Geometriae Dedicata最新文献

{"title":"Prym representations of the handlebody group","authors":"","doi":"10.1007/s10711-024-00911-5","DOIUrl":"https://doi.org/10.1007/s10711-024-00911-5","url":null,"abstract":"<h3>Abstract</h3> <p>Let <em>S</em> be an oriented, closed surface of genus <em>g</em>. The mapping class group of <em>S</em> is the group of orientation preserving homeomorphisms of <em>S</em> modulo isotopy. In 1997, Looijenga introduced the Prym representations, which are virtual representations of the mapping class group that depend on a finite, abelian group. Let <em>V</em> be a genus <em>g</em> handlebody with boundary <em>S</em>. The handlebody group is the subgroup of those mapping classes of <em>S</em> that extend over <em>V</em>. The twist group is the subgroup of the handlebody group generated by twists about meridians. Here, we restrict the Prym representations to the handlebody group and further to the twist group. We determine the image of the representations in the cyclic case.</p>","PeriodicalId":55103,"journal":{"name":"Geometriae Dedicata","volume":null,"pages":null},"PeriodicalIF":0.5,"publicationDate":"2024-03-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140577943","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Mirror stabilizers for lattice complex hyperbolic triangle groups","authors":"Martin Deraux","doi":"10.1007/s10711-024-00910-6","DOIUrl":"https://doi.org/10.1007/s10711-024-00910-6","url":null,"abstract":"<p>For each lattice complex hyperbolic triangle group, we study the Fuchsian stabilizers of (reprentatives of each group orbit of) mirrors of complex reflections. We give explicit generators for the stabilizers, and compute their signature in the sense of Fuchsian groups. For some groups, we also find explicit pairs of complex lines such that the union of their stabilizers generate the ambient lattice.</p>","PeriodicalId":55103,"journal":{"name":"Geometriae Dedicata","volume":null,"pages":null},"PeriodicalIF":0.5,"publicationDate":"2024-03-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140315858","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"K3 surfaces with two involutions and low Picard number","authors":"Dino Festi, Wim Nijgh, Daniel Platt","doi":"10.1007/s10711-024-00900-8","DOIUrl":"https://doi.org/10.1007/s10711-024-00900-8","url":null,"abstract":"<p>Let <i>X</i> be a complex algebraic K3 surface of degree 2<i>d</i> and with Picard number <span>(rho )</span>. Assume that <i>X</i> admits two commuting involutions: one holomorphic and one anti-holomorphic. In that case, <span>(rho ge 1)</span> when <span>(d=1)</span> and <span>(rho ge 2)</span> when <span>(d ge 2)</span>. For <span>(d=1)</span>, the first example defined over <span>({mathbb {Q}})</span> with <span>(rho =1)</span> was produced already in 2008 by Elsenhans and Jahnel. A K3 surface provided by Kondō, also defined over <span>({mathbb {Q}})</span>, can be used to realise the minimum <span>(rho =2)</span> for all <span>(dge 2)</span>. In these notes we construct new explicit examples of K3 surfaces over the rational numbers realising the minimum <span>(rho =2)</span> for <span>(d=2,3,4)</span>. We also show that a nodal quartic surface can be used to realise the minimum <span>(rho =2)</span> for infinitely many different values of <i>d</i>. Finally, we strengthen a result of Morrison by showing that for any even lattice <i>N</i> of rank <span>(1le r le 10)</span> and signature <span>((1,r-1))</span> there exists a K3 surface <i>Y</i> defined over <span>({mathbb {R}})</span> such that <span>({{,textrm{Pic},}}Y_{mathbb {C}}={{,textrm{Pic},}}Y cong N)</span>.\u0000</p>","PeriodicalId":55103,"journal":{"name":"Geometriae Dedicata","volume":null,"pages":null},"PeriodicalIF":0.5,"publicationDate":"2024-03-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140125414","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Left-invariant distributions diffeomorphic to flat distributions","authors":"Sebastiano Nicolussi Golo, Alessandro Ottazzi","doi":"10.1007/s10711-024-00905-3","DOIUrl":"https://doi.org/10.1007/s10711-024-00905-3","url":null,"abstract":"<p>For a stratified group <i>G</i>, we construct a class of Lie groups endowed with a left-invariant distribution locally diffeomorphic to the flat distribution of <i>G</i>. Vice versa, we show that all Lie groups with a left-invariant distribution that is locally diffeomorphic to the flat distribution of <i>G</i> belong to the class we constructed, if the Lie algebra of <i>G</i> has finite Tanaka prolongation.\u0000</p>","PeriodicalId":55103,"journal":{"name":"Geometriae Dedicata","volume":null,"pages":null},"PeriodicalIF":0.5,"publicationDate":"2024-03-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140125261","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"A geometric characterization of cyclic p-gonal surfaces","authors":"Daniel M. Gallo","doi":"10.1007/s10711-024-00902-6","DOIUrl":"https://doi.org/10.1007/s10711-024-00902-6","url":null,"abstract":"<p>A closed Riemann surface <i>S</i> of genus <span>(gge 2)</span> is called <i>cyclic p-gonal</i> if it has an automorphism <span>(rho )</span> of order <i>p</i>, where <i>p</i> is a prime, such that <span>(S/langle rho rangle )</span> has genus 0. For <span>(p=2)</span>, the surface is called hyperelliptic and <span>(rho )</span> is an involution with <span>(2g+2)</span> fixed points. Classicaly, cyclic p-gonal surfaces can be characterized using Fuchsian groups. In this paper we establish a geometric characterization of cyclic p-gonal surfaces. Specifically, this is determined by collections of simple geodesic arcs on the surfaces and graphs associated to these arcs. In previous work, the author has given a geometric characterization of hyperelliptic surfaces in terms of simple, closed geodesics and graphs associated to these. The present work may be seen as an extension. Involutions, however, have properties that do not generalize to arbitrary automorphisms of order <i>p</i>. Hence, the number of vertices needed in the graphs used here is larger than that of the hyperelliptic case.</p>","PeriodicalId":55103,"journal":{"name":"Geometriae Dedicata","volume":null,"pages":null},"PeriodicalIF":0.5,"publicationDate":"2024-03-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140125647","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Rupert property of some particular n-simplices and n-octahedrons","authors":"Pongbunthit Tonpho, Wacharin Wichiramala","doi":"10.1007/s10711-024-00894-3","DOIUrl":"https://doi.org/10.1007/s10711-024-00894-3","url":null,"abstract":"<p>Three hundred years ago, Prince Rupert of Rhine showed that a unit cube has the property that one copy of it can be passed through a suitable hole in another copy. Under this situation, we say that a unit cube has the Rupert property. In the past years, there are many research studying about the Rupert property of many solids in <span>(mathbb {R}^3)</span>. For higher dimensions, the <i>n</i>-dimensional cube and the regular <i>n</i>-simplex were studied to have the Rupert property. In this work, we focus on the Rupert property of some polyhedrons in <i>n</i> dimensions. In particular, we show that some particular <i>n</i>-dimensional simplices, generalized <i>n</i>-dimensional octahedrons and some related solids in <span>(mathbb {R}^n)</span> have the Rupert property using arbitrarily small rotations and translations.</p>","PeriodicalId":55103,"journal":{"name":"Geometriae Dedicata","volume":null,"pages":null},"PeriodicalIF":0.5,"publicationDate":"2024-03-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140045022","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Cusped Borel Anosov representations with positivity","authors":"Gye-Seon Lee, Tengren Zhang","doi":"10.1007/s10711-024-00895-2","DOIUrl":"https://doi.org/10.1007/s10711-024-00895-2","url":null,"abstract":"<p>We show that if a cusped Borel Anosov representation from a lattice <span>(Gamma subset textsf{PGL}_2({{,mathrm{mathbb {R}},}}))</span> to <span>(textsf{PGL}_d({{,mathrm{mathbb {R}},}}))</span> contains a unipotent element with a single Jordan block in its image, then it is necessarily a (cusped) Hitchin representation. We also show that the amalgamation of a Hitchin representation with a cusped Borel Anosov representation that is not Hitchin is never cusped Borel Anosov.\u0000</p>","PeriodicalId":55103,"journal":{"name":"Geometriae Dedicata","volume":null,"pages":null},"PeriodicalIF":0.5,"publicationDate":"2024-03-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140044804","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Curvature bounds on length-minimizing discs","authors":"","doi":"10.1007/s10711-024-00892-5","DOIUrl":"https://doi.org/10.1007/s10711-024-00892-5","url":null,"abstract":"<p>We show that a length-minimizing disk inherits the upper curvature bound of the target. As a consequence we prove that harmonic discs and ruled discs inherit the upper curvature bound from the ambient space.</p>","PeriodicalId":55103,"journal":{"name":"Geometriae Dedicata","volume":null,"pages":null},"PeriodicalIF":0.5,"publicationDate":"2024-03-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140045332","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Quasi-metric antipodal spaces and maximal Gromov hyperbolic spaces","authors":"Kingshook Biswas","doi":"10.1007/s10711-024-00903-5","DOIUrl":"https://doi.org/10.1007/s10711-024-00903-5","url":null,"abstract":"<p>Hyperbolic fillings of metric spaces are a well-known tool for proving results on extending quasi-Moebius maps between boundaries of Gromov hyperbolic spaces to quasi-isometries between the spaces. For a hyperbolic filling <i>Y</i> of the boundary of a Gromov hyperbolic space <i>X</i>, one has a quasi-Moebius identification between the boundaries <span>(partial Y)</span> and <span>(partial X)</span>. For CAT(-1) spaces, and more generally boundary continuous Gromov hyperbolic spaces, one can refine the quasi-Moebius structure on the boundary to a Moebius structure. It is then natural to ask whether there exists a functorial hyperbolic filling of the boundary by a boundary continuous Gromov hyperbolic space with an identification between boundaries which is not just quasi-Moebius, but in fact Moebius. The filling should be functorial in the sense that a Moebius homeomorphism between boundaries should induce an isometry between there fillings. We give a positive answer to this question for a large class of boundaries satisfying one crucial hypothesis, the <i>antipodal property</i>. This gives a class of compact spaces called <i>quasi-metric antipodal spaces</i>. For any such space <i>Z</i>, we give a functorial construction of a boundary continuous Gromov hyperbolic space <span>(mathcal {M}(Z))</span> together with a Moebius identification of its boundary with <i>Z</i>. The space <span>(mathcal {M}(Z))</span> is maximal amongst all fillings of <i>Z</i>. These spaces <span>(mathcal {M}(Z))</span> give in fact all examples of a natural class of spaces called <i>maximal Gromov hyperbolic spaces</i>. We prove an equivalence of categories between quasi-metric antipodal spaces and maximal Gromov hyperbolic spaces. This is part of a more general equivalence we prove between the larger categories of certain spaces called <i>antipodal spaces</i> and <i>maximal Gromov product spaces</i>. We prove that the injective hull of a Gromov product space <i>X</i> is isometric to the maximal Gromov product space <span>(mathcal {M}(Z))</span>, where <i>Z</i> is the boundary of <i>X</i>. We also show that a Gromov product space is injective if and only if it is maximal.</p>","PeriodicalId":55103,"journal":{"name":"Geometriae Dedicata","volume":null,"pages":null},"PeriodicalIF":0.5,"publicationDate":"2024-03-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140045025","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Homogeneous semisymmetric neutral 4-manifolds","authors":"","doi":"10.1007/s10711-024-00898-z","DOIUrl":"https://doi.org/10.1007/s10711-024-00898-z","url":null,"abstract":"<h3>Abstract</h3> <p>We determine homogeneous semi-symmetric neutral manifolds of dimension 4. We also describe all the possible semi-symmetric curvature tensors on four-dimensional neutral vector spaces.</p>","PeriodicalId":55103,"journal":{"name":"Geometriae Dedicata","volume":null,"pages":null},"PeriodicalIF":0.5,"publicationDate":"2024-03-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140044809","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}