Mathematical Sciences: Mapping Class Groups and TeichmuellerSpaces

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9401284 Ivanov The principal investigator, Nikolai Ivanov, intends to continue his work on the mapping class groups of surfaces, braid groups of surfaces, complexes of curves of surfaces, and Teichmueller spaces. The investigator intends to concentrate on the interrelationships between these notions in order to shed new light on the algebraic structure of the mapping class groups of surfaces on the one hand, and the global geometry of the Teichmueller spaces on the other hand. An important guiding principle here is a still fairly mysterious analogy between the mapping class groups and Teichmueller spaces on the one side and arithmetic groups and symmetric spaces on the other side, established in earlier works of the principal investigator and other mathematicians. In particular, the principal investigator intends to prove that braid groups of surfaces have only the obvious automorphisms, and that there are no nontrivial homomorphisms from the arithmetic groups of rank at least two into mapping class groups. The principal investigator intends to continue his study of the rigidity properties of the mapping class groups and Teichmueller spaces. He also intends to give a general explanation of various phenomena in the topology of surfaces exhibiting stabilization over the genus, including presentations of the mapping class groups by generators and relations and the quantum field theory representations. Finally, the investigator will continue his investigation of rigidity and stability phenomena in the universal Teichmueller spaces. Mapping class groups and Teichmueller spaces serve as a meeting ground for several of the main branches of mathematics, including topology, complex analysis, and algebraic geometry. Recently they have also proved to be important in some questions of theoretical physics. These objects in a subtle way both resemble and differ from the much more well understood arithmetic groups and symmetric spaces. Various approaches to m ade this analogy (and difference!) precise and explicit form the core of the project. While symmetric spaces form a well established pool of possible spatial images investigated in mathematics (and arithmetic groups to a big extent control their structure), Teichmueller spaces gradually emerge as a new, richer and equally important pattern (and mapping class groups are similarly expected to control their structure). The principal investigator intends to concentate on two of the most interesting and striking properties of these objects: rigidity and stability. Because of rigidity, the structure of these objects is expected to be completely determined by their shape only. Stability tells us that when these objects become more and more complicated with a natural parameter (genus) growing, their main characteristics cease to change at a certain point. The stability phenomena is expected to be relevant not only to the mathematical but also to the quantum physics applications of the mapping class groups and Teichmueller spaces. ***