Symmetry and Commensurability

对称性和可通约性

• 批准号: 0805908
• 负责人: Genevieve Walsh
• 金额: $ 9.92万
• 依托单位: Tufts University
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2008
• 资助国家: 美国
• 起止时间: 2008-08-01 至 2012-07-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0805908&HistoricalAwards=false
• 关键词: Symmetry; Commensurability

AbstractAward: DMS-0805908Principal Investigator: Genevieve S. WalshTwo n-manifolds are commensurable if they have homeomorphicfinite-sheeted covers. This notion is particularly useful for3-manifolds. For 3-manifolds which admit a geometry,commensurability respects this geometry. In light of the recentproof of geometrization for 3-manifolds, classification intocommensurability classes is a refinement of the classification of3-manifolds by which geometry they admit. This applies to3-manifolds which have been decomposed along 2-spheres and2-tori. Questions concerning virtual properties of$3$-manifolds, such as if a 3-manifold is virtually Haken orvirtually fibered, are also questions about its commensurabilityclass. Commensurability also respects certain number-theoreticproperties of 3-manifolds. A classification of hyperbolic3-manifolds up to commensurability would be very useful for thefield. A first step towards understanding commensurabilityclasses of hyperbolic 3-manifolds is understandingcommensurability classes of hyperbolic knot complements. The PIproposes to work on the conjecture that there are at most 3hyperbolic knot complements in a given commensurability class,and to understand how these commensurabilities canoccur. Commensurability is also a useful equivalence relation onhyperbolic surfaces, with the definition that two hyperbolic2-manifolds are commensurable if they have isometricfinite-sheeted covers. The PI proposes to further develop thistheory, in particular to understand commensurability classes ofsurfaces with large symmetry groups.3-manifolds are spaces which locally look like a 3-dimensionalball. In particular, our universe is a 3-manifold and we do notknow which one it is. A deeper understanding ofcommensurabilities amongst 3-manifolds would significantly refinegeometrization for 3-manifolds. Knot complements are a naturallyoccurring class of 3-manifolds, and thus good candidates forstudy. 2-manifolds, or surfaces, are spaces which locally looklike a 2-dimensional disc. Hyperbolic surfaces are the mostcommon type of surface and are used in an extremely wide varietyof scientific contexts. The study of symmetry andcommensurability is very useful for these surfaces. The PIregularly speaks on her work to diverse audiences. Any resultsfrom this research will be described in research seminars, postedon the arXiv, and broadly distributed.

AbstractAward：DMS-0805908原理研究人员：Genevieve S. Walshtwo n-manifolds如果具有同型磷脂封面，则可以相称。 这个概念特别有用，对于3个manifolds。对于接受几何形状的3个manifolds，可相当的性能尊重这种几何形状。鉴于3个策略的几何化的最新介绍，分类强度类别是对他们承认的几何形状的3个manifolds的分类的完善。这适用于已分解沿2个spheres和2-tori的3个manifolds。 关于$ 3 $ manifolds的虚拟属性的问题，例如几乎可以使用3个manifold的纤维纤维，也是关于其可高度频道的问题。 可怜性也尊重3个manifolds的一定数量理论。 双Bolic3-manifold的分类为可相称的性能对于Thefield非常有用。 理解双曲线3个manifolds的可高音级的第一步是了解双曲结互补的可承受性类别。 pip子在给定能力类别中最多有3个精细结的猜想，并了解这些不一致的能力如何conoccur。可相当性也是基因验力表面上的有用的等效关系，其定义是，如果有等距的芬特式覆盖层，则可以将两个双波利2个manifolds相称。 PI提议进一步发展这种理论，特别是了解具有大对称组的表面的可辨式类别。3-manifolds是当地看起来像3二维球的空间。 特别是，我们的宇宙是一个3个manifold，我们不知道它是哪一个。 3个manifolds之间对3型脉冲的更深入了解，将对3个manifolds进行显着完善地球化化。 结的互补是自然而然的3个杂志类别，因此出色的候选人。 2个manifolds或表面是局部看起来像二维光盘的空间。双曲线表面是最常见的表面类型，在非常多种的科学环境中使用。 对称性和对称性的研究对于这些表面非常有用。幽灵对她的作品进行了不同的观众的讲话。 该研究的任何结果都将在研究研讨会，发布ARXIV并广泛分布。