Colding and Gabai have given an effective version of Li's theorem that non-Haken hyperbolic 3-manifolds have finitely many irreducible Heegaard splittings. As a corollary of their work, we show that Haken hyperbolic 3-manifolds have a finite collection of strongly irreducible Heegaard surfaces $S_i$ and incompressible surfaces $K_j$ such that any strongly irreducible Heegaard surface is a Haken sum $S_i + sum_j n_j K_j$, up to one-sided associates of the Heegaard surfaces.
{"title":"Strongly irreducible Heegaard splittings of hyperbolic 3-manifolds","authors":"Tejas Kalelkar","doi":"10.1090/proc/15114","DOIUrl":"https://doi.org/10.1090/proc/15114","url":null,"abstract":"Colding and Gabai have given an effective version of Li's theorem that non-Haken hyperbolic 3-manifolds have finitely many irreducible Heegaard splittings. As a corollary of their work, we show that Haken hyperbolic 3-manifolds have a finite collection of strongly irreducible Heegaard surfaces $S_i$ and incompressible surfaces $K_j$ such that any strongly irreducible Heegaard surface is a Haken sum $S_i + sum_j n_j K_j$, up to one-sided associates of the Heegaard surfaces.","PeriodicalId":8454,"journal":{"name":"arXiv: Geometric Topology","volume":"60 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2019-11-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"77243736","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 : 2019-11-25DOI: 10.2140/agt.2020.20.2553
Zhenkun Li
In this paper, we explore the interplay between contact structures and sutured monopole Floer homology. First, we study the behavior of contact elements, which were defined by Baldwin and Sivek, under the operation of performing Floer excisions, which was introduced to the context of sutured monopole Floer homology by Kronheimer and Mrowka. We then compute the sutured monopole Floer homology of some special balanced sutured manifolds, using tools closely related to contact geometry. For application, we obtain an exact triangle for the oriented skein relation in monopole theory and derive a connected sum formula for sutured monopole Floer homology.
{"title":"Contact structures, excisions and sutured monopole Floer homology","authors":"Zhenkun Li","doi":"10.2140/agt.2020.20.2553","DOIUrl":"https://doi.org/10.2140/agt.2020.20.2553","url":null,"abstract":"In this paper, we explore the interplay between contact structures and sutured monopole Floer homology. First, we study the behavior of contact elements, which were defined by Baldwin and Sivek, under the operation of performing Floer excisions, which was introduced to the context of sutured monopole Floer homology by Kronheimer and Mrowka. We then compute the sutured monopole Floer homology of some special balanced sutured manifolds, using tools closely related to contact geometry. For application, we obtain an exact triangle for the oriented skein relation in monopole theory and derive a connected sum formula for sutured monopole Floer homology.","PeriodicalId":8454,"journal":{"name":"arXiv: Geometric Topology","volume":"69 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2019-11-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"86142885","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 : 2019-11-22DOI: 10.1142/s0129167x20500767
V. Turaev, A. Virelizier
Let G be a discrete group and C be an additive spherical G-fusion category. We prove that the state sum 3-dimensional HQFT derived from C is isomorphic to the surgery 3-dimensional HQFT derived from the G-center of C.
{"title":"On 3-dimensional homotopy quantum field theory III: Comparison of two approaches","authors":"V. Turaev, A. Virelizier","doi":"10.1142/s0129167x20500767","DOIUrl":"https://doi.org/10.1142/s0129167x20500767","url":null,"abstract":"Let G be a discrete group and C be an additive spherical G-fusion category. We prove that the state sum 3-dimensional HQFT derived from C is isomorphic to the surgery 3-dimensional HQFT derived from the G-center of C.","PeriodicalId":8454,"journal":{"name":"arXiv: Geometric Topology","volume":"29 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2019-11-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"82233992","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 compute all local limits of all connected subgroups of $SL_3(mathbb{R})$ in the Chabauty topology
我们计算了Chabauty拓扑中$SL_3(mathbb{R})$的所有连通子群的所有局部极限
{"title":"Local Limits of Connected Subgroups of SL 3 (ℝ)","authors":"Nir Lazarovich, Arielle Leitner","doi":"10.5802/CRMATH.160","DOIUrl":"https://doi.org/10.5802/CRMATH.160","url":null,"abstract":"We compute all local limits of all connected subgroups of $SL_3(mathbb{R})$ in the Chabauty topology","PeriodicalId":8454,"journal":{"name":"arXiv: Geometric Topology","volume":"28 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2019-11-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"76230363","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 : 2019-11-16DOI: 10.2140/PJM.2020.309.257
Samuel A. Ballas
In this paper we produce many examples of thin subgroups of special linear groups that are isomorphic to the fundamental groups of non-arithmetic hyperbolic manifolds. Specifically, we show that the non-arithmetic lattices in $mathrm{SO}(n,1)$ constructed by Gromov and Piateski-Shapiro can be embedded into $mathrm{SL}_{n+1}(mathbb{R})$ so that their images are thin subgroups
{"title":"Thin subgroups isomorphic to\u0000Gromov–Piatetski-Shapiro lattices","authors":"Samuel A. Ballas","doi":"10.2140/PJM.2020.309.257","DOIUrl":"https://doi.org/10.2140/PJM.2020.309.257","url":null,"abstract":"In this paper we produce many examples of thin subgroups of special linear groups that are isomorphic to the fundamental groups of non-arithmetic hyperbolic manifolds. Specifically, we show that the non-arithmetic lattices in $mathrm{SO}(n,1)$ constructed by Gromov and Piateski-Shapiro can be embedded into $mathrm{SL}_{n+1}(mathbb{R})$ so that their images are thin subgroups","PeriodicalId":8454,"journal":{"name":"arXiv: Geometric Topology","volume":"10 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2019-11-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"76142137","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 : 2019-11-15DOI: 10.2140/agt.2021.21.2899
Michael C. Harrison
A fibration of $mathbb{R}^3$ by oriented lines is given by a unit vector field $V : mathbb{R}^3 to S^2$, for which all of the integral curves are oriented lines. A line fibration is called skew if no two fibers are parallel. Skew fibrations have been the focus of recent study, in part due to their close relationships with great circle fibrations of $S^3$ and with tight contact structures on $mathbb{R}^3$. Both geometric and topological classifications of the space of skew fibrations have appeared; these classifications rely on certain rigid geometric properties exhibited by skew fibrations. Here we study these properties for line fibrations which are not necessarily skew, and we offer some partial answers to the question: in what sense do nonskew fibrations look and behave like skew fibrations? We develop and utilize a technique, called the parallel plane pushoff, for studying nonskew fibrations. In addition, we summarize the known relationship between line fibrations and contact structures, and we extend these results to give a complete correspondence. Finally, we develop a technique for generating nonskew fibrations and offer a number of examples.
{"title":"Fibrations of $mathbb{R}^3$ by oriented lines","authors":"Michael C. Harrison","doi":"10.2140/agt.2021.21.2899","DOIUrl":"https://doi.org/10.2140/agt.2021.21.2899","url":null,"abstract":"A fibration of $mathbb{R}^3$ by oriented lines is given by a unit vector field $V : mathbb{R}^3 to S^2$, for which all of the integral curves are oriented lines. A line fibration is called skew if no two fibers are parallel. Skew fibrations have been the focus of recent study, in part due to their close relationships with great circle fibrations of $S^3$ and with tight contact structures on $mathbb{R}^3$. Both geometric and topological classifications of the space of skew fibrations have appeared; these classifications rely on certain rigid geometric properties exhibited by skew fibrations. Here we study these properties for line fibrations which are not necessarily skew, and we offer some partial answers to the question: in what sense do nonskew fibrations look and behave like skew fibrations? We develop and utilize a technique, called the parallel plane pushoff, for studying nonskew fibrations. In addition, we summarize the known relationship between line fibrations and contact structures, and we extend these results to give a complete correspondence. Finally, we develop a technique for generating nonskew fibrations and offer a number of examples.","PeriodicalId":8454,"journal":{"name":"arXiv: Geometric Topology","volume":"23 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2019-11-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"90522409","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 consider the control problem where, given an orthonormal tangent frame in the hyperbolic plane or three dimensional hyperbolic space, one is allowed to transport the frame a fixed distance $r > 0$ along the geodesic in direction of the first vector, or rotate it in place a right angle. We characterize the values of $r > 0$ for which the set of orthonormal frames accessible using these transformations is discrete. In the hyperbolic plane this is equivalent to solving the discreteness problem for a particular one parameter family of two-generator subgroups of $PSL_2(mathbb{R})$. In the three dimensional case we solve this problem for a particular one parameter family of subgroups of the isometry group which have four generators.
{"title":"On the discreteness of states accessible via right-angled paths in hyperbolic space","authors":"E. García, Pablo Lessa","doi":"10.4171/LEM/66-3/4-4","DOIUrl":"https://doi.org/10.4171/LEM/66-3/4-4","url":null,"abstract":"We consider the control problem where, given an orthonormal tangent frame in the hyperbolic plane or three dimensional hyperbolic space, one is allowed to transport the frame a fixed distance $r > 0$ along the geodesic in direction of the first vector, or rotate it in place a right angle. We characterize the values of $r > 0$ for which the set of orthonormal frames accessible using these transformations is discrete. \u0000In the hyperbolic plane this is equivalent to solving the discreteness problem for a particular one parameter family of two-generator subgroups of $PSL_2(mathbb{R})$. In the three dimensional case we solve this problem for a particular one parameter family of subgroups of the isometry group which have four generators.","PeriodicalId":8454,"journal":{"name":"arXiv: Geometric Topology","volume":"99 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2019-11-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"85820041","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 : 2019-11-08DOI: 10.5666/KMJ.2022.62.2.363
Benjamin Peet
In two previous papers the author presented a general construction of finite, fiber- and orientation-preserving group actions on orientable Seifert manifolds. In this paper we restrict our attention to elliptic 3-manifolds. A proof is given that orientation-reversing and fiber-preserving diffeomorphisms of Seifert manifolds do not exist for nonzero Euler class, in particular elliptic 3-manifolds. Each type of elliptic 3-manifold is then considered and the possible group actions that fit the given construction. This is shown to be all but a few cases that have been considered elsewhere. Finally, a presentation for the quotient space under such an action is constructed and a specific example is generated.
{"title":"Finite, fiber-preserving group actions on elliptic 3-manifolds","authors":"Benjamin Peet","doi":"10.5666/KMJ.2022.62.2.363","DOIUrl":"https://doi.org/10.5666/KMJ.2022.62.2.363","url":null,"abstract":"In two previous papers the author presented a general construction of finite, fiber- and orientation-preserving group actions on orientable Seifert manifolds. In this paper we restrict our attention to elliptic 3-manifolds. A proof is given that orientation-reversing and fiber-preserving diffeomorphisms of Seifert manifolds do not exist for nonzero Euler class, in particular elliptic 3-manifolds. Each type of elliptic 3-manifold is then considered and the possible group actions that fit the given construction. This is shown to be all but a few cases that have been considered elsewhere. Finally, a presentation for the quotient space under such an action is constructed and a specific example is generated.","PeriodicalId":8454,"journal":{"name":"arXiv: Geometric Topology","volume":"67 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2019-11-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"72802377","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 : 2019-10-27DOI: 10.1142/s0218216520500285
Ayaka Shimizu, R. Takahashi
A region crossing change at a region of a spatial-graph diagram is a transformation changing every crossing on the boundary of the region. In this paper, it is shown that every spatial graph consisting of theta-curves can be unknotted by region crossing changes.
{"title":"Region crossing change on spatial theta-curves","authors":"Ayaka Shimizu, R. Takahashi","doi":"10.1142/s0218216520500285","DOIUrl":"https://doi.org/10.1142/s0218216520500285","url":null,"abstract":"A region crossing change at a region of a spatial-graph diagram is a transformation changing every crossing on the boundary of the region. In this paper, it is shown that every spatial graph consisting of theta-curves can be unknotted by region crossing changes.","PeriodicalId":8454,"journal":{"name":"arXiv: Geometric Topology","volume":"42 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2019-10-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"82076196","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 : 2019-10-25DOI: 10.2140/agt.2021.21.2929
Thomas Budzinski, N. Curien, Bram Petri
We determine the asymptotic growth rate of the diameter of the random hyperbolic surfaces constructed by Brooks and Makover. This model consists of a uniform gluing of $2n$ hyperbolic ideal triangles along their sides followed by a compactification to get a random hyperbolic surface of genus roughly $n/2$. We show that the diameter of those random surfaces is asymptotic to $2 log n$ in probability as $n to infty$.
我们确定了Brooks和Makover构造的随机双曲曲面直径的渐近增长率。该模型由$2n$双曲理想三角形沿其侧面的均匀胶合组成，然后进行紧化以获得一个大致为$n/2$的随机双曲曲面。我们证明了这些随机曲面的直径在概率上是渐近于$2 log n$的，如$n to infty$。
{"title":"The diameter of random Belyi surfaces","authors":"Thomas Budzinski, N. Curien, Bram Petri","doi":"10.2140/agt.2021.21.2929","DOIUrl":"https://doi.org/10.2140/agt.2021.21.2929","url":null,"abstract":"We determine the asymptotic growth rate of the diameter of the random hyperbolic surfaces constructed by Brooks and Makover. This model consists of a uniform gluing of $2n$ hyperbolic ideal triangles along their sides followed by a compactification to get a random hyperbolic surface of genus roughly $n/2$. We show that the diameter of those random surfaces is asymptotic to $2 log n$ in probability as $n to infty$.","PeriodicalId":8454,"journal":{"name":"arXiv: Geometric Topology","volume":"21 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2019-10-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"89691285","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}
京公网安备 11010802042870号
京ICP备2023020795号-1
扫码关注我们
求助内容：
应助结果提醒方式：