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Topological and dynamical properties of Torelli groups of partitioned surfaces 分割曲面托雷利群的拓扑和动力学特性
IF 0.5 4区 数学
Geometriae Dedicata Pub Date : 2024-02-07 DOI: 10.1007/s10711-024-00889-0
Hyungryul Baik, Hyunshik Shin, Philippe Tranchida
{"title":"Topological and dynamical properties of Torelli groups of partitioned surfaces","authors":"Hyungryul Baik, Hyunshik Shin, Philippe Tranchida","doi":"10.1007/s10711-024-00889-0","DOIUrl":"https://doi.org/10.1007/s10711-024-00889-0","url":null,"abstract":"<p>Putman introduced a notion of a partitioned surface which is a surface with boundary with decoration restricting how the surface can be embedded into larger surfaces, and defined the Torelli group of the partitioned surfaces. In this paper, we study some topological and dynamical aspects of the Torelli groups of partitioned surfaces. More precisely, first we obtain upper and lower bounds on the cohomological dimension of Torelli groups of partitioned surfaces and show that those two bounds coincide when at most three boundary components are grouped together in the partition of the boundary. Second, we study the asymptotic translation lengths of Torelli groups of partitioned surfaces on the corresponding curve complexes. We show that the minimal asymptotic translation length asymptotically behaves almost like the reciprocal of the Euler characteristic of the surface. This generalizes the previous result of the first and second authors on Torelli groups for closed surfaces.</p>","PeriodicalId":55103,"journal":{"name":"Geometriae Dedicata","volume":null,"pages":null},"PeriodicalIF":0.5,"publicationDate":"2024-02-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139752564","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
Counting conjugacy classes of fully irreducibles: double exponential growth 计算完全不可复数的共轭类：双指数增长
IF 0.5 4区 数学
Geometriae Dedicata Pub Date : 2024-02-07 DOI: 10.1007/s10711-024-00885-4
Ilya Kapovich, Catherine Pfaff
{"title":"Counting conjugacy classes of fully irreducibles: double exponential growth","authors":"Ilya Kapovich, Catherine Pfaff","doi":"10.1007/s10711-024-00885-4","DOIUrl":"https://doi.org/10.1007/s10711-024-00885-4","url":null,"abstract":"<p>Inspired by results of Eskin and Mirzakhani (J Mod Dyn 5(1):71–105, 2011) counting closed geodesics of length <span>(le L)</span> in the moduli space of a fixed closed surface, we consider a similar question in the <span>(Out (F_r))</span> setting. The Eskin-Mirzakhani result can be equivalently stated in terms of counting the number of conjugacy classes (within the mapping class group) of pseudo-Anosovs whose dilatations have natural logarithm <span>(le L)</span>. Let <span>({mathfrak {N}}_r(L))</span> denote the number of <span>(Out (F_r))</span>-conjugacy classes of fully irreducibles satisfying that the natural logarithm of their dilatation is <span>(le L)</span>. We prove for <span>(rge 3)</span> that as <span>(Lrightarrow infty )</span>, the number <span>({mathfrak {N}}_r(L))</span> has double exponential (in <i>L</i>) lower and upper bounds. These bounds reveal behavior not present in the surface setting or in classical hyperbolic dynamical systems.</p>","PeriodicalId":55103,"journal":{"name":"Geometriae Dedicata","volume":null,"pages":null},"PeriodicalIF":0.5,"publicationDate":"2024-02-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139752333","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
Real structures on root stacks and parabolic connections 根栈和抛物线连接上的实结构
IF 0.5 4区 数学
Geometriae Dedicata Pub Date : 2024-01-30 DOI: 10.1007/s10711-023-00880-1
Sujoy Chakraborty, Arjun Paul
{"title":"Real structures on root stacks and parabolic connections","authors":"Sujoy Chakraborty, Arjun Paul","doi":"10.1007/s10711-023-00880-1","DOIUrl":"https://doi.org/10.1007/s10711-023-00880-1","url":null,"abstract":"<p>Let <i>D</i> be a reduced effective strict normal crossing divisor on a smooth complex variety <i>X</i>, and let <span>(mathfrak {X}_D)</span> be the associated root stack over <span>(mathbb C)</span>. Suppose that <i>X</i> admits an anti-holomorphic involution (real structure) that keeps <i>D</i> invariant. We show that the root stack <span>(mathfrak {X}_D)</span> naturally admits a real structure compatible with <i>X</i>. We also establish an equivalence of categories between the category of real logarithmic connections on this root stack and the category of real parabolic connections on <i>X</i>.\u0000</p>","PeriodicalId":55103,"journal":{"name":"Geometriae Dedicata","volume":null,"pages":null},"PeriodicalIF":0.5,"publicationDate":"2024-01-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139644594","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
Discrete groups of packed, non-positively curved, Gromov hyperbolic metric spaces 包装、非正曲、格罗莫夫双曲度量空间的离散群
IF 0.5 4区 数学
Geometriae Dedicata Pub Date : 2024-01-30 DOI: 10.1007/s10711-023-00874-z
Nicola Cavallucci, Andrea Sambusetti
{"title":"Discrete groups of packed, non-positively curved, Gromov hyperbolic metric spaces","authors":"Nicola Cavallucci, Andrea Sambusetti","doi":"10.1007/s10711-023-00874-z","DOIUrl":"https://doi.org/10.1007/s10711-023-00874-z","url":null,"abstract":"<p>We prove a quantitative version of the classical Tits’ alternative for discrete groups acting on packed Gromov-hyperbolic spaces supporting a convex geodesic bicombing. Some geometric consequences, as uniform estimates on systole, diastole, algebraic entropy and critical exponent of the groups, will be presented. Finally we will study the behaviour of these group actions under limits, providing new examples of compact classes of metric spaces.</p>","PeriodicalId":55103,"journal":{"name":"Geometriae Dedicata","volume":null,"pages":null},"PeriodicalIF":0.5,"publicationDate":"2024-01-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139644583","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
Circumcenter extension maps for non-positively curved spaces 非正曲线空间的圆心扩展映射
IF 0.5 4区 数学
Geometriae Dedicata Pub Date : 2024-01-30 DOI: 10.1007/s10711-023-00881-0
Merlin Incerti-Medici
{"title":"Circumcenter extension maps for non-positively curved spaces","authors":"Merlin Incerti-Medici","doi":"10.1007/s10711-023-00881-0","DOIUrl":"https://doi.org/10.1007/s10711-023-00881-0","url":null,"abstract":"<p>We show that every cross ratio preserving homeomorphism between boundaries of Hadamard manifolds extends to a map, called circumcenter extension, provided that the manifolds satisfy certain visibility conditions. We describe regions on which this map is Hölder-continuous. Furthermore, we show that this map is a rough isometry, whenever the manifolds admit cocompact group actions by isometries and we improve previously known quasi-isometry constants, provided by Biswas, in the case of 2-dimensional <span>(mathrm {CAT(-1)})</span> manifolds. Finally, we provide a sufficient condition for this map to be an isometry in the case of Hadamard surfaces.\u0000</p>","PeriodicalId":55103,"journal":{"name":"Geometriae Dedicata","volume":null,"pages":null},"PeriodicalIF":0.5,"publicationDate":"2024-01-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139644942","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
Entropy of real rational surface automorphisms: actions on the fundamental groups 实有理曲面自形的熵：基本群上的作用
IF 0.5 4区 数学
Geometriae Dedicata Pub Date : 2024-01-30 DOI: 10.1007/s10711-024-00884-5
Kyounghee Kim, Eric P. Klassen
{"title":"Entropy of real rational surface automorphisms: actions on the fundamental groups","authors":"Kyounghee Kim, Eric P. Klassen","doi":"10.1007/s10711-024-00884-5","DOIUrl":"https://doi.org/10.1007/s10711-024-00884-5","url":null,"abstract":"<p>This article develops a method to compute the action induced on the fundamental group by an automorphism of a real rational surface. Then, it uses this method to compute the induced actions for a certain family of basic quadratic real automorphisms. By utilizing an invariant set in the fundamental group, we introduce a method to estimate a lower bound of the action’s growth rate on the fundamental group. The growth rate of this induced action provides a lower bound for the entropy of the real surface automorphism. These calculations are carried out for an important family of real surface automorphisms, and new lower bounds are obtained for the entropy of these automorphisms.\u0000</p>","PeriodicalId":55103,"journal":{"name":"Geometriae Dedicata","volume":null,"pages":null},"PeriodicalIF":0.5,"publicationDate":"2024-01-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139644945","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
Homotopy equivalent boundaries of cube complexes 立方体复合物的同调等效边界
IF 0.5 4区 数学
Geometriae Dedicata Pub Date : 2024-01-27 DOI: 10.1007/s10711-023-00877-w
Talia Fernós, David Futer, Mark Hagen
{"title":"Homotopy equivalent boundaries of cube complexes","authors":"Talia Fernós, David Futer, Mark Hagen","doi":"10.1007/s10711-023-00877-w","DOIUrl":"https://doi.org/10.1007/s10711-023-00877-w","url":null,"abstract":"<p>A finite-dimensional CAT(0) cube complex <i>X</i> is equipped with several well-studied boundaries. These include the <i>Tits boundary</i> <span>(partial _TX)</span> (which depends on the CAT(0) metric), the <i>Roller boundary</i> <span>({partial _R}X)</span> (which depends only on the combinatorial structure), and the <i>simplicial boundary</i> <span>(partial _triangle X)</span> (which also depends only on the combinatorial structure). We use a partial order on a certain quotient of <span>({partial _R}X)</span> to define a simplicial Roller boundary <span>({mathfrak {R}}_triangle X)</span>. Then, we show that <span>(partial _TX)</span>, <span>(partial _triangle X)</span>, and <span>({mathfrak {R}}_triangle X)</span> are all homotopy equivalent, <span>(text {Aut}(X))</span>-equivariantly up to homotopy. As an application, we deduce that the perturbations of the CAT(0) metric introduced by Qing do not affect the equivariant homotopy type of the Tits boundary. Along the way, we develop a self-contained exposition providing a dictionary among different perspectives on cube complexes.\u0000</p>","PeriodicalId":55103,"journal":{"name":"Geometriae Dedicata","volume":null,"pages":null},"PeriodicalIF":0.5,"publicationDate":"2024-01-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139582817","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
Intersection theory and volumes of moduli spaces of flat metrics on the sphere 球面上平面度量的交点理论和模量空间的体积
IF 0.5 4区 数学
Geometriae Dedicata Pub Date : 2024-01-27 DOI: 10.1007/s10711-023-00883-y
Duc-Manh Nguyen, Vincent Koziarz
{"title":"Intersection theory and volumes of moduli spaces of flat metrics on the sphere","authors":"Duc-Manh Nguyen, Vincent Koziarz","doi":"10.1007/s10711-023-00883-y","DOIUrl":"https://doi.org/10.1007/s10711-023-00883-y","url":null,"abstract":"<p>Let <span>(mathbb {P}Omega ^dmathcal {M}_{0,n}(kappa ))</span>, where <span>(kappa =(k_1,dots ,k_n))</span>, be a stratum of (projectivized) <i>d</i>-differentials in genus 0. We prove a recursive formula which relates the volume of <span>(mathbb {P}Omega ^dmathcal {M}_{0,n}(kappa ))</span> to the volumes of other strata of lower dimensions in the case where none of the <span>(k_i)</span> is divisible by <i>d</i>. As an application, we give a new proof of the Kontsevich’s formula for the volumes of strata of quadratic differentials with simple poles and zeros of odd order, which was originally proved by Athreya–Eskin–Zorich. In another application, we show that up to some power of <span>(pi )</span>, the volume of the moduli spaces of flat metrics on the sphere with prescribed cone angles is a continuous piecewise polynomial with rational coefficients function of the angles, provided none of the angles is an integral multiple of <span>(2pi )</span>. This generalizes the results of Koziarz and Nguyen (Ann Sci l’Éc Normale Supér 51(6):1549–1597, 2018) and McMullen (Am J Math 139(1):261–291, 2017).\u0000</p>","PeriodicalId":55103,"journal":{"name":"Geometriae Dedicata","volume":null,"pages":null},"PeriodicalIF":0.5,"publicationDate":"2024-01-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139582919","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 Tholozan’s volume formula for closed anti-de-Sitter 3-manifolds 论封闭反德西特3-manifolds的索洛赞体积公式
IF 0.5 4区 数学
Geometriae Dedicata Pub Date : 2024-01-23 DOI: 10.1007/s10711-023-00878-9
François Labourie
{"title":"On Tholozan’s volume formula for closed anti-de-Sitter 3-manifolds","authors":"François Labourie","doi":"10.1007/s10711-023-00878-9","DOIUrl":"https://doi.org/10.1007/s10711-023-00878-9","url":null,"abstract":"","PeriodicalId":55103,"journal":{"name":"Geometriae Dedicata","volume":null,"pages":null},"PeriodicalIF":0.5,"publicationDate":"2024-01-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139551714","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
From $$L^p$$ bounds to Gromov–Hausdorff convergence of Riemannian manifolds 从 $$L^p$$ 边界到黎曼流形的格罗莫夫-豪斯多夫收敛性
IF 0.5 4区 数学
Geometriae Dedicata Pub Date : 2024-01-03 DOI: 10.1007/s10711-023-00875-y
Brian Allen
{"title":"From $$L^p$$ bounds to Gromov–Hausdorff convergence of Riemannian manifolds","authors":"Brian Allen","doi":"10.1007/s10711-023-00875-y","DOIUrl":"https://doi.org/10.1007/s10711-023-00875-y","url":null,"abstract":"<p>In this paper we provide a way of taking <span>(L^p)</span>, <span>(p &gt; frac{m}{2})</span> bounds on a <span>(m-)</span> dimensional Riemannian metric and transforming that into Hölder bounds for the corresponding distance function. One can think of this new estimate as a type of Morrey inequality for Riemannian manifolds where one thinks of a Riemannian metric as the gradient of the corresponding distance function so that the <span>(L^p)</span>, <span>(p &gt; frac{m}{2})</span> bound analogously implies Hölder control on the distance function. This new estimate is then used to state a compactness theorem, another theorem which guarantees convergence to a particular Riemmanian manifold, and a new scalar torus stability result. We expect these results to be useful for proving geometric stability results in the presence of scalar curvature bounds when Gromov–Hausdorff convergence can be achieved.\u0000</p>","PeriodicalId":55103,"journal":{"name":"Geometriae Dedicata","volume":null,"pages":null},"PeriodicalIF":0.5,"publicationDate":"2024-01-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139082688","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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