Symmetry and Commensurability

摘要奖：DMS-0805908主要研究者：Genevive S.Walsh如果两个n-流形具有同胚有限片覆盖，则它们是可公度的。这个概念对3-流形特别有用。对于允许一个几何的3-流形，可公度性尊重这个几何。根据最近对3-流形几何化的证明，分类为可公度类是对3-流形分类的一种改进，通过它们所允许的几何。这适用于已沿2-球面和2-环面分解的3-流形。关于$3$-流形的虚拟性质的问题，例如一个3-流形是否是虚拟Haken或虚拟纤维的，也是关于它的可公度性的问题。可公度性还尊重3-流形的某些数论性质。双曲3-流形的公度性分类对这一领域非常有用。理解双曲3-流形的可公度性类的第一步是理解双曲纽结补的可公度性类。PI建议致力于猜测在给定的可公度类中最多有3个双曲线结补，并理解这些可公度性是如何发生的。可公度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2-流形具有等距有限片覆盖，则它们是可公度的。PI建议进一步发展这一理论，特别是理解具有大对称群的曲面的可公度类。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3-流形，我们不知道它是哪一个。更深入地理解三维流形之间的可公度性将大大提高三维流形的公度化程度。纽结补是一类自然存在的3-流形，因此是很好的学习对象。2-流形或曲面是局部看起来像2维圆盘的空间。双曲曲面是最常见的曲面类型，在非常广泛的科学背景下使用。对于这些曲面，对称性和可公度性的研究是非常有用的。PI经常向不同的观众谈论她的工作。这项研究的任何结果都将在研究研讨会上描述，在arxiv上张贴，并广泛分发。

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.