We show that the isometry group of a finite-volume hyperbolic 3-manifold acts simply transitively on many of its closed geodesics. Combining this observation with the Virtual Special Theorems of the first author and Wise, we show that every non-arithmetic lattice in PSL$(2,mathbb{C})$ is the full group of orientation-preserving isometries for some other lattice and that the orientation-preserving isometry group of a finite-volume hyperbolic 3-manifold acts non-trivially on the homology of some finite-sheeted cover.
{"title":"Simply transitive geodesics and omnipotence of lattices in PSL$(2,mathbb{C})$","authors":"Ian Agol, Tam Cheetham-West, Yair Minsky","doi":"arxiv-2409.08418","DOIUrl":"https://doi.org/arxiv-2409.08418","url":null,"abstract":"We show that the isometry group of a finite-volume hyperbolic 3-manifold acts\u0000simply transitively on many of its closed geodesics. Combining this observation\u0000with the Virtual Special Theorems of the first author and Wise, we show that\u0000every non-arithmetic lattice in PSL$(2,mathbb{C})$ is the full group of\u0000orientation-preserving isometries for some other lattice and that the\u0000orientation-preserving isometry group of a finite-volume hyperbolic 3-manifold\u0000acts non-trivially on the homology of some finite-sheeted cover.","PeriodicalId":501037,"journal":{"name":"arXiv - MATH - Group Theory","volume":"5 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142253761","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 a special Moufang sets with abelian root subgroups derive from a quadratic Jordan division algebra if a certain finiteness condition is satisfied.
我们证明，如果满足一定的有限性条件，具有无性根子群的特殊牟方集派生自二次乔丹除法代数。
{"title":"Special Moufang sets of finite dimension","authors":"Matthias Grüninger","doi":"arxiv-2409.07445","DOIUrl":"https://doi.org/arxiv-2409.07445","url":null,"abstract":"We prove that a special Moufang sets with abelian root subgroups derive from\u0000a quadratic Jordan division algebra if a certain finiteness condition is\u0000satisfied.","PeriodicalId":501037,"journal":{"name":"arXiv - MATH - Group Theory","volume":"4 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142206460","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 positive integer such that $p$ does not divide $m$ where $p$ is prime. In this paper we find the number of conjugacy classes of completely reducible cyclic subgroups in GL$(2, q)$ of order $m$, where $q$ is a power of $p$.
{"title":"Conjugacy classes of completely reducible cyclic subgroups of GL$(2, q)$","authors":"Prashun Kumar, Geetha Venkataraman","doi":"arxiv-2409.07244","DOIUrl":"https://doi.org/arxiv-2409.07244","url":null,"abstract":"Let $m$ be a positive integer such that $p$ does not divide $m$ where $p$ is\u0000prime. In this paper we find the number of conjugacy classes of completely\u0000reducible cyclic subgroups in GL$(2, q)$ of order $m$, where $q$ is a power of\u0000$p$.","PeriodicalId":501037,"journal":{"name":"arXiv - MATH - Group Theory","volume":"18 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142206433","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 necessary and sufficient condition for the restricted wreath product $Awr B$ to be $mathcal{C}$-hereditarily conjugacy separable where $mathcal{C}$ is an extension-closed pseudo-variety of finite groups. Moreover, we prove that the Grigorchuk group is 2-hereditarily conjugacy separable.
{"title":"$mathcal{C}$-Hereditarily conjugacy separable groups and wreath products","authors":"Alexander Bishop, Michal Ferov, Mark Pengitore","doi":"arxiv-2409.06200","DOIUrl":"https://doi.org/arxiv-2409.06200","url":null,"abstract":"We provide a necessary and sufficient condition for the restricted wreath\u0000product $Awr B$ to be $mathcal{C}$-hereditarily conjugacy separable where\u0000$mathcal{C}$ is an extension-closed pseudo-variety of finite groups. Moreover,\u0000we prove that the Grigorchuk group is 2-hereditarily conjugacy separable.","PeriodicalId":501037,"journal":{"name":"arXiv - MATH - Group Theory","volume":"71 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142206438","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}
The extended mapping class group of a surface $Sigma$ is defined to be the group of isotopy classes of (not necessarily orientation-preserving) homeomorphisms of $Sigma$. We are able to show that the extended mapping class group of an $n$-punctured sphere is generated by two elements of finite order exactly when $nnot=4$. We use this result to prove that the extended mapping class group of a genus 2 surface is generated by two elements of finite order.
{"title":"Generating Extended Mapping Class Groups with Two Periodic Elements","authors":"Reid Harris","doi":"arxiv-2409.06350","DOIUrl":"https://doi.org/arxiv-2409.06350","url":null,"abstract":"The extended mapping class group of a surface $Sigma$ is defined to be the\u0000group of isotopy classes of (not necessarily orientation-preserving)\u0000homeomorphisms of $Sigma$. We are able to show that the extended mapping class\u0000group of an $n$-punctured sphere is generated by two elements of finite order\u0000exactly when $nnot=4$. We use this result to prove that the extended mapping\u0000class group of a genus 2 surface is generated by two elements of finite order.","PeriodicalId":501037,"journal":{"name":"arXiv - MATH - Group Theory","volume":"70 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142206439","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 for any surface of genus at least 3 equipped with any choice of framing, the graph of non-separating curves with winding number 0 with respect to the framing is hierarchically hyperbolic but not Gromov hyperbolic. We also describe how to build analogues of the curve graph for marked strata of abelian differentials that capture the combinatorics of their boundaries, analogous to how the curve graph captures the combinatorics of the augmented Teichmueller space. These curve graph analogues are also shown to be hierarchically, but not Gromov, hyperbolic.
{"title":"Hierarchical hyperbolicity of admissible curve graphs and the boundary of marked strata","authors":"Aaron Calderon, Jacob Russell","doi":"arxiv-2409.06798","DOIUrl":"https://doi.org/arxiv-2409.06798","url":null,"abstract":"We show that for any surface of genus at least 3 equipped with any choice of\u0000framing, the graph of non-separating curves with winding number 0 with respect\u0000to the framing is hierarchically hyperbolic but not Gromov hyperbolic. We also\u0000describe how to build analogues of the curve graph for marked strata of abelian\u0000differentials that capture the combinatorics of their boundaries, analogous to\u0000how the curve graph captures the combinatorics of the augmented Teichmueller\u0000space. These curve graph analogues are also shown to be hierarchically, but not\u0000Gromov, hyperbolic.","PeriodicalId":501037,"journal":{"name":"arXiv - MATH - Group Theory","volume":"62 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142206435","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}
Jonas BeyrerUNISTRA, Olivier GuichardUNISTRA, François LabourieUniCA, Beatrice Pozzetti, Anna Wienhard
We prove that $Theta$-positive representations of fundamental groups of surfaces (possibly cusped or of infinite type) satisfy a collar lemma, and their associated cross-ratios are positive. As a consequence we deduce that $Theta$-positive representations form closed subsets of the representation variety.
{"title":"Positivity, cross-ratios and the Collar Lemma","authors":"Jonas BeyrerUNISTRA, Olivier GuichardUNISTRA, François LabourieUniCA, Beatrice Pozzetti, Anna Wienhard","doi":"arxiv-2409.06294","DOIUrl":"https://doi.org/arxiv-2409.06294","url":null,"abstract":"We prove that $Theta$-positive representations of fundamental groups of\u0000surfaces (possibly cusped or of infinite type) satisfy a collar lemma, and\u0000their associated cross-ratios are positive. As a consequence we deduce that\u0000$Theta$-positive representations form closed subsets of the representation\u0000variety.","PeriodicalId":501037,"journal":{"name":"arXiv - MATH - Group Theory","volume":"13 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142206442","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 describe the structure of simple inverse Hausdorff semitopological $omega$-semigroups with compact maximal subgroups. In particular we show that if $S$ is a simple inverse Hausdorff semitopological $omega$-semigroups with compact maximal subgroups, then $S$ is topologically isomorphic to the Bruck--Reilly extension $left(textbf{BR}(T,theta),tau_{textbf{BR}}^{oplus}right)$ of a finite semilattice $T=left[E;G_alpha,varphi_{alpha,beta}right]$ of compact groups $G_alpha$ in the class of topological inverse semigroups, where $tau_{textbf{BR}}^{oplus}$ is the sum direct topology on $textbf{BR}(T,theta)$. Also we prove that every Hausdorff locally compact shift-continuous topology on the simple inverse Hausdorff semitopological $omega$-semigroups with compact maximal subgroups with adjoined zero is either compact or discrete.
{"title":"On semitopological simple inverse $ω$-semigroups with compact maximal subgroups","authors":"Oleg Gutik, Kateryna Maksymyk","doi":"arxiv-2409.06344","DOIUrl":"https://doi.org/arxiv-2409.06344","url":null,"abstract":"We describe the structure of simple inverse Hausdorff semitopological\u0000$omega$-semigroups with compact maximal subgroups. In particular we show that\u0000if $S$ is a simple inverse Hausdorff semitopological $omega$-semigroups with\u0000compact maximal subgroups, then $S$ is topologically isomorphic to the\u0000Bruck--Reilly extension\u0000$left(textbf{BR}(T,theta),tau_{textbf{BR}}^{oplus}right)$ of a finite\u0000semilattice $T=left[E;G_alpha,varphi_{alpha,beta}right]$ of compact\u0000groups $G_alpha$ in the class of topological inverse semigroups, where\u0000$tau_{textbf{BR}}^{oplus}$ is the sum direct topology on\u0000$textbf{BR}(T,theta)$. Also we prove that every Hausdorff locally compact\u0000shift-continuous topology on the simple inverse Hausdorff semitopological\u0000$omega$-semigroups with compact maximal subgroups with adjoined zero is either\u0000compact or discrete.","PeriodicalId":501037,"journal":{"name":"arXiv - MATH - Group Theory","volume":"25 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142206434","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}
Montserrat Casals-Ruiz, Ilya Kazachkov, Mallika Roy
In this note, we characterise when the kernel of a rational character of a right-anlged Artin group, also known as generalised Bestiva-Brady group, is finitely generated and finitely presented. In these cases, we exhibit a finite generating set and a presentation. These results generalise Dicks and Leary's presentations of Bestina-Brady kernels and provide an algebraic proof for the results proven by Meier, Meinert, and VanWyk.
{"title":"Presentation of kernels of rational characters of right-angled Artin groups","authors":"Montserrat Casals-Ruiz, Ilya Kazachkov, Mallika Roy","doi":"arxiv-2409.06315","DOIUrl":"https://doi.org/arxiv-2409.06315","url":null,"abstract":"In this note, we characterise when the kernel of a rational character of a\u0000right-anlged Artin group, also known as generalised Bestiva-Brady group, is\u0000finitely generated and finitely presented. In these cases, we exhibit a finite\u0000generating set and a presentation. These results generalise Dicks and Leary's\u0000presentations of Bestina-Brady kernels and provide an algebraic proof for the\u0000results proven by Meier, Meinert, and VanWyk.","PeriodicalId":501037,"journal":{"name":"arXiv - MATH - Group Theory","volume":"13 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142206436","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 describe train track automata for large classes of fully irreducible elements of Out($F_r$), and their associated geodesics in Culler-Vogtmann Outer Space.
{"title":"Out($F_r$) train track automata I: Proper full fold decompositions","authors":"Catherine Eva Pfaff","doi":"arxiv-2409.05599","DOIUrl":"https://doi.org/arxiv-2409.05599","url":null,"abstract":"We describe train track automata for large classes of fully irreducible\u0000elements of Out($F_r$), and their associated geodesics in Culler-Vogtmann Outer\u0000Space.","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":"142206443","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}
京公网安备 11010802042870号
京ICP备2023020795号-1
扫码关注我们
求助内容：
应助结果提醒方式：