Kevin Li, Clara Loeh, Marco Moraschini, Roman Sauer, Matthias Uschold
We present an axiomatic approach to combination theorems for various homological properties of groups and, more generally, of chain complexes. Examples of such properties include algebraic finiteness properties, $ell^2$-invisibility, $ell^2$-acyclicity, lower bounds for Novikov--Shubin invariants, and vanishing of homology growth. We introduce an algebraic version of Ab'ert--Bergeron--Frk{a}czyk--Gaboriau's cheap rebuilding property that implies vanishing of torsion homology growth and admits a combination theorem. As an application, we show that certain graphs of groups with amenable vertex groups and elementary amenable edge groups have vanishing torsion homology growth.
{"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":"25 1","pages":""},"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}
We study smooth actions by lattices $Gamma$ in higher-rank simple Lie groups $G$ assuming one element of the action acts with positive topological entropy and prove a number of new rigidity results. For lattices $Gamma$ in $mathrm{SL}(n,mathbb{R})$ acting on $n$-manifolds, if the action has positive topological entropy we show the lattice must be commensurable with $mathrm{SL}(n,mathbb{Z})$. Moreover, such actions admit an absolutely continuous probability measure with positive metric entropy; adapting arguments by Katok and Rodriguez Hertz, we show such actions are measurably conjugate to affine actions on (infra)tori. In our main technical arguments, we study families of probability measures invariant under sub-actions of the induced $G$-action on an associated fiber bundle. To control entropy properties of such measures, in the appendix we establish certain upper semicontinuity of entropy under weak-$*$ convergence, adapting classical results of Yomdin and Newhouse.
{"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":"13 1","pages":""},"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}
Javier Aramayona, Rodrigo De Pool, Rachel Skipper, Jing Tao, Nicholas G. Vlamis, Xiaolei Wu
We show that continuous epimorphisms between a class of subgroups of mapping class groups of orientable infinite-genus 2-manifolds with no planar ends are always induced by homeomorphisms. This class of subgroups includes the pure mapping class group, the closure of the compactly supported mapping classes, and the full mapping class group in the case that the underlying manifold has a finite number of ends or is perfectly self-similar. As a corollary, these groups are Hopfian topological groups.
{"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":"2 1","pages":""},"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}
In this article, we investigate symmetric designs admitting a flag-transitive and point-primitive affine automorphism group. We prove that if an automorphism group $G$ of a symmetric $(v,k,lambda)$ design with $lambda$ prime is point-primitive of affine type, then $G=2^{6}{:}mathrm{S}_{6}$ and $(v,k,lambda)=(16,6,2)$, or $G$ is a subgroup of $mathrm{AGamma L}_{1}(q)$ for some odd prime power $q$. In conclusion, we present a classification of flag-transitive and point-primitive symmetric designs with $lambda$ prime, which says that such an incidence structure is a projective space $mathrm{PG}(n,q)$, it has parameter set $(15,7,3)$, $(7, 4, 2)$, $(11, 5, 2)$, $(11, 6, 2)$, $(16,6,2)$ or $(45, 12, 3)$, or $v=p^d$ where $p$ is an odd prime and the automorphism group is a subgroup of $mathrm{AGamma L}_{1}(q)$.
{"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":"17 1","pages":""},"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}
We provide a brief overview of our upcoming work identifying all the thin Heckoid groups in $PSL(2,mathbb{C})$. Here we give a complete list of the $55$ thin generalised triangle groups of slope $1/2$. This work was presented at the conference Computational Aspects of Thin Groups, IMSS, Singapore and presents an application of joint work initiated with Colin Maclachlan
{"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":"71 1","pages":""},"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}
Nestor Colin, Rita Jiménez Rolland, Porfirio L. León Álvarez, Luis Jorge Sánchez Saldaña
In cite{Ha86} Harer explicitly constructed a spine for the decorated Teichm"uller space of orientable surfaces with at least one puncture and negative Euler characteristic. In this paper we point out some instances where his computation of the dimension of this spine is off by $1$ and give the correct dimension.
{"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":"2 1","pages":""},"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}
We show that if a Sylow $p$-subgroup of a finite group $G$ is nilpotent of class at most $p$, then the $p$-part of the Bogomolov multiplier of $G$ is locally controlled.
{"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":"70 1","pages":""},"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}
Let $mathscr{C}_{+}(p,q)^0$ and $mathscr{C}_{-}(p,q)^0$ be the semigroups $mathscr{C}_{+}(a,b)$ and $mathscr{C}_{-}(a,b)$ with the adjoined zero. We show that the semigroups $mathscr{C}_{+}(p,q)^0$ and $mathscr{C}_{-}(p,q)^0$ admit continuum many different Hausdorff locally compact shift-continuous topologies up to topological isomorphism.
{"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":"1 1","pages":""},"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}
It is known that an abelian group $A$ and a $2$-cocycle $c:A times A to C$ yield a group ${mathscr{H}}(A,C,c)$ which we call a Heisenberg group. This group, a central extension of $A$, is the archetype of a class~$2$ nilpotent group. In this note, we prove that under mild conditions, any class~$2$ nilpotent group $G$ is equivalent as an extension of $G/[G,G]$ to a Heisenberg group ${mathscr{H}}(G/[G,G], [G,G], c')$ whose $2$-cocycle $c'$ is bimultiplicative.
众所周知，一个无性群 $A$ 和一个 $2$ 循环 $c:A times A to C$ 产生一个群 ${mathscr{H}}(A,C,c)$ 我们称之为海森堡群。这个群是 $A$ 的中心扩展，是类~$2$ 无穷群的原型。在本论文中，我们将证明在温和的条件下，任何一个类~$2$无穷群 $G$都等价于$G/[G,G]$的一个扩展，即一个海森堡群 ${mathscr{H}}(G/[G,G],[G,G],c')$，其$2$循环 $c'$是二乘的。
{"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":"71 1","pages":""},"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}
We prove that if $G$ is a transitive permutation group of sufficiently large degree $n$, then either $G$ is primitive and Frobenius, or the proportion of derangements in $G$ is larger than $1/(2n^{1/2})$. This is sharp, generalizes substantially bounds of Cameron--Cohen and Guralnick--Wan, and settles a conjecture of Guralnick--Tiep in large degree. We also give an application to coverings of varieties over finite fields.
{"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":"70 1","pages":""},"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}
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