We give sufficient conditions on $p$-blocks of $p$-nilpotent groups over $mathbb{F}_p$ to be splendidly Rickard equivalent and $p$-permutation equivalent to their Brauer correspondents. The paper also contains Galois descent results on $p$-permutation modules and $p$-permutation equivalences that hold for arbitrary groups.
{"title":"Galois descent of equivalences between blocks of 𝑝-nilpotent groups","authors":"Robert Boltje, D. Yılmaz","doi":"10.1090/proc/15704","DOIUrl":"https://doi.org/10.1090/proc/15704","url":null,"abstract":"We give sufficient conditions on $p$-blocks of $p$-nilpotent groups over $mathbb{F}_p$ to be splendidly Rickard equivalent and $p$-permutation equivalent to their Brauer correspondents. The paper also contains Galois descent results on $p$-permutation modules and $p$-permutation equivalences that hold for arbitrary groups.","PeriodicalId":8427,"journal":{"name":"arXiv: Group Theory","volume":"209 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2021-02-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"85508049","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 : 2020-12-29DOI: 10.46298/jgcc.2021.13.1.7036
Sebastia Mijares, E. Ventura
An extension of subgroups $Hleqslant Kleqslant F_A$ of the free group of rank $|A|=rgeqslant 2$ is called onto when, for every ambient free basis $A'$, the Stallings graph $Gamma_{A'}(K)$ is a quotient of $Gamma_{A'}(H)$. Algebraic extensions are onto and the converse implication was conjectured by Miasnikov-Ventura-Weil, and resolved in the negative, first by Parzanchevski-Puder for rank $r=2$, and recently by Kolodner for general rank. In this note we study properties of this new type of extension among free groups (as well as the fully onto variant), and investigate their corresponding closure operators. Interestingly, the natural attempt for a dual notion -- into extensions -- becomes trivial, making a Takahasi type theorem not possible in this setting.
{"title":"Onto extensions of free groups.","authors":"Sebastia Mijares, E. Ventura","doi":"10.46298/jgcc.2021.13.1.7036","DOIUrl":"https://doi.org/10.46298/jgcc.2021.13.1.7036","url":null,"abstract":"An extension of subgroups $Hleqslant Kleqslant F_A$ of the free group of rank $|A|=rgeqslant 2$ is called onto when, for every ambient free basis $A'$, the Stallings graph $Gamma_{A'}(K)$ is a quotient of $Gamma_{A'}(H)$. Algebraic extensions are onto and the converse implication was conjectured by Miasnikov-Ventura-Weil, and resolved in the negative, first by Parzanchevski-Puder for rank $r=2$, and recently by Kolodner for general rank. In this note we study properties of this new type of extension among free groups (as well as the fully onto variant), and investigate their corresponding closure operators. Interestingly, the natural attempt for a dual notion -- into extensions -- becomes trivial, making a Takahasi type theorem not possible in this setting.","PeriodicalId":8427,"journal":{"name":"arXiv: Group Theory","volume":"6 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2020-12-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"82505420","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 : 2020-12-03DOI: 10.21538/0134-4889-2021-27-1-240-245
D. Churikov, C. Praeger
For a positive integer $k$, a group $G$ is said to be totally $k$-closed if in each of its faithful permutation representations, say on a set $Omega$, $G$ is the largest subgroup of $operatorname{Sym}(Omega)$ which leaves invariant each of the $G$-orbits in the induced action on $Omegatimesdotstimes Omega=Omega^k$. We prove that every abelian group $G$ is totally $(n(G)+1)$-closed, but is not totally $n(G)$-closed, where $n(G)$ is the number of invariant factors in the invariant factor decomposition of $G$. In particular, we prove that for each $kgeq2$ and each prime $p$, there are infinitely many finite abelian $p$-groups which are totally $k$-closed but not totally $(k-1)$-closed. This result in the special case $k=2$ is due to Abdollahi and Arezoomand. We pose several open questions about total $k$-closure.
{"title":"Finite totally k-closed groups","authors":"D. Churikov, C. Praeger","doi":"10.21538/0134-4889-2021-27-1-240-245","DOIUrl":"https://doi.org/10.21538/0134-4889-2021-27-1-240-245","url":null,"abstract":"For a positive integer $k$, a group $G$ is said to be totally $k$-closed if in each of its faithful permutation representations, say on a set $Omega$, $G$ is the largest subgroup of $operatorname{Sym}(Omega)$ which leaves invariant each of the $G$-orbits in the induced action on $Omegatimesdotstimes Omega=Omega^k$. We prove that every abelian group $G$ is totally $(n(G)+1)$-closed, but is not totally $n(G)$-closed, where $n(G)$ is the number of invariant factors in the invariant factor decomposition of $G$. In particular, we prove that for each $kgeq2$ and each prime $p$, there are infinitely many finite abelian $p$-groups which are totally $k$-closed but not totally $(k-1)$-closed. This result in the special case $k=2$ is due to Abdollahi and Arezoomand. We pose several open questions about total $k$-closure.","PeriodicalId":8427,"journal":{"name":"arXiv: Group Theory","volume":"70 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2020-12-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"74070478","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 : 2020-12-02DOI: 10.1142/S0218216521500279
Linjun Li
We consider a natural generalization of braids which we call shrinking braids. We state the relations of shrinking braids and use them to define algebraically the monoid $R$. We endow a subset of $R$ with a emph{left distributive monoid} structure and use it to extend the Dehornoy order on $B_{infty}$ to an order on $R$. By using this order, we prove that $R$ is isomorphic to the monoid which is generated (geometrically) by shrinking braids.
{"title":"Shrinking braids and left distributive monoid","authors":"Linjun Li","doi":"10.1142/S0218216521500279","DOIUrl":"https://doi.org/10.1142/S0218216521500279","url":null,"abstract":"We consider a natural generalization of braids which we call shrinking braids. We state the relations of shrinking braids and use them to define algebraically the monoid $R$. We endow a subset of $R$ with a emph{left distributive monoid} structure and use it to extend the Dehornoy order on $B_{infty}$ to an order on $R$. By using this order, we prove that $R$ is isomorphic to the monoid which is generated (geometrically) by shrinking braids.","PeriodicalId":8427,"journal":{"name":"arXiv: Group Theory","volume":"29 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2020-12-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"84157576","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 : 2020-12-02DOI: 10.1007/978-981-13-2047-7_5
A. Hulpke
{"title":"Calculating Subgroups with GAP","authors":"A. Hulpke","doi":"10.1007/978-981-13-2047-7_5","DOIUrl":"https://doi.org/10.1007/978-981-13-2047-7_5","url":null,"abstract":"","PeriodicalId":8427,"journal":{"name":"arXiv: Group Theory","volume":"1 1","pages":"91-106"},"PeriodicalIF":0.0,"publicationDate":"2020-12-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"88747198","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 exhibit a topological group G with property (T) acting non-elementarily and continuously on the circle. This group is an uncountable totally disconnected closed subgroup of Homeo + (S 1). It has a large unitary dual since it separates points. It comes from homeomorphisms of dendrites and a kaleidoscopic construction. Alternatively, it can be seen as the group of elements preserving some specific geodesic lamination of the hyperbolic disk. We also prove that this action is unique up to conjugation and that it can't be smoothened in any way. Finally, we determine the universal minimal flow of the group G.
{"title":"A group with Property (T) acting on the circle","authors":"Bruno Duchesne","doi":"10.1093/imrn/rnac136","DOIUrl":"https://doi.org/10.1093/imrn/rnac136","url":null,"abstract":"We exhibit a topological group G with property (T) acting non-elementarily and continuously on the circle. This group is an uncountable totally disconnected closed subgroup of Homeo + (S 1). It has a large unitary dual since it separates points. It comes from homeomorphisms of dendrites and a kaleidoscopic construction. Alternatively, it can be seen as the group of elements preserving some specific geodesic lamination of the hyperbolic disk. We also prove that this action is unique up to conjugation and that it can't be smoothened in any way. Finally, we determine the universal minimal flow of the group G.","PeriodicalId":8427,"journal":{"name":"arXiv: Group Theory","volume":"25 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2020-11-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"73082944","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}
Gamma-semigroup is introduced as a generalization of semigroups by M. K Sen and Saha. In this paper we describe amalgam of two Gamma-semigroups and discuss the embeddability of this amalgam. Further we obtained a necessary condition for the embeddability of completely alpha-regular Gamma-semigroup amalgam.
{"title":"On the embedding of $ Gamma $-semigroup Amalgam","authors":"A. SmishaM, P. G. Romeo","doi":"10.26637/MJM0901/0075","DOIUrl":"https://doi.org/10.26637/MJM0901/0075","url":null,"abstract":"Gamma-semigroup is introduced as a generalization of semigroups by M. K Sen and Saha. In this paper we describe amalgam of two Gamma-semigroups and discuss the embeddability of this amalgam. Further we obtained a necessary condition for the embeddability of completely alpha-regular Gamma-semigroup amalgam.","PeriodicalId":8427,"journal":{"name":"arXiv: Group Theory","volume":"36 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2020-10-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"79249529","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 : 2020-10-25DOI: 10.1016/J.JALGEBRA.2021.06.039
A. Clay, I. Ba
{"title":"The space of circular orderings and semiconjugacy.","authors":"A. Clay, I. Ba","doi":"10.1016/J.JALGEBRA.2021.06.039","DOIUrl":"https://doi.org/10.1016/J.JALGEBRA.2021.06.039","url":null,"abstract":"","PeriodicalId":8427,"journal":{"name":"arXiv: Group Theory","volume":"31 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2020-10-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"83695774","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}
A group $Gamma$ is said to be uniformly HS stable if any map $varphi : Gamma to U(n)$ that is almost a unitary representation (w.r.t. the Hilbert Schmidt norm) is close to a genuine unitary representation of the same dimension. We present a complete classification of uniformly HS stable groups among finitely generated residually finite ones. Necessity of the residual finiteness assumption is discussed. A similar result is shown to hold assuming only amenability.
如果任何映射$varphi : Gamma to U(n)$几乎是酉表示(w.r.t.希尔伯特施密特范数)接近于相同维度的真正酉表示，则群$Gamma$被称为一致HS稳定。给出了在有限生成的剩余有限群中一致HS稳定群的完全分类。讨论了剩余有限假设的必要性。一个类似的结果显示，只假设顺从。
{"title":"On uniform Hilbert Schmidt stability of groups","authors":"D. Akhtiamov, Alon Dogon","doi":"10.1090/proc/15772","DOIUrl":"https://doi.org/10.1090/proc/15772","url":null,"abstract":"A group $Gamma$ is said to be uniformly HS stable if any map $varphi : Gamma to U(n)$ that is almost a unitary representation (w.r.t. the Hilbert Schmidt norm) is close to a genuine unitary representation of the same dimension. We present a complete classification of uniformly HS stable groups among finitely generated residually finite ones. Necessity of the residual finiteness assumption is discussed. A similar result is shown to hold assuming only amenability.","PeriodicalId":8427,"journal":{"name":"arXiv: Group Theory","volume":"27 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2020-10-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"89446232","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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