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On bounded paradoxical sets and Lie groups 论有界悖论集和李群
IF 0.5 4区 数学
Geometriae Dedicata Pub Date : 2024-04-14 DOI: 10.1007/s10711-024-00923-1
Grzegorz Tomkowicz
{"title":"On bounded paradoxical sets and Lie groups","authors":"Grzegorz Tomkowicz","doi":"10.1007/s10711-024-00923-1","DOIUrl":"https://doi.org/10.1007/s10711-024-00923-1","url":null,"abstract":"<p>We will prove that any non-empty open set in every complete connected metric space (<i>X</i>, <i>d</i>), where balls have compact closures, contains a paradoxical (uncountable) set relative to a non-supramenable connected Lie group that acts continuously and transitively on <i>X</i>.</p>","PeriodicalId":55103,"journal":{"name":"Geometriae Dedicata","volume":"61 1","pages":""},"PeriodicalIF":0.5,"publicationDate":"2024-04-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140577829","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
Trisections obtained by trivially regluing surface-knots 通过表面节点的琐碎回归获得的三段论
IF 0.5 4区 数学
Geometriae Dedicata Pub Date : 2024-04-13 DOI: 10.1007/s10711-024-00919-x
Tsukasa Isoshima
{"title":"Trisections obtained by trivially regluing surface-knots","authors":"Tsukasa Isoshima","doi":"10.1007/s10711-024-00919-x","DOIUrl":"https://doi.org/10.1007/s10711-024-00919-x","url":null,"abstract":"<p>Let <i>S</i> be a <span>(P^2)</span>-knot which is the connected sum of a 2-knot with normal Euler number 0 and an unknotted <span>(P^2)</span>-knot with normal Euler number <span>({pm }{2})</span> in a closed 4-manifold <i>X</i> with trisection <span>(T_{X})</span>. Then, we show that the trisection of <i>X</i> obtained by the trivial gluing of relative trisections of <span>(overline{nu (S)})</span> and <span>(X-nu (S))</span> is diffeomorphic to a stabilization of <span>(T_{X})</span>. It should be noted that this result is not obvious since boundary-stabilizations introduced by Kim and Miller are used to construct a relative trisection of <span>(X-nu (S))</span>. As a corollary, if <span>(X=S^4)</span> and <span>(T_X)</span> was the genus 0 trisection of <span>(S^4)</span>, the resulting trisection is diffeomorphic to a stabilization of the genus 0 trisection of <span>(S^4)</span>. This result is related to the conjecture that is a 4-dimensional analogue of Waldhausen’s theorem on Heegaard splittings.</p>","PeriodicalId":55103,"journal":{"name":"Geometriae Dedicata","volume":"84 1","pages":""},"PeriodicalIF":0.5,"publicationDate":"2024-04-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140577837","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
Betti numbers of nearly $$G_2$$ and nearly Kähler 6-manifolds with Weyl curvature bounds 近 $$G_2$$ 和近 Kähler 6-manifolds 的贝蒂数与韦尔曲率边界
IF 0.5 4区 数学
Geometriae Dedicata Pub Date : 2024-04-12 DOI: 10.1007/s10711-024-00920-4
Anton Iliashenko
{"title":"Betti numbers of nearly $$G_2$$ and nearly Kähler 6-manifolds with Weyl curvature bounds","authors":"Anton Iliashenko","doi":"10.1007/s10711-024-00920-4","DOIUrl":"https://doi.org/10.1007/s10711-024-00920-4","url":null,"abstract":"<p>In this paper we use the Weitzenböck formulas to get information about the Betti numbers of compact nearly <span>(G_2)</span> and compact nearly Kähler 6-manifolds. First, we establish estimates on two curvature-type self adjoint operators on particular spaces assuming bounds on the sectional curvature. Then using the Weitzenböck formulas on harmonic forms, we get results of the form: if certain lower bounds hold for these curvature operators then certain Betti numbers are zero. Finally, we combine both steps above to get sufficient conditions of vanishing of certain Betti numbers based on the bounds on the sectional curvature.\u0000</p>","PeriodicalId":55103,"journal":{"name":"Geometriae Dedicata","volume":"15 1","pages":""},"PeriodicalIF":0.5,"publicationDate":"2024-04-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140578032","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
Branching laws of Klein four-symmetric pairs for $$textrm{Sp}(n,mathbb {R})$$ $$textrm{Sp}(n,mathbb {R})$$ 的克莱因四对称对的分支规律
IF 0.5 4区 数学
Geometriae Dedicata Pub Date : 2024-04-11 DOI: 10.1007/s10711-024-00922-2
Jiaying Ding, Haian He, Huangyuan Pan, Lifu Wang
{"title":"Branching laws of Klein four-symmetric pairs for $$textrm{Sp}(n,mathbb {R})$$","authors":"Jiaying Ding, Haian He, Huangyuan Pan, Lifu Wang","doi":"10.1007/s10711-024-00922-2","DOIUrl":"https://doi.org/10.1007/s10711-024-00922-2","url":null,"abstract":"<p>For the real symplectic groups <span>(G=textrm{Sp}(n,mathbb {R}))</span>, we classify all the Klein four-symmetric pairs <span>((G,G^Gamma ))</span>, and determine whether there exist infinite-dimensional irreducible <span>((mathfrak {g},K))</span>-modules discretely decomposable upon restriction to <span>(G^Gamma )</span>. As a consequence, we obtain a similar result to Chen and He (Int J Math 34(1):2250094, 2023, Corollary 21).</p>","PeriodicalId":55103,"journal":{"name":"Geometriae Dedicata","volume":"84 1","pages":""},"PeriodicalIF":0.5,"publicationDate":"2024-04-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140577828","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
Topological aspects of the space of metric measure spaces 度量空间的拓扑方面
IF 0.5 4区 数学
Geometriae Dedicata Pub Date : 2024-04-11 DOI: 10.1007/s10711-024-00921-3
Daisuke Kazukawa, Hiroki Nakajima, Takashi Shioya
{"title":"Topological aspects of the space of metric measure spaces","authors":"Daisuke Kazukawa, Hiroki Nakajima, Takashi Shioya","doi":"10.1007/s10711-024-00921-3","DOIUrl":"https://doi.org/10.1007/s10711-024-00921-3","url":null,"abstract":"<p>Gromov introduced two distance functions, the box distance and the observable distance, on the space of isomorphism classes of metric measure spaces and developed the convergence theory of metric measure spaces. We investigate several topological properties on the space equipped with these distance functions toward a deep understanding of convergence theory.</p>","PeriodicalId":55103,"journal":{"name":"Geometriae Dedicata","volume":"6 1","pages":""},"PeriodicalIF":0.5,"publicationDate":"2024-04-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140577951","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 Nielsen realization problem for high degree del Pezzo surfaces 高阶德尔佩佐曲面的尼尔森实现问题
IF 0.5 4区 数学
Geometriae Dedicata Pub Date : 2024-04-08 DOI: 10.1007/s10711-024-00912-4
Seraphina Eun Bi Lee
{"title":"The Nielsen realization problem for high degree del Pezzo surfaces","authors":"Seraphina Eun Bi Lee","doi":"10.1007/s10711-024-00912-4","DOIUrl":"https://doi.org/10.1007/s10711-024-00912-4","url":null,"abstract":"<p>Let <i>M</i> be a smooth 4-manifold underlying some del Pezzo surface of degree <span>(d ge 6)</span>. We consider the smooth Nielsen realization problem for <i>M</i>: which finite subgroups of <span>({{,textrm{Mod},}}(M) = pi _0({{,textrm{Homeo},}}^+(M)))</span> have lifts to <span>({{,textrm{Diff},}}^+(M) le {{,textrm{Homeo},}}^+(M))</span> under the quotient map <span>(pi : {{,textrm{Homeo},}}^+(M) rightarrow {{,textrm{Mod},}}(M))</span>? We give a complete classification of such finite subgroups of <span>({{,textrm{Mod},}}(M))</span> for <span>(d ge 7)</span> and a partial answer for <span>(d = 6)</span>. For the cases <span>(d ge 8)</span>, the quotient map <span>(pi )</span> admits a section with image contained in <span>({{,textrm{Diff},}}^+(M))</span>. For the case <span>(d = 7)</span>, we show that all finite order elements of <span>({{,textrm{Mod},}}(M))</span> have lifts to <span>({{,textrm{Diff},}}^+(M))</span>, but there are finite subgroups of <span>({{,textrm{Mod},}}(M))</span> that do not lift to <span>({{,textrm{Diff},}}^+(M))</span>. We prove that the condition of whether a finite subgroup <span>(G le {{,textrm{Mod},}}(M))</span> lifts to <span>({{,textrm{Diff},}}^+(M))</span> is equivalent to the existence of a certain equivariant connected sum realizing <i>G</i>. For the case <span>(d = 6)</span>, we show this equivalence for all maximal finite subgroups <span>(G le {{,textrm{Mod},}}(M))</span>.\u0000</p>","PeriodicalId":55103,"journal":{"name":"Geometriae Dedicata","volume":"149 1","pages":""},"PeriodicalIF":0.5,"publicationDate":"2024-04-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140577830","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
On the Teichmüller stack of compact quotients of $${text {SL}}_2({mathbb {C}})$$ 论 $${text {SL}}_2({mathbb {C}})$$ 的紧凑商的泰希米勒堆栈
IF 0.5 4区 数学
Geometriae Dedicata Pub Date : 2024-04-04 DOI: 10.1007/s10711-024-00916-0
Théo Jamin
{"title":"On the Teichmüller stack of compact quotients of $${text {SL}}_2({mathbb {C}})$$","authors":"Théo Jamin","doi":"10.1007/s10711-024-00916-0","DOIUrl":"https://doi.org/10.1007/s10711-024-00916-0","url":null,"abstract":"<p>This article aims to pursue and generalize, by using the global point of view offered by the stacks, the local study made by <span>Ghys</span> (J für die reine und angewandte Mathematik 468:113–138, 1995) concerning the deformations of complex structures of compact quotients of <span>({text {SL}}_2({mathbb {C}}))</span>. In his article, <span>Ghys</span> showed that the analytic germ of the representation variety <span>({mathcal {R}}(varGamma ):={text {Hom}}(varGamma ,{text {SL}}_2({mathbb {C}})))</span> of <span>(varGamma )</span> in <span>({text {SL}}_2({mathbb {C}}))</span>, pointed at the trivial morphism, determines the Kuranishi space of <span>({text {SL}}_2({mathbb {C}})/varGamma )</span>. In this note, we show that the tautological family above a Zariski analytic open subset <i>V</i> in <span>({mathcal {R}}(varGamma ))</span> remains complete. Moreover, the computation of the isotropy group of a complex structure in Teichmüller space, allows us to affirm that the quotient stack <span>([V/{text {SL}}_2({mathbb {C}})])</span> is an open substack of the Teichmüller stack of <span>({text {SL}}_2({mathbb {C}})/varGamma )</span>.</p>","PeriodicalId":55103,"journal":{"name":"Geometriae Dedicata","volume":"32 1","pages":""},"PeriodicalIF":0.5,"publicationDate":"2024-04-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140577949","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
Boundary metric of Epstein-Penner convex hull and discrete conformality 爱泼斯坦-彭纳凸壳的边界度量和离散保角性
IF 0.5 4区 数学
Geometriae Dedicata Pub Date : 2024-04-03 DOI: 10.1007/s10711-024-00901-7
Xin Nie
{"title":"Boundary metric of Epstein-Penner convex hull and discrete conformality","authors":"Xin Nie","doi":"10.1007/s10711-024-00901-7","DOIUrl":"https://doi.org/10.1007/s10711-024-00901-7","url":null,"abstract":"<p>The Epstein-Penner convex hull construction associates to every decorated punctured hyperbolic surface a convex set in the Minkowski space. It works in the de Sitter and anti-de Sitter spaces as well. In these three spaces, the quotient of the spacelike boundary part of the convex set has an induced Euclidean, spherical and hyperbolic metric, respectively, with conical singularities. We show that this gives a bijection from the decorated Teichmüller space to a moduli space of such metrics in the Euclidean and hyperbolic cases, as well as a bijection between specific subspaces of them in the spherical case. Moreover, varying the decoration of a fixed hyperbolic surface corresponds to a discrete conformal change of the metric. This gives a new 3-dimensional interpretation of discrete conformality which is in a sense inverse to the Bobenko-Pinkall-Springborn interpretation.</p>","PeriodicalId":55103,"journal":{"name":"Geometriae Dedicata","volume":"54 1","pages":""},"PeriodicalIF":0.5,"publicationDate":"2024-04-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140577732","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
Variation of holonomy for projective structures and an application to drilling hyperbolic 3-manifolds 投影结构的整体性变化及其在钻孔双曲3-manifolds中的应用
IF 0.5 4区 数学
Geometriae Dedicata Pub Date : 2024-04-03 DOI: 10.1007/s10711-024-00908-0
Martin Bridgeman, Kenneth Bromberg
{"title":"Variation of holonomy for projective structures and an application to drilling hyperbolic 3-manifolds","authors":"Martin Bridgeman, Kenneth Bromberg","doi":"10.1007/s10711-024-00908-0","DOIUrl":"https://doi.org/10.1007/s10711-024-00908-0","url":null,"abstract":"<p>We bound the derivative of complex length of a geodesic under variation of the projective structure on a closed surface in terms of the norm of the Schwarzian in a neighborhood of the geodesic. One application is to cone-manifold deformations of acylindrical hyperbolic 3-manifolds.\u0000</p>","PeriodicalId":55103,"journal":{"name":"Geometriae Dedicata","volume":"90 1","pages":""},"PeriodicalIF":0.5,"publicationDate":"2024-04-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140577738","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
Halfspaces and hypersurfaces in the bidisk 双盘中的半空间和超曲面
IF 0.5 4区 数学
Geometriae Dedicata Pub Date : 2024-04-03 DOI: 10.1007/s10711-024-00917-z
Virginie Charette, Youngju Kim
{"title":"Halfspaces and hypersurfaces in the bidisk","authors":"Virginie Charette, Youngju Kim","doi":"10.1007/s10711-024-00917-z","DOIUrl":"https://doi.org/10.1007/s10711-024-00917-z","url":null,"abstract":"<p>We construct a halfspace in the bidisk, whose boundary acts like a bisector. As an application, we build a fundamental domain consisting of such halfspaces for the action of groups that project to Schottky groups in both factors.</p>","PeriodicalId":55103,"journal":{"name":"Geometriae Dedicata","volume":"31 1","pages":""},"PeriodicalIF":0.5,"publicationDate":"2024-04-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140577740","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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