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.

摘要奖项：DMS-0805908 首席研究员：Genevieve S. Walsh 如果两个 n 流形具有同胚有限片盖，则它们是可通约的。 这个概念对于 3 流形特别有用。对于承认几何形状的 3-流形，可通约性尊重该几何形状。根据最近对 3-流形几何化的证明，可通约性类别的分类是对 3-流形所承认的几何分类的改进。这适用于已沿 2-球体和 2-环面分解的 3-流形。 关于 3 美元流形的虚拟属性的问题，例如 3 流形是否实际上是 Haken 或虚拟光纤，也是关于其可通约性类的问题。 可通约性还遵循 3-流形的某些数论性质。 双曲流形的可通约性分类对于该领域将非常有用。 理解双曲 3-流形的通约性类的第一步是理解双曲结补的通约性类。 PI 提议研究在给定的可通约性类别中最多有 3 个双曲结补的猜想，并理解这些可通约性是如何发生的。可通约性也是双曲曲面上的一种有用的等价关系，其定义是如果两个双曲流形具有等距有限片覆盖，则它们是可通约的。 PI 建议进一步发展这一理论，特别是理解具有大对称群的表面的可通约类别。3-流形是局部看起来像 3 维球的空间。 特别是，我们的宇宙是一个三流形，但我们不知道它是哪一个。 对 3 流形之间的可通约性的更深入理解将显着改进 3 流形的几何化。 结补体是一类自然存在的 3 流形，因此是很好的研究对象。 二维流形或表面是局部看起来像二维圆盘的空间。双曲曲面是最常见的曲面类型，用于极其广泛的科学环境中。 对称性和可通约性的研究对于这些表面非常有用。 PI 定期向不同的受众讲述她的工作。 这项研究的任何结果都将在研究研讨会上进行描述，发布在 arXiv 上并广泛分发。