Mathematical Sciences: Mapping Class Groups and Teichmueller Spaces

这个项目是关于曲面的映射类群的代数结构和Teichmueller空间的几何结构。众所周知，它们之间有着密切的联系，也与曲面曲线复合体的组合结构有关。研究者尼古拉·伊万诺夫（Nikolai Ivanov）在过去已经成功地利用了这些相互关系，并打算继续这方面的研究。首先，他计划通过映射类群和算术群之间的类比以及R. Schwartz， B. Farb， A. Eskin， B. Kleiner和B. Leeb最近关于算术群的结果来探索映射类群是准等距刚性的可能性。该项目的另一部分涉及映射类组的子组结构，这些组通常由伪anosov元素生成。希望是Teichmueller空间是“足够双曲的”（正如最近H. Masur和Y. Minsky关于曲线复合体双曲性的定理所证明的那样），以便这样的子群大量存在。最后，研究者计划在Teichmueller空间上探索两种最新的几何结构，即Thurston半度量和Kerckhoff几何。特别是，他计划找到这些结构的所有自同构。对表面及其上各种几何结构的全球研究可以追溯到上个世纪，由高斯、莫比斯和黎曼奠定了基础。映射类群编码了所有可能的曲面对称，而泰奇穆勒空间收集了一个曲面上所有可能的特定类型的几何结构。事实证明，表面理论的这两个方面是密切相关的，而该项目侧重于这种关系的几个方面。在过去的十年里，由于一种很有前途的物理“万有理论”，即所谓的弦理论，表面理论获得了额外的意义。在这个理论中，物质的基本成分不是点状粒子，而是在运动过程中扫过表面的环（“弦”）的集合。事实证明，表面上的各种几何结构，以及它们的对称性，在这里都是至关重要的。＊＊＊

9704817 Ivanov This project is concerned with the algebraic structure of the mapping class groups of surfaces and the geometric structure of Teichmueller spaces. As is well known, they are deeply related, and also related to the combinatorial structure of complexes of curves of surfaces. The investigator, Nikolai Ivanov, has successfully exploited these interrelations in the past, and intends to continue this line of research. First, he is planning to explore the possibility that the mapping class groups are quasi-isometrically rigid, guided by the analogy between mapping class groups and arithmetic groups and recent results about arithmetic groups due to R. Schwartz, B. Farb, A. Eskin, B. Kleiner and B. Leeb. Another part of the project is concerned with the structure of subgroups of the mapping class groups normally generated by pseudo-Anosov elements. The hope is that the Teichmueller spaces are "sufficiently hyperbolic" (as evidenced by the recent theorem of H. Masur and Y. Minsky about hyperbolicity of complexes of curves) in order for such subgroups to exist in abundance. Finally, the investigator is planning to explore two more recent geometric structures on Teichmueller spaces, namely, the Thurston semi-metric and the Kerckhoff geometry. In particular, he is planning to find all automorphisms of these structures. The global study of surfaces and of various geometric structures on them goes back to the last century, with the foundations laid down by Gauss, Moebius, and Riemann. The mapping class groups encode all possible symmetries of surfaces, while Teichmueller spaces collect together all possible geometric structures of a particular sort on a surface. It turns out that these two aspects of the theory of surfaces are deeply interrelated, and the project focuses on several aspects of this relationship. In the last decade, the theory of surfaces acquired an additional significance as a result of a promising physical "theory of everything ," namely the so-called string theory. In this theory, the elementary constituents of matter are not point-like particles, but rather collections of loops ("strings") sweeping a surface during their motion. It turns out that both various geometric structures on surfaces, as well as their symmetries, are of fundamental importance here. ***