Geometry of Teichmüller space

在他过去的工作中，我提供了沿Teichmuller测地线的短曲线的组合分类。这类似于从Kleinian曲面群中得到的双曲$3$-流形中的短曲线分类（部分原因是）。从那时起，我的工作表明，探索曲线复合体和Teichmuller空间之间的联系是非常富有成效的。它使我们更好地理解了在Teichmuller空间中一些几何上定义的线，即Teichmuller测地线、极小线和嫁接射线，以及这些对象之间的关系。通过提供Teichmuller距离的组合模型和计算测地线的发散率，它还可以更好地理解Teichmuller空间的大尺度几何。我们的问题有两个主要主题。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比。这种类比是提出一些问题的动机，即刚性和计数问题。另一个主题是理解Teichmuller空间中不同度量之间的关系。在Teichmuller空间中有几个不同的度量，许多问题对一个度量都有答案，但对另一个却没有。我们提出了在Teichmuller空间上研究测地线在Lipschitz度量中的行为。这有一个额外的优势，即利普希茨度规可以作为Teichmuller度规和外太空的利普希茨度规之间的桥梁，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很好地解决了所提出的问题。

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.