Large Scale Geometry and Compactifications of Arithmetic Groups, Symmetric Spaces and Buildings

大尺度几何和算术群、对称空间和建筑物的紧化

• 批准号: 0405884
• 负责人: Lizhen Ji
• 金额: $ 3.5万
• 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: 2004
• 资助国家: 美国
• 起止时间: 2004-11-15 至 2006-10-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0405884&HistoricalAwards=false
• 关键词: Large; Scale; Geometry; Compactifications; Arithmetic

Symmetries have played an important role in sciences and artsand are described in terms of group theory. Two important types ofgroups are discrete groups and Lie groups.Lie groups are closely related to homogeneous spaces.An important class of homogeneous spaces consists ofsymmetric spaces, a very distinguished class of special Riemannianmanifolds. Examples of symmetric spaces include the Euclidean spacesand the spheres in them. Another type of symmetric spacesis given by hyperbolic spaces, in which there are infinitelymany nonparallel lines which do not intersect with each other.Discrete groups acting on symmetric spaces give rise to locallysymmetric spaces; for example, surfaces with constant curvatureare locally symmetric spaces.The interplay between the topology, geometry and group theoryof discrete groups, Lie groups and locally symmetric spaceshas been intensively studied in mathematics.In this proposal, the PI proposes to study the Novikov conjecturesfor arithmetic groups using the large scale geometry andcompactifications of symmetric spaces of noncompact type.Specifically, an important invariant of the asymptotic geometryis the asymptotic dimension, the finiteness of which is closelyrelated to the Novikov conjectures. For a torsion free arithmeticsubgroup of a semisimple algebraic group,the partial Borel-Serre compactificationof the associated symmetric space is the universal coveringof the classifying space of the arithmetic group. For applicationsto the Novikov conjectures, we need a large compactification of thepartial Borel-Serre compactification. To study S-arithmetic subgroups,a generalization of arithmetic groups, we also need compactificationsof Bruhat-Tits buildings. Compactifications of symmetricspaces and buildings are also important for other purposes.

对称性在科学和艺术中扮演着重要的角色，并在群论中得到描述。两种重要的群是离散群和李群。李群与齐性空间密切相关。一类重要的齐性空间由对称空间组成，对称空间是一类非常特殊的黎曼流形。对称空间的例子包括欧几里德空间和其中的球面。另一类对称空间是双曲空间，其中有无穷多条互不相交的非平行线，离散群作用在对称空间上，产生局部对称空间;例如，常曲率的曲面是局部对称空间。拓扑学、几何学和离散群的群论之间的相互作用，李群和局部对称空间在数学中已经得到了深入的研究。在这个提议中，PI建议使用非紧型对称空间的大尺度几何和紧化来研究算术群的Novikov逼近。具体地说，渐近几何的一个重要不变量是渐近维数，它的有限性与Novikov定理密切相关。对于半单代数群的无挠算术子群，其对应对称空间的部分Borel-Serre紧化是该算术群的分类空间的泛覆盖.对于Novikov图的应用，我们需要部分Borel-Serre紧化的一个大的紧化。S-等差子群是等差群的推广，为了研究S-等差子群，我们还需要Bruhat-Tits建筑的紧化。建筑空间和建筑物的紧凑化对于其他目的也很重要。