Geometry of Teichmüller space

Teichmüller 空间的几何

• 批准号: 435885-2013
• 负责人: Rafi, Kasra
• 金额: $ 2.11万
• 依托单位: University of Toronto
• 依托单位国家: 加拿大
• 项目类别: Discovery Grants Program - Individual
• 财政年份: 2017
• 资助国家: 加拿大
• 起止时间: 2017-01-01 至 2018-12-31
• 项目状态: 已结题
• 来源: https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=635357
• 关键词: Geometry; Teichm; ller; space

In his past work, I have provided a combinatorial classification ofshort curves along a Teichmuller geodesic. This is similar to (and partly motivated by) classification of short curves in the hyperbolic$3$--manifold obtained from a Kleinian surface group. Since then,the my work has shown that exploring the connections between the curve complex and Teichmuller space can be very fruitful. It has resulted in better understanding of some geometrically defined lines in Teichmuller space, namely, Teichmuller geodesics, lines of minima and grafting rays, as well as of the relationships among these objects. It has also allowed for better understandingof the large-scale geometry of Teichmuller space by providing a combinatorial model for the Teichmuller distance and computing the divergence rate of geodesics.There are two major themes for our problems. First, we can think ofthe action of the mapping class group on Teichmuller space asan analogue of the action of a lattice on a Lie group. This analogy is the motivation of some of the proposed problems, namely the rigidity andthe counting problems. The other theme is to understand the relation between different metricson Teichmuller space. There are several different metrics of interest on Teichmuller space and many questions are answered for one metric but not for another. We propose the study of the behavior of geodesics in the Lipschitz metric on Teichmuller space. This has the added advantage that the Lipschitz metric can act as a bridge between the Teichmuller metric and the Lipschitz metric in the Outer space which is the other focus of this proposal. The topics discussed in this proposal are similar to problems the PI has tackled successfully in the past, and the techniques developed in the earlier workby the PI have proven to be effective. Thus the PI is well positioned to engage the proposed problems.

在他过去的工作中，我提供了一个组合分类的短曲线沿着Teichmuller测地线。这是类似的（和部分动机）分类的短曲线的双曲3 $-流形从Kleinian表面组。从那时起，我的工作表明，探索曲线复形和Teichmuller空间之间的联系可以非常富有成效。它导致了更好地了解一些几何定义的线在Teichmuller空间，即Teichmuller测地线，线的极小和嫁接射线，以及这些对象之间的关系。它还允许更好地理解大规模几何Teichmuller空间通过提供一个组合模型的Teichmuller距离和计算的分歧率测地线。首先，我们可以把映射类群在Teichmuller空间上的作用看作是格在李群上的作用的类似物。这种类比是一些被提出的问题的动机，即刚性和计数问题。另一个主题是理解不同度量的Teichmuller空间之间的关系。Teichmuller空间有几个不同的度量，许多问题只针对一个度量而不是另一个度量。本文研究了Teichmuller空间上测地线在Lipschitz度量下的行为。这有一个额外的优点，即Lipschitz度量可以作为Teichmuller度量和Lipschitz度量之间的桥梁，这是本提案的另一个重点。本提案中讨论的主题与PI过去成功解决的问题相似，PI在早期工作中开发的技术已被证明是有效的。因此，PI很好地定位于解决所提出的问题。