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

对称空间、建筑物和 S 算术群的紧化以及积分诺维科夫猜想

• 批准号: 0604878
• 负责人: Lizhen Ji
• 金额: $ 11.85万
• 依托单位: Regents of the University of Michigan - Ann Arbor
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2006
• 资助国家: 美国
• 起止时间: 2006-08-01 至 2009-07-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0604878&HistoricalAwards=false
• 关键词: Compactifications; Symmetric; Spaces; Buildings; Arithmetic

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.

算术群是自然产生的，在数论、几何和拓扑学等许多数学领域都发挥了重要作用。我们熟悉的群如Z和SL（2， Z）是等差群。一个更大的类由s算术群组成，如SL（2, Z[1/p]）。在本课题中，IP计划研究s -算术群的大尺度几何，并为它们证明外科理论和k理论中的积分Novikov猜想。由于许多自然s算术群如SL（n, Z[1/p]）包含非平凡扭转元，因此本文强调广义积分Novikov猜想。s -算术群自然地作用于对称空间和Bruhat-Tits建筑的产物上。PI还建议研究Bruhat-Tits建筑的密实性。另一个密切相关的类是映射类群的类，它们作用于Teichmuller空间。PI还计划利用适当的Teichmuller空间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，它在现代物理学中起着举足轻重的作用。在无限离散群中，算术群是一个特殊而重要的问题。例如，构成瓷砖、壁纸和水晶对称性的群构成了一类算术群。由于算术群与许多不同领域的联系，人们对它们进行了深入的研究，并成功地应用了它们。等差群类的自然推广是s等差群类。本文将研究s -算术群的大尺度几何，并为它们证明拓扑学中的一个重要猜想——诺维科夫猜想。