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Geometriae Dedicata最新文献

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The Menger curve and spherical CR uniformization of a closed hyperbolic 3-orbifold 封闭双曲 3 轨道的门格尔曲线和球面 CR 均匀化
IF 0.5 4区 数学
Geometriae Dedicata Pub Date : 2024-06-17 DOI: 10.1007/s10711-024-00934-y
Jiming Ma, Baohua Xie
{"title":"The Menger curve and spherical CR uniformization of a closed hyperbolic 3-orbifold","authors":"Jiming Ma, Baohua Xie","doi":"10.1007/s10711-024-00934-y","DOIUrl":"https://doi.org/10.1007/s10711-024-00934-y","url":null,"abstract":"<p>Let <span>(G_{6,3})</span> be a hyperbolic polygon-group with boundary the Menger curve. Granier (Groupes discrets en géométrie hyperbolique—aspects effectifs, Université de Fribourg, 2015) constructed a discrete, convex cocompact and faithful representation <span>(rho )</span> of <span>(G_{6,3})</span> into <span>(textbf{PU}(2,1))</span>. We show the 3-orbifold at infinity of <span>(rho (G_{6,3}))</span> is a closed hyperbolic 3-orbifold, with underlying space the 3-sphere and singular locus the <span>({mathbb {Z}}_3)</span>-coned chain-link <span>(C(6,-2))</span>. This answers the second part of Kapovich’s Conjecture 10.6 in Kapovich (in: In the tradition of thurston II. Geometry and groups, Springer, Cham, 2022), and it also provides the second explicit example of a closed hyperbolic 3-orbifold that admits a uniformizable spherical CR-structure after Schwartz’s first example in Schwartz (Invent Math 151(2):221–295, 2003).</p>","PeriodicalId":55103,"journal":{"name":"Geometriae Dedicata","volume":"131 1","pages":""},"PeriodicalIF":0.5,"publicationDate":"2024-06-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141509589","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}
引用次数: 0
Isoperiodic families of Poncelet polygons inscribed in a circle and circumscribed about conics from a confocal pencil 从共焦点铅笔看刻画在圆内和圆锥周边的庞塞莱多边形的等周期族
IF 0.5 4区 数学
Geometriae Dedicata Pub Date : 2024-05-14 DOI: 10.1007/s10711-024-00929-9
Vladimir Dragović, Milena Radnović
{"title":"Isoperiodic families of Poncelet polygons inscribed in a circle and circumscribed about conics from a confocal pencil","authors":"Vladimir Dragović, Milena Radnović","doi":"10.1007/s10711-024-00929-9","DOIUrl":"https://doi.org/10.1007/s10711-024-00929-9","url":null,"abstract":"<p>Poncelet polygons inscribed in a circle and circumscribed about conics from a confocal family naturally arise in the analysis of the numerical range and Blaschke products. We examine the behaviour of such polygons when the inscribed conic varies through a confocal pencil and discover cases when each conic from the confocal family is inscribed in an <i>n</i>-polygon, which is inscribed in the circle, with the same <i>n</i>. Complete geometric characterization of such cases for <span>(nin {4,6})</span> is given and proved that this cannot happen for other values of <i>n</i>. We establish a relationship of such families of Poncelet quadrangles and hexagons to solutions of a Painlevé VI equation.</p>","PeriodicalId":55103,"journal":{"name":"Geometriae Dedicata","volume":"121 1","pages":""},"PeriodicalIF":0.5,"publicationDate":"2024-05-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141060280","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}
引用次数: 0
Fundamental groups and group presentations with bounded relator lengths 有界关系长度的基群和群呈现
IF 0.5 4区 数学
Geometriae Dedicata Pub Date : 2024-05-10 DOI: 10.1007/s10711-024-00915-1
Sergio Zamora
{"title":"Fundamental groups and group presentations with bounded relator lengths","authors":"Sergio Zamora","doi":"10.1007/s10711-024-00915-1","DOIUrl":"https://doi.org/10.1007/s10711-024-00915-1","url":null,"abstract":"<p>We study the geometry of compact geodesic spaces with trivial first Betti number admitting large finite groups of isometries. We show that if a finite group <i>G</i> acts by isometries on a compact geodesic space <i>X</i> whose first Betti number vanishes, then <span>({text {diam}}(X) / {text {diam}}(X / G ) le 4 sqrt{ vert G vert })</span>. For a group <i>G</i> and a finite symmetric generating set <i>S</i>, <span>(P_k(varGamma (G, S)))</span> denotes the 2-dimensional CW-complex whose 1-skeleton is the Cayley graph <span>(varGamma )</span> of <i>G</i> with respect to <i>S</i> and whose 2-cells are <i>m</i>-gons for <span>(0 le m le k)</span>, defined by the simple graph loops of length <i>m</i> in <span>(varGamma )</span>, up to cyclic permutations. Let <i>G</i> be a finite abelian group with <span>(vert G vert ge 3)</span> and <i>S</i> a symmetric set of generators for which <span>(P_k(varGamma (G,S)))</span> has trivial first Betti number. We show that the first nontrivial eigenvalue <span>(-lambda _1)</span> of the Laplacian on the Cayley graph satisfies <span>(lambda _1 ge 2 - 2 cos ( 2 pi / k ) )</span>. We also give an explicit upper bound on the diameter of the Cayley graph of <i>G</i> with respect to <i>S</i> of the form <span>(O (k^2 vert S vert log vert G vert ))</span>. Related explicit bounds for the Cheeger constant and Kazhdan constant of the pair (<i>G</i>, <i>S</i>) are also obtained.</p>","PeriodicalId":55103,"journal":{"name":"Geometriae Dedicata","volume":"40 1","pages":""},"PeriodicalIF":0.5,"publicationDate":"2024-05-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140929899","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}
引用次数: 0
The Liouville current of holomorphic quadratic differential metrics 全态二次微分度量的柳维尔电流
IF 0.5 4区 数学
Geometriae Dedicata Pub Date : 2024-05-07 DOI: 10.1007/s10711-024-00928-w
Jiajun Shi
{"title":"The Liouville current of holomorphic quadratic differential metrics","authors":"Jiajun Shi","doi":"10.1007/s10711-024-00928-w","DOIUrl":"https://doi.org/10.1007/s10711-024-00928-w","url":null,"abstract":"<p>In this paper we study the Liouville current of flat cone metrics coming from a holomorphic quadratic differential. Anja Bankovic and Christopher J. Leininger proved in Bankovic and Leininger (Trans Am Math Soc 370:1867–1884, 2018) that for a fixed closed surface, there is an injection map from the space of flat cone metrics to the space of geodesic currents. We manage to show that metrics coming from holomorphic quadratic differentials can be distinguished from other flat metrics by just looking at the geodesic currents. The key idea is to analyze the support of Liouville current, which is a topological invariant independent of the metric, and get information about cone angles and holonomy. The holonomy part involves some subtlety of relationship between singular foliation and geodesic lamination. We also obtain a new proof of a classical result that almost all simple geodesics of a quadratic differential metric will be dense in the surface. Furthermore, for other flat cone metrics, there is no simple dense geodesic.</p>","PeriodicalId":55103,"journal":{"name":"Geometriae Dedicata","volume":"25 1","pages":""},"PeriodicalIF":0.5,"publicationDate":"2024-05-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140884557","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}
引用次数: 0
An extension of the Thurston metric to projective filling currents 瑟斯顿度量向投影填充流的扩展
IF 0.5 4区 数学
Geometriae Dedicata Pub Date : 2024-05-06 DOI: 10.1007/s10711-024-00914-2
Jenya Sapir
{"title":"An extension of the Thurston metric to projective filling currents","authors":"Jenya Sapir","doi":"10.1007/s10711-024-00914-2","DOIUrl":"https://doi.org/10.1007/s10711-024-00914-2","url":null,"abstract":"<p>We study the geometry of the space of projectivized filling geodesic currents <span>(mathbb {P}mathcal {C}_{fill}(S))</span>. Bonahon showed that Teichmüller space, <span>(mathcal {T}(S))</span> embeds into <span>(mathbb {P}mathcal {C}_{fill}(S))</span>. We extend the symmetrized Thurston metric from <span>(mathcal {T}(S))</span> to the entire (projectivized) space of filling currents, and we show that <span>(mathcal {T}(S))</span> is isometrically embedded into the bigger space. Moreover, we show that there is no quasi-isometric projection back down to <span>(mathcal {T}(S))</span>. Lastly, we study the geometry of a length-minimizing projection from <span>(mathbb {P}mathcal {C}_{fill}(S))</span> to <span>(mathcal {T}(S))</span> defined previously by Hensel and the author.</p>","PeriodicalId":55103,"journal":{"name":"Geometriae Dedicata","volume":"23 1","pages":""},"PeriodicalIF":0.5,"publicationDate":"2024-05-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140884251","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}
引用次数: 0
Temperedness of locally symmetric spaces: the product case 局部对称空间的可变性：乘积情况
IF 0.5 4区 数学
Geometriae Dedicata Pub Date : 2024-05-03 DOI: 10.1007/s10711-024-00904-4
Tobias Weich, Lasse L. Wolf
{"title":"Temperedness of locally symmetric spaces: the product case","authors":"Tobias Weich, Lasse L. Wolf","doi":"10.1007/s10711-024-00904-4","DOIUrl":"https://doi.org/10.1007/s10711-024-00904-4","url":null,"abstract":"<p>Let <span>(X=X_1times X_2)</span> be a product of two rank one symmetric spaces of non-compact type and <span>(Gamma )</span> a torsion-free discrete subgroup in <span>(G_1times G_2)</span>. We show that the spectrum of <span>(Gamma backslash (X_1times X_2))</span> is related to the asymptotic growth of <span>(Gamma )</span> in the two directions defined by the two factors. We obtain that <span>(L^2(Gamma backslash (G_1 times G_2)))</span> is tempered for a large class of <span>(Gamma )</span>.\u0000</p>","PeriodicalId":55103,"journal":{"name":"Geometriae Dedicata","volume":"62 1","pages":""},"PeriodicalIF":0.5,"publicationDate":"2024-05-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140884250","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}
引用次数: 0
Minimality and unique ergodicity of Veech 1969 type interval exchange transformations 维奇 1969 型区间交换变换的最小性和唯一遍历性
IF 0.5 4区 数学
Geometriae Dedicata Pub Date : 2024-05-03 DOI: 10.1007/s10711-024-00888-1
Sébastien Ferenczi, Pascal Hubert
{"title":"Minimality and unique ergodicity of Veech 1969 type interval exchange transformations","authors":"Sébastien Ferenczi, Pascal Hubert","doi":"10.1007/s10711-024-00888-1","DOIUrl":"https://doi.org/10.1007/s10711-024-00888-1","url":null,"abstract":"<p>We give conditions for minimality of <span>({mathbb {Z}}/N{mathbb {Z}})</span> extensions of a rotation of angle <span>(alpha )</span> with one marked point, solving the problem for any prime <i>N</i>: for <span>(N=2)</span>, these correspond to the Veech 1969 examples, for which a necessary and sufficient condition was not known yet. We provide also a word combinatorial criterion of minimality valid for general interval exchange transformations, which applies to <span>({mathbb {Z}}/N{mathbb {Z}})</span> extensions of any interval exchange transformation with any number of marked points. Then we give a condition for unique ergodicity of these extensions when the initial interval exchange transformation is linearly recurrent and there are one or two marked points.\u0000</p>","PeriodicalId":55103,"journal":{"name":"Geometriae Dedicata","volume":"4 1","pages":""},"PeriodicalIF":0.5,"publicationDate":"2024-05-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140884252","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}
引用次数: 0
Distribution of periodic orbits in the homology group of a knot complement 绳结补集同调群中周期轨道的分布
IF 0.5 4区 数学
Geometriae Dedicata Pub Date : 2024-04-29 DOI: 10.1007/s10711-024-00927-x
Solly Coles, Richard Sharp
{"title":"Distribution of periodic orbits in the homology group of a knot complement","authors":"Solly Coles, Richard Sharp","doi":"10.1007/s10711-024-00927-x","DOIUrl":"https://doi.org/10.1007/s10711-024-00927-x","url":null,"abstract":"<p>Consider a transitive Anosov flow on a closed 3-manifold. After removing a finite set of null-homologous periodic orbits, we study the distribution of the remaining periodic orbits in the homology of the knot complement.\u0000</p>","PeriodicalId":55103,"journal":{"name":"Geometriae Dedicata","volume":"12 1","pages":""},"PeriodicalIF":0.5,"publicationDate":"2024-04-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140839802","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}
引用次数: 0
Decomposition complexity growth of finitely generated groups 有限生成群的分解复杂性增长
IF 0.5 4区 数学
Geometriae Dedicata Pub Date : 2024-04-29 DOI: 10.1007/s10711-024-00924-0
Trevor Davila
{"title":"Decomposition complexity growth of finitely generated groups","authors":"Trevor Davila","doi":"10.1007/s10711-024-00924-0","DOIUrl":"https://doi.org/10.1007/s10711-024-00924-0","url":null,"abstract":"<p>Finite decomposition complexity and asymptotic dimension growth are two generalizations of M. Gromov’s asymptotic dimension which can be used to prove property A for large classes of finitely generated groups of infinite asymptotic dimension. In this paper, we introduce the notion of decomposition complexity growth, which is a quasi-isometry invariant generalizing both finite decomposition complexity and dimension growth. We show that subexponential decomposition complexity growth implies property A, and is preserved by certain group and metric constructions.</p>","PeriodicalId":55103,"journal":{"name":"Geometriae Dedicata","volume":"14 1","pages":""},"PeriodicalIF":0.5,"publicationDate":"2024-04-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140839893","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}
引用次数: 0
Scalar curvature along the Ricci flow 沿利玛窦流的标量曲率
IF 0.5 4区 数学
Geometriae Dedicata Pub Date : 2024-04-15 DOI: 10.1007/s10711-024-00913-3
Yi Li
{"title":"Scalar curvature along the Ricci flow","authors":"Yi Li","doi":"10.1007/s10711-024-00913-3","DOIUrl":"https://doi.org/10.1007/s10711-024-00913-3","url":null,"abstract":"<p>In this note, we prove a well-known conjecture on the Ricci flow under a curvature condition, which is a pinching between the Ricci and Weyl tensors divided by suitably translated scalar curvature, motivated by Cao’s result (Commun Anal Geom 19(5):975–990, 2011).\u0000</p>","PeriodicalId":55103,"journal":{"name":"Geometriae Dedicata","volume":"43 1","pages":""},"PeriodicalIF":0.5,"publicationDate":"2024-04-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140578049","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}
引用次数: 0
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