Geometry of Teichmüller space

点击翻译按钮获取中文摘要

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.