Compactifications of Symmetric Spaces, Buildings and S-Arithmetic Groups, and Integral Novikov Conjecture

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Arithmetic groups arise naturally and have played a important role in many areas of mathematics such as number theory, geometry and topology. The familiar groups such as Z and SL(2, Z) are arithmetic groups. A larger class consists of S-arithmetic groups such as SL(2, Z[1/p]). In this proposal, the IP plans to study the large scale geometry of S-arithmetic groups and to prove the integral Novikov conjecture in surgery theory and K-theory for them. Since many natural S-arithmetic groups such as SL(n, Z[1/p]) contain nontrivial torsion elements, this proposal emphasizes a generalized integral Novikov conjecture. S-arithemtic groups act naturally on products of symmetric spaces and Bruhat-Tits buildings. The PI also proposes to study compactifications of Bruhat-Tits buildings. Another closely related class is the class of mapping class groups, which act on the Teichmuller spaces. The PI also plans to study the integral Novikov conjecture for the mapping class groups by using suitable compactifications of the Teichmuller spaces. Symmetry is a fundamental notion in science and art. In fact, it has played a pivotal role in modern physics. Among infinite discrete groups, arithmetic groups are special and important. For example, the groups underlying the symmetry of tiles, wallpapers and crystals form a class of arithmetic groups. Due to their connections with many different areas, arithmetic groups have been intensively studied and applied with success. A natural generalization of the class of arithmetic groups is the class of S-arithmetic groups. This proposal will study the large scale geometry of S-arithmetic groups and prove an important conjecture in topology, Novikov conjecture, for them.