Pub Date : 2019-05-20DOI: 10.1142/S0218216519500974
J. Mostovoy, Christopher Roque-M'arquez
The group of planar (or flat) pure braids on $n$ strands, also known as the pure twin group, is the fundamental group of the configuration space $F_{n,3}(mathbb{R})$ of $n$ labelled points in $mathbb{R}$ no three of which coincide. The planar pure braid groups on 3, 4 and 5 strands are free. In this note we describe the planar pure braid group on 6 strands: it is a free product of the free group on 71 generators and 20 copies of the free abelian group of rank two.
{"title":"Planar pure braids on six strands","authors":"J. Mostovoy, Christopher Roque-M'arquez","doi":"10.1142/S0218216519500974","DOIUrl":"https://doi.org/10.1142/S0218216519500974","url":null,"abstract":"The group of planar (or flat) pure braids on $n$ strands, also known as the pure twin group, is the fundamental group of the configuration space $F_{n,3}(mathbb{R})$ of $n$ labelled points in $mathbb{R}$ no three of which coincide. The planar pure braid groups on 3, 4 and 5 strands are free. In this note we describe the planar pure braid group on 6 strands: it is a free product of the free group on 71 generators and 20 copies of the free abelian group of rank two.","PeriodicalId":8427,"journal":{"name":"arXiv: Group Theory","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2019-05-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"74823741","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}
To any finite graph $X$ (viewed as a topological space) we assosiate some explicit compact metric space ${cal X}^r(X)$ which we call {it the reflection tree of graphs $X$}. This space is of topological dimension $le1$ and its connected components are locally connected. We show that if $X$ is appropriately triangulated (as a simplicial graph $Gamma$ for which $X$ is the geometric realization) then the visual boundary $partial_infty(W,S)$ of the right angled Coxeter system $(W,S)$ with the nerve isomorphic to $Gamma$ is homeomorphic to ${cal X}^r(X)$. For each $X$, this yields in particular many word hyperbolic groups with Gromov boundary homeomorphic to the space ${cal X}^r(X)$.
{"title":"Reflection trees of graphs as boundaries of Coxeter groups","authors":"Jacek 'Swikatkowski","doi":"10.2140/AGT.2021.21.351","DOIUrl":"https://doi.org/10.2140/AGT.2021.21.351","url":null,"abstract":"To any finite graph $X$ (viewed as a topological space) we assosiate some explicit compact metric space ${cal X}^r(X)$ which we call {it the reflection tree of graphs $X$}. This space is of topological dimension $le1$ and its connected components are locally connected. We show that if $X$ is appropriately triangulated (as a simplicial graph $Gamma$ for which $X$ is the geometric realization) then the visual boundary $partial_infty(W,S)$ of the right angled Coxeter system $(W,S)$ with the nerve isomorphic to $Gamma$ is homeomorphic to ${cal X}^r(X)$. For each $X$, this yields in particular many word hyperbolic groups with Gromov boundary homeomorphic to the space ${cal X}^r(X)$.","PeriodicalId":8427,"journal":{"name":"arXiv: Group Theory","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2019-05-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"75846761","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 construct asynchronous and sometimes synchronous automatic structures for amalgamated products and HNN extensions of groups that are strongly asynchronously (or synchronously) coset automatic with respect to the associated automatic subgroups, subject to further geometric conditions. These results are proved in the general context of fundamental groups of graphs of groups. The hypotheses of our closure results are satisfied in a variety of examples such as Artin groups of sufficiently large type, Coxeter groups, virtually abelian groups, and groups that are hyperbolic relative to virtually abelian subgroups.
{"title":"Automaticity for graphs of groups","authors":"S. Hermiller, D. Holt, T. Susse, Sarah Rees","doi":"10.4171/GGD/605","DOIUrl":"https://doi.org/10.4171/GGD/605","url":null,"abstract":"In this article we construct asynchronous and sometimes synchronous automatic structures for amalgamated products and HNN extensions of groups that are strongly asynchronously (or synchronously) coset automatic with respect to the associated automatic subgroups, subject to further geometric conditions. These results are proved in the general context of fundamental groups of graphs of groups. The hypotheses of our closure results are satisfied in a variety of examples such as Artin groups of sufficiently large type, Coxeter groups, virtually abelian groups, and groups that are hyperbolic relative to virtually abelian subgroups.","PeriodicalId":8427,"journal":{"name":"arXiv: Group Theory","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2019-05-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"90181590","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 present a powerful theorem for proving the irreducibility of tempered unitary representations of the free group.
给出了一个证明自由群的缓变酉表示的不可约性的有力定理。
{"title":"Free group representations: Duplicity on the boundary","authors":"W. Hebisch, M. Kuhn, T. Steger","doi":"10.1090/tran/8546","DOIUrl":"https://doi.org/10.1090/tran/8546","url":null,"abstract":"We present a powerful theorem for proving the irreducibility of tempered unitary representations of the free group.","PeriodicalId":8427,"journal":{"name":"arXiv: Group Theory","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2019-05-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"83935982","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}
Author(s): Dey, Subhadip; Kapovich, Michael | Abstract: We extend several notions and results from the classical Patterson-Sullivan theory to the setting of Anosov subgroups of higher rank semisimple Lie groups, working primarily with invariant Finsler metrics on associated symmetric spaces. In particular, we prove the equality between the Hausdorff dimensions of flag limit sets, computed with respect to a suitable Gromov (pre-)metric on the flag manifold, and the Finsler critical exponents of Anosov subgroups.
{"title":"Patterson-Sullivan theory for Anosov subgroups","authors":"S. Dey, M. Kapovich","doi":"10.1090/tran/8713","DOIUrl":"https://doi.org/10.1090/tran/8713","url":null,"abstract":"Author(s): Dey, Subhadip; Kapovich, Michael | Abstract: We extend several notions and results from the classical Patterson-Sullivan theory to the setting of Anosov subgroups of higher rank semisimple Lie groups, working primarily with invariant Finsler metrics on associated symmetric spaces. In particular, we prove the equality between the Hausdorff dimensions of flag limit sets, computed with respect to a suitable Gromov (pre-)metric on the flag manifold, and the Finsler critical exponents of Anosov subgroups.","PeriodicalId":8427,"journal":{"name":"arXiv: Group Theory","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2019-04-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"90064109","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 investigate groups whose Cayley graphs have poor-ly connected subgraphs. We prove that a finitely generated group has bounded separation in the sense of Benjamini--Schramm--Timar if and only if it is virtually free. We then prove a gap theorem for connectivity of finitely presented groups, and prove that there is no comparable theorem for all finitely generated groups. Finally, we formulate a connectivity version of the conjecture that every group of type $F$ with no Baumslag-Solitar subgroup is hyperbolic, and prove it for groups with at most quadratic Dehn function.
{"title":"Poorly connected groups","authors":"David Hume, J. M. Mackay","doi":"10.1090/PROC/15128","DOIUrl":"https://doi.org/10.1090/PROC/15128","url":null,"abstract":"We investigate groups whose Cayley graphs have poor-ly connected subgraphs. We prove that a finitely generated group has bounded separation in the sense of Benjamini--Schramm--Timar if and only if it is virtually free. We then prove a gap theorem for connectivity of finitely presented groups, and prove that there is no comparable theorem for all finitely generated groups. Finally, we formulate a connectivity version of the conjecture that every group of type $F$ with no Baumslag-Solitar subgroup is hyperbolic, and prove it for groups with at most quadratic Dehn function.","PeriodicalId":8427,"journal":{"name":"arXiv: Group Theory","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2019-04-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"88309454","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}
Ostrom and Wagner (1959) proved that if the automorphism group $G$ of a finite projective plane $pi$ acts $2$-transitively on the points of $pi$, then $pi$ is isomorphic to the Desarguesian projective plane and $G$ is isomorphic to $mathrm{PGamma L}(3,q)$ (for some prime-power $q$). In the more general case of a finite rank $2$ irreducible spherical building, also known as a emph{generalized polygon}, the theorem of Fong and Seitz (1973) gave a classification of the emph{Moufang} examples. A conjecture of Kantor, made in print in 1991, says that there are only two non-classical examples of flag-transitive generalized quadrangles up to duality. Recently, the authors made progress toward this conjecture by classifying those finite generalized quadrangles which have an automorphism group $G$ acting transitively on antiflags. In this paper, we take this classification much further by weakening the hypothesis to $G$ being transitive on ordered pairs of collinear points and ordered pairs of concurrent lines.
{"title":"A classification of finite locally 2-transitive generalized quadrangles","authors":"J. Bamberg, Caiheng Li, Eric Swartz","doi":"10.1090/tran/8236","DOIUrl":"https://doi.org/10.1090/tran/8236","url":null,"abstract":"Ostrom and Wagner (1959) proved that if the automorphism group $G$ of a finite projective plane $pi$ acts $2$-transitively on the points of $pi$, then $pi$ is isomorphic to the Desarguesian projective plane and $G$ is isomorphic to $mathrm{PGamma L}(3,q)$ (for some prime-power $q$). In the more general case of a finite rank $2$ irreducible spherical building, also known as a emph{generalized polygon}, the theorem of Fong and Seitz (1973) gave a classification of the emph{Moufang} examples. A conjecture of Kantor, made in print in 1991, says that there are only two non-classical examples of flag-transitive generalized quadrangles up to duality. Recently, the authors made progress toward this conjecture by classifying those finite generalized quadrangles which have an automorphism group $G$ acting transitively on antiflags. In this paper, we take this classification much further by weakening the hypothesis to $G$ being transitive on ordered pairs of collinear points and ordered pairs of concurrent lines.","PeriodicalId":8427,"journal":{"name":"arXiv: Group Theory","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2019-03-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"90580101","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}
Motivated by recent activity in low-dimensional topology, we provide a new criterion for left-orderability of a group under the assumption that the group is circularly-orderable: A group $G$ is left-orderable if and only if $G times mathbb{Z}/nmathbb{Z}$ is circularly-orderable for all $n > 1$. This implies that every circularly-orderable group which is not left-orderable gives rise to a collection of positive integers that exactly encode the obstruction to left-orderability, which we call the obstruction spectrum. We precisely describe the behaviour of the obstruction spectrum with respect to torsion, and show that this same behaviour can be mirrored by torsion-free groups, whose obstruction spectra are in general more complex.
{"title":"Promoting circular-orderability to left-orderability","authors":"J. Bell, A. Clay, Tyrone Ghaswala","doi":"10.5802/AIF.3394","DOIUrl":"https://doi.org/10.5802/AIF.3394","url":null,"abstract":"Motivated by recent activity in low-dimensional topology, we provide a new criterion for left-orderability of a group under the assumption that the group is circularly-orderable: A group $G$ is left-orderable if and only if $G times mathbb{Z}/nmathbb{Z}$ is circularly-orderable for all $n > 1$. This implies that every circularly-orderable group which is not left-orderable gives rise to a collection of positive integers that exactly encode the obstruction to left-orderability, which we call the obstruction spectrum. We precisely describe the behaviour of the obstruction spectrum with respect to torsion, and show that this same behaviour can be mirrored by torsion-free groups, whose obstruction spectra are in general more complex.","PeriodicalId":8427,"journal":{"name":"arXiv: Group Theory","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2019-03-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"87470143","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 : 2019-03-07DOI: 10.1007/978-3-030-40822-0_5
S. Eliahou, J. Fromentin
{"title":"Gapsets of Small Multiplicity","authors":"S. Eliahou, J. Fromentin","doi":"10.1007/978-3-030-40822-0_5","DOIUrl":"https://doi.org/10.1007/978-3-030-40822-0_5","url":null,"abstract":"","PeriodicalId":8427,"journal":{"name":"arXiv: Group Theory","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2019-03-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"81525841","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 will give a definition of quadratic forms on bimodules and prove the sandwich classification theorem for subgroups of the general linear group $mathrm{GL}(P)$ normalized by the elementary unitary group $mathrm{EU}(P)$ if $P$ is a nondegenerate bimodule with large enough hyperbolic part.
{"title":"Groups normalized by the odd unitary group","authors":"E. Voronetsky","doi":"10.1090/spmj/1630","DOIUrl":"https://doi.org/10.1090/spmj/1630","url":null,"abstract":"We will give a definition of quadratic forms on bimodules and prove the sandwich classification theorem for subgroups of the general linear group $mathrm{GL}(P)$ normalized by the elementary unitary group $mathrm{EU}(P)$ if $P$ is a nondegenerate bimodule with large enough hyperbolic part.","PeriodicalId":8427,"journal":{"name":"arXiv: Group Theory","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2019-02-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"75218989","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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