Symmetry and Commensurability

摘要/ abstract摘要：dms -0805908主要研究者：Genevieve S. walsh两个n-流形具有同胚有限层盖是可通约的。这个概念对3流形特别有用。对于承认几何的3流形，可通约性尊重这种几何。根据最近对3流形几何化的证明，将3流形分类为可通约性类是对它们承认几何的3流形分类的改进。这适用于沿2球和2环面分解的3流形。关于$3$流形的虚拟性质的问题，例如如果一个3-流形是虚拟Haken或虚拟光纤，也是关于其可通约性类的问题。可通约性也尊重3流形的某些数论性质。对双曲流形进行可通约性的分类将对该领域非常有用。理解双曲3流形的可通约性类的第一步是理解双曲结补的可通约性类。pii提出在给定的可通约性类中最多有3个双曲结互补的猜想，并了解这些可通约性是如何发生的。可通约性也是双曲曲面上一个有用的等价关系，它的定义是两个双曲- 2流形如果具有等距有限层盖是可通约性的。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通约性类。三维流形是局部看起来像三维球的空间。特别是，我们的宇宙是一个三流形，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的理解将显著地细化3流形的几何化。结补是一种自然发生的3流形，因此是很好的研究对象。二维流形或曲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，在各种各样的科学背景中都有广泛的应用。对称性和可通约性的研究对这些曲面非常有用。这位艺术家经常向不同的听众讲述她的作品。本研究的任何结果将在研究研讨会上进行描述，发表在arXiv上，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.