In this article, we associate to isometries of CAT(0) cube complexes specific subspaces, referred to as emph{median sets}, which play a similar role as minimising sets of semisimple isometries in CAT(0) spaces. Various applications are deduced, including a cubulation of centralisers, a splitting theorem, a proof that Dehn twists in mapping class groups must be elliptic for every action on a CAT(0) cube complex, a cubical version of the flat torus theorem, and a structural theorem about polycyclic groups acting on CAT(0) cube complexes.
{"title":"Median Sets of Isometries in CAT(0) Cube Complexes and Some Applications","authors":"A. Genevois","doi":"10.1307/MMJ/20195823","DOIUrl":"https://doi.org/10.1307/MMJ/20195823","url":null,"abstract":"In this article, we associate to isometries of CAT(0) cube complexes specific subspaces, referred to as emph{median sets}, which play a similar role as minimising sets of semisimple isometries in CAT(0) spaces. Various applications are deduced, including a cubulation of centralisers, a splitting theorem, a proof that Dehn twists in mapping class groups must be elliptic for every action on a CAT(0) cube complex, a cubical version of the flat torus theorem, and a structural theorem about polycyclic groups acting on CAT(0) cube complexes.","PeriodicalId":8427,"journal":{"name":"arXiv: Group Theory","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2019-02-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"79291511","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 : 2019-02-01DOI: 10.22108/IJGT.2020.123551.1628
Hayder Abbas Janabi, T. Breuer, E. Horváth
We show that for each positive integer $n$, there are a group $G$ and a subgroup $H$ such that the ordinary depth is $d(H, G) = 2n$. This solves the open problem posed by Lars Kadison whether even ordinary depth larger than $6$ can occur.
{"title":"Subgroups of arbitrary even ordinary depth","authors":"Hayder Abbas Janabi, T. Breuer, E. Horváth","doi":"10.22108/IJGT.2020.123551.1628","DOIUrl":"https://doi.org/10.22108/IJGT.2020.123551.1628","url":null,"abstract":"We show that for each positive integer $n$, there are a group $G$ and a subgroup $H$ such that the ordinary depth is $d(H, G) = 2n$. This solves the open problem posed by Lars Kadison whether even ordinary depth larger than $6$ can occur.","PeriodicalId":8427,"journal":{"name":"arXiv: Group Theory","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2019-02-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"73670058","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 : 2019-01-23DOI: 10.22108/TOC.2021.127225.1817
S. Mirafzal, M. Ziaee
Let $Omega$ be a $m$-set, where $m>1$, is an integer. The Hamming graph $H(n,m)$, has $Omega ^{n}$ as its vertex-set, with two vertices are adjacent if and only if they differ in exactly one coordinate. In this paper, we provide a proof on the automorphism group of the Hamming graph $H(n,m)$, by using elementary facts of group theory and graph theory.
{"title":"A note on the automorphism group of the Hamming graph","authors":"S. Mirafzal, M. Ziaee","doi":"10.22108/TOC.2021.127225.1817","DOIUrl":"https://doi.org/10.22108/TOC.2021.127225.1817","url":null,"abstract":"Let $Omega$ be a $m$-set, where $m>1$, is an integer. The Hamming graph $H(n,m)$, has $Omega ^{n}$ as its vertex-set, with two vertices are adjacent if and only if they differ in exactly one coordinate. In this paper, we provide a proof on the automorphism group of the Hamming graph $H(n,m)$, by using elementary facts of group theory and graph theory.","PeriodicalId":8427,"journal":{"name":"arXiv: Group Theory","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2019-01-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"77913922","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}
Liedtke (2008) has introduced group functors $K$ and $tilde K$, which are used in the context of describing certain invariants for complex algebraic surfaces. He proved that these functors are connected to the theory of central extensions and Schur multipliers. In this work we relate $K$ and $tilde K$ to a group functor $tau$ arising in the construction of the non-abelian exterior square of a group. In contrast to $tilde K$, there exist efficient algorithms for constructing $tau$, especially for polycyclic groups. Supported by computations with the computer algebra system GAP, we investigate when $K(G,3)$ is a quotient of $tau(G)$, and when $tau(G)$ and $tilde K(G,3)$ are isomorphic.
{"title":"On a Group Functor Describing Invariants of Algebraic Surfaces","authors":"H. Dietrich, P. Moravec","doi":"10.14760/OWP-2019-08","DOIUrl":"https://doi.org/10.14760/OWP-2019-08","url":null,"abstract":"Liedtke (2008) has introduced group functors $K$ and $tilde K$, which are used in the context of describing certain invariants for complex algebraic surfaces. He proved that these functors are connected to the theory of central extensions and Schur multipliers. In this work we relate $K$ and $tilde K$ to a group functor $tau$ arising in the construction of the non-abelian exterior square of a group. In contrast to $tilde K$, there exist efficient algorithms for constructing $tau$, especially for polycyclic groups. Supported by computations with the computer algebra system GAP, we investigate when $K(G,3)$ is a quotient of $tau(G)$, and when $tau(G)$ and $tilde K(G,3)$ are isomorphic.","PeriodicalId":8427,"journal":{"name":"arXiv: Group Theory","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2019-01-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"88028310","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 : 2018-11-02DOI: 10.22108/IJGT.2019.113217.1507
James Williams
In this note we show that for any powerful $p$-group $G$, the subgroup $Omega_{i}(G^{p^{j}})$ is powerfully nilpotent for all $i,jgeq1$ when $p$ is an odd prime, and $igeq1$, $jgeq2$ when $p=2$. We provide an example to show why this modification is needed in the case $p=2$. Furthermore we obtain a bound on the powerful nilpotency class of $Omega_{i}(G^{p^{j}})$. We give an example to show that powerfully nilpotent characteristic subgroups of powerful $p$-groups need not be strongly powerful.
{"title":"Omegas of Agemos in Powerful Groups.","authors":"James Williams","doi":"10.22108/IJGT.2019.113217.1507","DOIUrl":"https://doi.org/10.22108/IJGT.2019.113217.1507","url":null,"abstract":"In this note we show that for any powerful $p$-group $G$, the subgroup $Omega_{i}(G^{p^{j}})$ is powerfully nilpotent for all $i,jgeq1$ when $p$ is an odd prime, and $igeq1$, $jgeq2$ when $p=2$. We provide an example to show why this modification is needed in the case $p=2$. Furthermore we obtain a bound on the powerful nilpotency class of $Omega_{i}(G^{p^{j}})$. We give an example to show that powerfully nilpotent characteristic subgroups of powerful $p$-groups need not be strongly powerful.","PeriodicalId":8427,"journal":{"name":"arXiv: Group Theory","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2018-11-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"77667523","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 : 2018-10-30DOI: 10.2140/agt.2021.21.2995
Kasra Rafi, Y. Verberne
We construct explicit examples of geodesics in the mapping class group and show that the shadow of a geodesic in mapping class group to the curve graph does not have to be a quasi-geodesic. We also show that the quasi-axis of a pseudo-Anosov element of the mapping class group may not have the strong contractibility property. Specifically, we show that, after choosing a generating set carefully, one can find a pseudo-Anosov homeomorphism f, a sequence of points w_k and a sequence of radii r_k so that the ball B(w_k, r_k) is disjoint from a quasi-axis a of f, but for any projection map from mapping class group to a, the diameter of the image of B(w_k, r_k) grows like log(r_k).
{"title":"Geodesics in the mapping class group","authors":"Kasra Rafi, Y. Verberne","doi":"10.2140/agt.2021.21.2995","DOIUrl":"https://doi.org/10.2140/agt.2021.21.2995","url":null,"abstract":"We construct explicit examples of geodesics in the mapping class group and show that the shadow of a geodesic in mapping class group to the curve graph does not have to be a quasi-geodesic. We also show that the quasi-axis of a pseudo-Anosov element of the mapping class group may not have the strong contractibility property. Specifically, we show that, after choosing a generating set carefully, one can find a pseudo-Anosov homeomorphism f, a sequence of points w_k and a sequence of radii r_k so that the ball B(w_k, r_k) is disjoint from a quasi-axis a of f, but for any projection map from mapping class group to a, the diameter of the image of B(w_k, r_k) grows like log(r_k).","PeriodicalId":8427,"journal":{"name":"arXiv: Group Theory","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2018-10-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"81573741","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 : 2018-10-29DOI: 10.17377/smzh.2019.60.208
A. Vasil’ev, D. Churikov
Let $G$ be a permutation group on a finite set $Omega$. The $k$-closure $G^{(k)}$ of the group $G$ is the largest subgroup of $operatorname{Sym}(Omega)$ having the same orbits as $G$ on the $k$-th Cartesian power $Omega^k$ of $Omega$. A group $G$ is called $frac{3}{2}$-transitive if its transitive and the orbits of a point stabilizer $G_alpha$ on the set $Omegasetminus{alpha}$ are of the same size greater than one. We prove that the $2$-closure $G^{(2)}$ of a $frac{3}{2}$-transitive permutation group $G$ can be found in polynomial time in size of $Omega$. In addition, if the group $G$ is not $2$-transitive, then for every positive integer $k$ its $k$-closure can be found within the same time. Applying the result, we prove the existence of a polynomial-time algorithm for solving the isomorphism problem for schurian $frac{3}{2}$-homogeneous coherent configurations, that is the configurations naturally associated with $frac{3}{2}$-transitive groups.
{"title":"$mathbf{2}$-Closure of $mathbf{frac{3}{2}}$-transitive group in polynomial time.","authors":"A. Vasil’ev, D. Churikov","doi":"10.17377/smzh.2019.60.208","DOIUrl":"https://doi.org/10.17377/smzh.2019.60.208","url":null,"abstract":"Let $G$ be a permutation group on a finite set $Omega$. The $k$-closure $G^{(k)}$ of the group $G$ is the largest subgroup of $operatorname{Sym}(Omega)$ having the same orbits as $G$ on the $k$-th Cartesian power $Omega^k$ of $Omega$. A group $G$ is called $frac{3}{2}$-transitive if its transitive and the orbits of a point stabilizer $G_alpha$ on the set $Omegasetminus{alpha}$ are of the same size greater than one. We prove that the $2$-closure $G^{(2)}$ of a $frac{3}{2}$-transitive permutation group $G$ can be found in polynomial time in size of $Omega$. In addition, if the group $G$ is not $2$-transitive, then for every positive integer $k$ its $k$-closure can be found within the same time. Applying the result, we prove the existence of a polynomial-time algorithm for solving the isomorphism problem for schurian $frac{3}{2}$-homogeneous coherent configurations, that is the configurations naturally associated with $frac{3}{2}$-transitive groups.","PeriodicalId":8427,"journal":{"name":"arXiv: Group Theory","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2018-10-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"86960573","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 homomorphism $i: H to G$ is a localization of $H$ if for every homomorphism $varphi: Hrightarrow G$ there exists a unique endomorphism $psi: Grightarrow G$, such that $i psi=varphi$ (maps are acting on the right). G"{o}bel and Trlifaj asked in cite[Problem 30.4(4), p. 831]{GT12} which abelian groups are centers of localizations of simple groups. Approaching this question we show that every countable abelian group is indeed the center of some localization of a quasi-simple group, i.e. a central extension of a simple group. The proof uses Obraztsov and Ol'shanskii's construction of infinite simple groups with a special subgroup lattice and also extensions of results on localizations of finite simple groups by the second author and Scherer, Th'{e}venaz and Viruel.
群同态$i: H to G$是$H$的一个局部化，如果对于每个同态$varphi: Hrightarrow G$存在一个唯一的自同态$psi: Grightarrow G$，例如$i psi=varphi$(映射作用于右侧)。Göbel和Trlifaj在cite[Problem 30.4(4), p. 831]{GT12}问哪些阿贝尔群是简单群的定域中心。针对这个问题，我们证明了每一个可数阿贝尔群确实是一个拟简单群的某个定域的中心，即一个简单群的中心扩展。该证明利用了Obraztsov和Ol’shanskii关于具有特殊子群格的无限简单群的构造，并推广了第二作者和Scherer、thsamvenaz和Viruel关于有限简单群的局域化的结果。
{"title":"On localizations of quasi-simple groups with given countable center","authors":"Ramón Flores, Jos'e L. Rodr'iguez","doi":"10.4171/ggd/573","DOIUrl":"https://doi.org/10.4171/ggd/573","url":null,"abstract":"A group homomorphism $i: H to G$ is a localization of $H$ if for every homomorphism $varphi: Hrightarrow G$ there exists a unique endomorphism $psi: Grightarrow G$, such that $i psi=varphi$ (maps are acting on the right). G\"{o}bel and Trlifaj asked in cite[Problem 30.4(4), p. 831]{GT12} which abelian groups are centers of localizations of simple groups. Approaching this question we show that every countable abelian group is indeed the center of some localization of a quasi-simple group, i.e. a central extension of a simple group. The proof uses Obraztsov and Ol'shanskii's construction of infinite simple groups with a special subgroup lattice and also extensions of results on localizations of finite simple groups by the second author and Scherer, Th'{e}venaz and Viruel.","PeriodicalId":8427,"journal":{"name":"arXiv: Group Theory","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2018-10-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"78834594","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 deduce from Sageev's results that whenever a group acts locally elliptically on a finite dimensional CAT(0) cube complex, then it must fix a point. As an application, we give an example of a group G such that G does not have property (T), but G and all its finitely generated subgroups can not act without a fixed point on a finite dimensional CAT(0) cube complex, answering a question by Barnhill and Chatterji.
{"title":"A note on locally elliptic actions on cube complexes","authors":"Nils Leder, Olga Varghese","doi":"10.2140/IIG.2020.18.1","DOIUrl":"https://doi.org/10.2140/IIG.2020.18.1","url":null,"abstract":"We deduce from Sageev's results that whenever a group acts locally elliptically on a finite dimensional CAT(0) cube complex, then it must fix a point. As an application, we give an example of a group G such that G does not have property (T), but G and all its finitely generated subgroups can not act without a fixed point on a finite dimensional CAT(0) cube complex, answering a question by Barnhill and Chatterji.","PeriodicalId":8427,"journal":{"name":"arXiv: Group Theory","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2018-10-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"77126790","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}
This paper shows that groups of order $64$ are uniquely determined up to isomorphism by their Tables of Marks. This then resolves a previously posed question about whether all groups of order less than $96$ are determined by their Tables of Marks.
{"title":"Groups of order 64 are determined by their Tables of Marks","authors":"Peter Bonart","doi":"10.12988/PMS.2018.8910","DOIUrl":"https://doi.org/10.12988/PMS.2018.8910","url":null,"abstract":"This paper shows that groups of order $64$ are uniquely determined up to isomorphism by their Tables of Marks. This then resolves a previously posed question about whether all groups of order less than $96$ are determined by their Tables of Marks.","PeriodicalId":8427,"journal":{"name":"arXiv: Group Theory","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2018-10-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"76646016","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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