mathrm{S}_{6}$ and\u0000$(v,k,lambda)=(16,6,2)$, or $G$ is a subgroup of $mathrm{AGamma L}_{1}(q)$\u0000for some odd prime power $q$. In conclusion, we present a classification of\u0000flag-transitive and point-primitive symmetric designs with $lambda$ prime,\u0000which says that such an incidence structure is a projective space\u0000$mathrm{PG}(n,q)$, it has parameter set $(15,7,3)$, $(7, 4, 2)$, $(11, 5, 2)$,\u0000$(11, 6, 2)$, $(16,6,2)$ or $(45, 12, 3)$, or $v=p^d$ where $p$ is an odd prime\u0000and the automorphism group is a subgroup of $mathrm{AGamma L}_{1}(q)$.","PeriodicalId":501037,"journal":{"name":"arXiv - MATH - Group Theory","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2024-09-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142206459","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 Thin Heckoid and Generalised Triangle Groups in $PSL(2,mathbb{C})$} 论$PSL(2,mathbb{C})$中的薄赫克洛德群和广义三角形群｝
arXiv - MATH - Group Theory Pub Date : 2024-09-06 DOI: arxiv-2409.04438
Alex Elzenaar, Gaven Martin, Jeroen Schillewaert
{"title":"On Thin Heckoid and Generalised Triangle Groups in $PSL(2,mathbb{C})$}","authors":"Alex Elzenaar, Gaven Martin, Jeroen Schillewaert","doi":"arxiv-2409.04438","DOIUrl":"https://doi.org/arxiv-2409.04438","url":null,"abstract":"We provide a brief overview of our upcoming work identifying all the thin\u0000Heckoid groups in $PSL(2,mathbb{C})$. Here we give a complete list of the $55$\u0000thin generalised triangle groups of slope $1/2$. This work was presented at the\u0000conference Computational Aspects of Thin Groups, IMSS, Singapore and presents\u0000an application of joint work initiated with Colin Maclachlan","PeriodicalId":501037,"journal":{"name":"arXiv - MATH - Group Theory","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2024-09-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142206461","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 the dimension of Harer's spine for the decorated Teichmüller space 论装饰的泰希米勒空间的哈勒脊线维度
arXiv - MATH - Group Theory Pub Date : 2024-09-06 DOI: arxiv-2409.04392
Nestor Colin, Rita Jiménez Rolland, Porfirio L. León Álvarez, Luis Jorge Sánchez Saldaña
{"title":"On the dimension of Harer's spine for the decorated Teichmüller space","authors":"Nestor Colin, Rita Jiménez Rolland, Porfirio L. León Álvarez, Luis Jorge Sánchez Saldaña","doi":"arxiv-2409.04392","DOIUrl":"https://doi.org/arxiv-2409.04392","url":null,"abstract":"In cite{Ha86} Harer explicitly constructed a spine for the decorated\u0000Teichm\"uller space of orientable surfaces with at least one puncture and\u0000negative Euler characteristic. In this paper we point out some instances where\u0000his computation of the dimension of this spine is off by $1$ and give the\u0000correct dimension.","PeriodicalId":501037,"journal":{"name":"arXiv - MATH - Group Theory","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2024-09-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142206463","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
Local control and Bogomolov multipliers of finite groups 有限群的局部控制和博戈莫洛夫乘数
arXiv - MATH - Group Theory Pub Date : 2024-09-06 DOI: arxiv-2409.04274
Primoz Moravec
{"title":"Local control and Bogomolov multipliers of finite groups","authors":"Primoz Moravec","doi":"arxiv-2409.04274","DOIUrl":"https://doi.org/arxiv-2409.04274","url":null,"abstract":"We show that if a Sylow $p$-subgroup of a finite group $G$ is nilpotent of\u0000class at most $p$, then the $p$-part of the Bogomolov multiplier of $G$ is\u0000locally controlled.","PeriodicalId":501037,"journal":{"name":"arXiv - MATH - Group Theory","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2024-09-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142206462","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 locally compact shift-continuous topologies on semigroups $mathscr{C}_{+}(a,b)$ and $mathscr{C}_{-}(a,b)$ with adjoined zero 关于半群$mathscr{C}_{+}(a,b)$和$mathscr{C}_{-}(a,b)$上有邻接零的局部紧凑移位连续拓扑学
arXiv - MATH - Group Theory Pub Date : 2024-09-05 DOI: arxiv-2409.03490
Oleg Gutik
{"title":"On locally compact shift-continuous topologies on semigroups $mathscr{C}_{+}(a,b)$ and $mathscr{C}_{-}(a,b)$ with adjoined zero","authors":"Oleg Gutik","doi":"arxiv-2409.03490","DOIUrl":"https://doi.org/arxiv-2409.03490","url":null,"abstract":"Let $mathscr{C}_{+}(p,q)^0$ and $mathscr{C}_{-}(p,q)^0$ be the semigroups\u0000$mathscr{C}_{+}(a,b)$ and $mathscr{C}_{-}(a,b)$ with the adjoined zero. We\u0000show that the semigroups $mathscr{C}_{+}(p,q)^0$ and $mathscr{C}_{-}(p,q)^0$\u0000admit continuum many different Hausdorff locally compact shift-continuous\u0000topologies up to topological isomorphism.","PeriodicalId":501037,"journal":{"name":"arXiv - MATH - Group Theory","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2024-09-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142206465","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 Heisenberg groups 关于海森堡群
arXiv - MATH - Group Theory Pub Date : 2024-09-05 DOI: arxiv-2409.03399
Florian L. Deloup
{"title":"On Heisenberg groups","authors":"Florian L. Deloup","doi":"arxiv-2409.03399","DOIUrl":"https://doi.org/arxiv-2409.03399","url":null,"abstract":"It is known that an abelian group $A$ and a $2$-cocycle $c:A times A to C$\u0000yield a group ${mathscr{H}}(A,C,c)$ which we call a Heisenberg group. This\u0000group, a central extension of $A$, is the archetype of a class~$2$ nilpotent\u0000group. In this note, we prove that under mild conditions, any class~$2$\u0000nilpotent group $G$ is equivalent as an extension of $G/[G,G]$ to a Heisenberg\u0000group ${mathscr{H}}(G/[G,G], [G,G], c')$ whose $2$-cocycle $c'$ is\u0000bimultiplicative.","PeriodicalId":501037,"journal":{"name":"arXiv - MATH - Group Theory","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2024-09-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142206466","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
Derangements in non-Frobenius groups 非弗罗贝纽斯群中的异变
arXiv - MATH - Group Theory Pub Date : 2024-09-05 DOI: arxiv-2409.03305
Daniele Garzoni
{"title":"Derangements in non-Frobenius groups","authors":"Daniele Garzoni","doi":"arxiv-2409.03305","DOIUrl":"https://doi.org/arxiv-2409.03305","url":null,"abstract":"We prove that if $G$ is a transitive permutation group of sufficiently large\u0000degree $n$, then either $G$ is primitive and Frobenius, or the proportion of\u0000derangements in $G$ is larger than $1/(2n^{1/2})$. This is sharp, generalizes\u0000substantially bounds of Cameron--Cohen and Guralnick--Wan, and settles a\u0000conjecture of Guralnick--Tiep in large degree. We also give an application to\u0000coverings of varieties over finite fields.","PeriodicalId":501037,"journal":{"name":"arXiv - MATH - Group Theory","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2024-09-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142206468","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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