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

算术群是自然而然产生的，在数论、几何学和拓扑学等许多数学领域发挥了重要作用。常见的群如Z和SL(2，Z)都是算术群。一个较大的类由S算术群组成，如SL(2，Z[1/p])。在这个方案中，IP计划研究S算术群的大规模几何，并证明它们在外科理论和K理论中的积分诺维科夫猜想。由于许多自然的S算术群，如SL(n，Z[1/p])都含有非平凡的扭元，所以这个建议强调了一个广义积分Novikov猜想。S-代数群自然作用于对称空间和Bruhat-Tits建筑物的乘积。该协会还建议研究Bruhat-Tits建筑物的紧凑化问题。另一个密切相关的类是作用在TeichMuller空间上的映射类群。PI还计划利用TeichMuller空间的适当紧化来研究映射类群的积分Novikov猜想。对称是科学和艺术中的一个基本概念。事实上，它在现代物理学中发挥了关键作用。在无限离散群中，算术群是特殊而重要的。例如，瓷砖、墙纸和水晶的对称性背后的群形成了一类算术群。由于算术群与许多不同领域的联系，算术群得到了深入的研究和成功的应用。算术群类的一个自然推广是S算术群类。本文将研究S算术群的大规模几何，并为其证明拓扑学中的一个重要猜想--诺维科夫猜想。

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.