Mathematical Sciences: Mapping Class Groups and Teichmueller Spaces

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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. ***