Geometry of Teichmüller space

在他过去的工作中，我提供了一个组合分类， 沿着Teichmuller测地线沿着的短曲线。 这类似于 (and部分动机）的分类短曲线的双曲 $3$--从Kleinian曲面群得到的流形。从那以后， 我的研究表明，探索 曲线复形和Teichmuller空间可以非常丰富。它有 导致更好地理解一些几何定义的线， Teichmuller空间，即Teichmuller测地线， 最小线和嫁接射线，以及之间的关系 这些物体。这也让我们更好地理解 的大规模几何Teichmuller空间提供了一个组合 Teichmuller距离模型和计算发散度 测地线的速率 我们的问题有两大主题。首先，我们可以想到 映射类群在Teichmuller空间上的作用为 李群上晶格作用的类似物。这个类比是 提出的一些问题的动机，即刚性和 计数的问题。 另一个主题是了解不同指标之间的关系 Teichmuller空间有几个不同的感兴趣的指标 Teichmuller空间和许多问题的答案为一个度量 但不是为了另一个人我们建议研究测地线的行为 在Teichmuller空间上的Lipschitz度量。这有一个额外的好处， Lipschitz度量可以作为Teichmuller度量 以及外层空间的Lipschitz度量，这是本提案的另一个重点。 本提案中讨论的主题与PI处理的问题相似 在过去成功地，在早期的工作中开发的技术 已经证明是有效的因此，PI处于有利位置， 提出的问题。

In his past work, I have provided a combinatorial classification of short 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 understanding of 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 of the action of the mapping class group on Teichmuller space as an 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 and the counting problems. The other theme is to understand the relation between different metrics on 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 work by the PI have proven to be effective. Thus the PI is well positioned to engage the proposed problems.