本书的内容是关于楼（building）理论及其在几何和拓扑中的应用。楼作为一种组合和几何结构由Jacques Tits引入，作为理解任意域上保距还原线性代数群结构的一种方法，Tits因此项工作获得2008年Abel奖。楼理论是研究代数群及其表示的必要工具，在几个相当不同的领域中具有重要应用。本书的第一部分是作者专为国内学生学习楼理论准备的导读资料，其中特别注重利用例子说明问题，可读性很强；第二部分则综述了楼理论在几何与拓扑方面的应用，不仅总结了近些年楼理论研究的成就，还提出了未来的研究方向。本书是一本观点较高、极具学术价值的数学学习资料，可供我国高等院校代数及相关专业作为教学参考书使用。 Symmetry is an essential concept in mathematics, science and daily life, and an effective mathematical tool to describe symmetry is the notion of groups. For example, the symmetries of the regular solids (or Platonic solids) are described by the finite subgroups of the rotation group SO(3). Therefore, finding the symmetry group of a geometric object or space is a classical and important problem. On the other hand, given a group, how to find a natural geometric space which realizes the group as its symmetries is also interesting and fruitful. One of the most useful or beautiful class of groups consists of algebraic groups, and their corresponding geometric spaces are given by Tits buildings. Originally introduced by Tits to give a geometric description of exceptional simple algebraic groups, buildings have turned out to be extremely useful in a broad range of subjects in contemporary mathematics, including algebra, geometry, topology, number theory, and analysis etc. Since the theory of algebraic groups is complicated, the theory of buildings can be technical and demanding by itself. This book gives an accessible approach by using elementary and concrete examples and by emphasizing many applications in many seemingly unrelated subjects. The reader will learn from this book what buildings are, why they are useful, and how they can be used.

前辅文
Part 1 Buildings and Groups
  1. Combinatorics
   1.1 Geometry
   1.2 Coxeter group
   1.3 Chamber systems
   1.4 Chamber complexes
   1.5 Conclusion
  2. Chevalley Groups
   2.1 (B;N) pairs
   2.2 Simple Lie algebras
   2.3 Classical groups
   2.4 Chevalley groups and (B;N) pairs
   2.5 Chevalley groups over local fields
   2.6 Examples
   2.7 Conclusion
  3. Reductive Groups over Local Fields
   3.1 Root data
   3.2 Reductive group
   3.3 Apartments
   3.4 Building of a reductive group
   3.5 Compactification of buildings
   3.6 Congruence subgroup
   3.7 Bounded subgroups
   3.8 Hecke algebra
   3.9 Sheaves on buildings
  4. Rigid Analytic Spaces
   4.1 Rigid analytic space and formal schemes
   4.2 Theorems of Mumford and Drinfeld
   4.3 Geometric invariant theory
   4.4 Mumford prolongation
   4.5 Formal schemes from flag varieties
   4.6 Analytic generic fiber
  Bibliography
Part 2 Buildings and Their Applications in Geometry and Topology
  5. Introduction and History of Buildings
   5.1 Summary
   5.2 History of buildings and outline of this part
   5.3 Acknowledgments and dedication
  6. Spherical Tits Buildings
   6.1 Definition of buildings as chamber complexes and Solomon-Tits theorem
   6.2 Semisimple Lie groups and buildings
   6.3 BN-pairs or Tits systems, and buildings
   6.4 Other definitions of and approaches to buildings
   6.5 Rigidity of Tits buildings
  7. Geometric Realizations and Applications of Spherical Tits Buildings
   7.1 Geodesic compactification of symmetric spaces
   7.2 Buildings and compactifications of symmetric spaces
   7.3 Topological spherical Tits buildings and Moufang buildings
   7.4 Mostow strong rigidity
   7.5 Rank rigidity of manifolds of nonpositive curvature
   7.6 Rank rigidity for CAT(0)-spaces and CAT(0)-groups
   7.7 Classification of isoparametric submanifolds
   7.8 Spherical buildings and compactifications of locally symmetric spaces
   7.9 Geodesic compactification, Gromov compactification and large scale geometry
   7.10 Cohomology of arithmetic groups
   7.11 Vanishing of simplicial volume of high rank locally symmetric spaces
   7.12 Generalizations of buildings: curve complexes and applications
  8. Euclidean Buildings
   8.1 Definitions and basic properties
   8.2 Semisimple p-adic groups and Euclidean buildings
   8.3 Compactification of Euclidean buildings by spherical buildings
   8.4 Satake compactifications of Bruhat-Tits buildings
  9. Applications of Euclidean Buildings
   9.1 p-adic curvature and vanishing of cohomology of lattices
   9.2 Super-rigidity and harmonic maps into Euclidean buildings
   9.3 Applications to S-arithmetic groups
   9.4 Applications to harmonic analysis and representation theories
  10. R-trees and R-buildings
   10.1 Definition of R-trees and basic properties
   10.2 Applications of R-trees in topology
   10.3 R-Euclidean buildings
   10.4 Quasi-isometry rigidity and tangent cones at infinity of symmetric spaces
  11. Twin Buildings and Kac-Moody Groups
   11.1 Twin buildings
   11.2 Kac-Moody algebras and Kac-Moody groups
   11.3 Kac-Moody groups as lattices and groups arising from buildings in geometric group theory
  12. Other Applications of Buildings
   12.1 Applications in algebraic geometry
   12.2 Random walks and the Martin boundary
   12.3 Finite groups
   12.4 Finite geometry
   12.5 Algebraic K-groups
   12.6 Algebraic combinatorics
   12.7 Expanders and Ramanujan graphs
  Bibliography
Index