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

对称性在科学和艺术中扮演着重要的角色，并在群论中被描述。群的两种重要类型是离散群和李群。李群与齐次空间密切相关。一类重要的齐次空间是由对称空间组成的，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得空间和其中的球体。另一种对称空间是由双曲空间给出的，在双曲空间中有无限多条互不相交的非平行线。作用于对称空间的离散群得到局部对称空间；例如，曲率恒定的曲面是局部对称空间。离散群、李群和局部对称空间的拓扑、几何和群论之间的相互作用在数学中得到了广泛的研究。在这个提议中，PI提出利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猜想。具体来说，渐近几何的一个重要不变量是渐近维数，其有限性与诺维科夫猜想密切相关。对于半简单代数群的无扭转算术子群，其相关对称空间的部分Borel-Serre紧化是该算术群的分类空间的全称覆盖。对于Novikov猜想的应用，我们需要局部Borel-Serre紧化的一个大紧化。为了研究s -算术子群，即算术群的一种推广，我们还需要对Bruhat-Tits结构进行紧化。对称空间和建筑的紧凑化对于其他目的也很重要。

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.