Symmetry and Commensurability

摘要奖：DMS-0805908主要研究者：Genevieve S. WalshTwo n-流形是可积的，如果它们有同胚有限片覆盖。 这个概念对于三维流形特别有用。对于允许一个几何的3-流形，可积性尊重这个几何。根据三维流形的几何化的最近证明，分类成可积类是三维流形分类的一个改进，三维流形的几何化是由三维流形的几何化所决定的。这适用于沿沿着2-球面和2-环面分解的3-流形。 关于3维流形的虚性质的问题，如3维流形是虚哈肯流形还是虚纤维流形，也是关于3维流形的可解性类的问题。 可公度性也尊重三维流形的某些数论性质。 双曲3-流形的一个分类是非常有用的领域。 理解双曲三维流形的可积类的第一步是理解双曲纽结补的可积类。 PI提出了一个猜想，即在给定的双曲纽结可积类中至多有3个双曲纽结补，并研究了这些可积类是如何产生的。可公度性也是双曲曲面上的一个有用的等价关系，定义了两个双曲2-流形是可公度的，如果它们有等距有限片覆盖。 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扩展性类。3-流形是局部看起来像3维球的空间。 特别是，我们的宇宙是一个三维流形，我们不知道它是哪一个。 深入理解三维流形之间的可积性，将对三维流形的几何化有重要意义。 纽结补是一类自然出现的三维流形，因此是研究的好对象。 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，在科学领域中应用非常广泛。 对这些曲面的对称性和可展性的研究是非常有用的。PI定期向不同的观众讲述她的作品。 这项研究的任何结果都将在研究研讨会上进行描述，张贴在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.