Buildings and their Applications in Geometry and Topology
Buildings and their Applications in Geometry and Topology
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建筑物及其在几何和拓扑学中的应用
DOI：
10.4310/ajm.2006.v10.n1.a5
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发表时间：
2006
影响因子：
0.6
通讯作者：
L. Ji
中科院分区：
文献类型：
--
作者：
L. Ji
建筑物最早是由J. Tits在20世纪50年代引入的，它给出了例外李群的系统几何解释，并以各种方式推广：欧几里德建筑物（Bruhat-Tits建筑物），拓扑建筑物，R-建筑物，特别是R-树。它们可用于各种学科的许多不同应用：代数群、有限群、有限几何、局部域上的表示理论、代数几何、算术变量的Arakelov交、代数K-理论、组合群论、整体几何和代数拓扑，特别是算术群和S-算术群的上同调群，半单李群和非正曲流形的余有限子群的刚性，Rn中高余维等参子流形的分类，Thurston几何化程序中三维流形上双曲结构的存在性.本文综述了建筑物在微分几何和几何拓扑中的几个应用。这些应用程序中有四个基本主题：1。建筑物通常描述对称空间和局部对称空间的无穷远处的几何形状，在度量的退化或缩放下也表现为限制对象。2.欧几里德建筑物是定义在局部域及其离散子群上的半单群的对称空间的类似物。3.较高等级的建筑物是刚性的，因此包含或诱导较高等级建筑物的物体往往是刚性的。4.建筑物上的附加结构，例如拓扑建筑物，在无限群的应用中很重要。
Buildings were first introduced by J. Tits in 1950s to give systematic geometric inter- pretations of exceptional Lie groups and have been generalized in various ways: Euclidean buildings (Bruhat-Tits buildings), topological buildings, R-buildings, in particular R-trees. They are useful for many different applications in various subjects: algebraic groups, finite groups, finite geometry, representation theory over local fields, algebraic geometry, Arakelov intersection for arithmetic va- rieties, algebraic K-theories, combinatorial group theory, global geometry and algebraic topology, in particular cohomology groups, of arithmetic groups and S-arithmetic groups, rigidity of cofinite subgroups of semisimple Lie groups and nonpositively curved manifolds, classification of isoparamet- ric submanifolds in R n of high codimension, existence of hyperbolic structures on three dimensional manifolds in Thurston's geometrization program. In this paper, we survey several applications of buildings in differential geometry and geometric topology. There are four underlying themes in these applications: 1. Buildings often describe the geometry at infinity of symmetric spaces and locally symmetric spaces and also appear as limiting objects under degeneration or scaling of metrics. 2. Euclidean buildings are analogues of symmetric spaces for semisimple groups defined over local fields and their discrete subgroups. 3. Buildings of higher rank are rigid and hence objects which contain or induce higher rank buildings tend to be rigid. 4. Additional structures on buildings, for example, topological buildings, are important in applications for infinite groups.