Buildings and their applications in geometry and topology Lizhen Ji∗ September 21, 2011 Abstract In this paper, we briefly introduce different types of buildings such as spherical buildings, Euclidean buildings, twin buildings, R-buildings and de- scribe some of their applications to many subjects: (1) differential geom- etry such as Mostow strong rigidity, rank rigidity of nonpositively curved manifolds, Margulis superrigidity, quasi-isometry rigidity, classification of isoparametric manifolds, compactifications of symmetric spaces and locally symmetric spaces, (2) topology such as cohomology and duality properties of arithmetic groups, simplicial volume of locally symmetric spaces and Novikov conjectures, (3) analysis such as harmonic maps, harmonic analysis and rep- resentation theory of p-adic Lie groups, (4) algebra such as finite simple groups, infinite simple groups, algebraic K-theory and representations of al- gebraic groups over finite fields. By putting together many different types of buildings and applications in seemly unrelated topics, we hope to present an overview of rich structures and applications of buildings and the underlying groups. Keywords. Tits building, Bruhat-Tits building, Bruhat-Tits theory, spherical building, Euclidean, topological building, twin building, arithmetic group, Mostow strong rigidity, Margulis super-rigidity, BN-pair, compacti- fication, quasi-isometry rigidity, Kac-Moody group, CAT(0)-space, CAT(0)- group, Solomon-Tits Theorem, cohomology group, cohomology dimension, generalized Poincar´eduality, simplicial volume, Coxeter complex, incidence geometry, finite geometry, simplicial complex, tree, geometrization conjec- ture of Thurston, curve complex, Teichm¨ullerspace, mapping class group. AMS subject classification numbers. 20E45, 53C24, 11F75. ∗Department of Mathematics, University of Michigan, Ann Arbor, MI 48109. Email address, [email protected] 2 Lizhen Ji Contents 1 Introduction and history of buildings 3 1.1 Summary . 3 1.2 History of buildings and outline of the paper . 5 1.3 Acknowledgments and dedication . 14 2 Spherical Tits buildings 15 2.1 Definition of buildings as chamber complexes and Solomon-Tits the- orem . 15 2.2 Semisimple Lie groups and buildings . 17 2.3 BN-pairs or Tits systems, and buildings . 18 2.4 Other definitions of and approaches to buildings . 19 2.5 Rigidity of Tits buildings . 21 3 Geometric realizations and applications of spherical Tits build- ings 23 3.1 Geodesic compactification of symmetric spaces . 23 3.2 Buildings and compactifications of symmetric spaces . 25 3.3 Topological spherical Tits buildings and Moufang buildings . 27 3.4 Mostow strong rigidity . 30 3.5 Rank rigidity of manifolds of nonpositive curvature . 35 3.6 Rank rigidity for CAT(0)-spaces and CAT(0)-groups . 37 3.7 Classification of isoparametric submanifolds . 41 3.8 Spherical buildings and compactifications of locally symmetric spaces 45 3.9 Geodesic compactification, Gromov compactification and large scale geometry . 49 3.10 Cohomology of arithmetic groups . 50 3.11 Vanishing of simplicial volume of high rank locally symmetric spaces 55 3.12 Generalizations of buildings: curve complexes and applications . 57 4 Euclidean buildings 58 4.1 Definitions and basic properties . 58 4.2 Semisimple p-adic groups and Euclidean buildings . 59 4.3 Compactification of Euclidean buildings by spherical buildings . 63 4.4 Satake compactifications of Bruhat-Tits buildings . 66 5 Applications of Euclidean buildings 67 5.1 p-adic curvature and vanishing of cohomology of lattices . 67 5.2 Super-rigidity and harmonic maps into Euclidean buildings . 69 5.3 Applications to S-arithmetic groups . 74 5.4 Applications to harmonic analysis and representation theories . 78 Buildings and their applications in geometry and topology 3 6 R-trees and R-buildings 80 6.1 Definition of R-trees and basic properties . 81 6.2 Applications of R-trees in topology . 82 6.3 R-Euclidean buildings . 88 6.4 Quasi-isometry rigidity and tangent cones at infinity of symmetric spaces . 89 7 Twin buildings and Kac-Moody groups 90 7.1 Twin buildings . 91 7.2 Kac-Moody algebras and Kac-Moody groups . 92 7.3 Kac-Moody groups as lattices and groups arising from buildings in geometric group theory . 94 8 Other applications of buildings 96 8.1 Applications in algebraic geometry . 96 8.2 Random walks and the Martin boundary . 98 8.3 Finite groups . 99 8.4 Finite geometry . 99 8.5 Algebraic K-groups . 100 8.6 Algebraic combinatorics . 100 8.7 Expanders and Ramanujan graphs . 100 1 Introduction and history of buildings 1.1 Summary Classical simple Lie groups over C are the symmetry groups of quadratic forms or sesquilinear forms of finite dimensional vector spaces over C. The same construc- tion works for vector spaces over finite fields and produces classical finite groups of Lie type, but this method does not extend to exceptional simple Lie groups. Buildings were originally introduced by Tits in 1950s in order to realize excep- tional Lie groups as the symmetry groups of spaces (or geometry) so that one can construct geometrically analogues of exceptional simple Lie groups over arbitrary fields.1 The first buildings were spherical buildings, and they have been generalized in various ways: Euclidean buildings (Bruhat-Tits buildings), topological buildings, R-buildings, in particular R-trees, twin buildings and hyperbolic buildings. Besides the combinatorial properties, they often carry additional structures from geome- try and group theory. They are useful for many different applications in various subjects such as algebraic groups, group theory and in particular finite groups, fi- nite geometry, representation theory over local fields, algebraic geometry, Arakelov 1According to [RT, p. 292], \... it is perhaps worth remarking that one of the initial motiva- tions for the theory of buildings, at a time when Chevalley's fundamental \Tohoku paper" had not yet appeared, was the search for a geometric way of obtaining algebraic analogues of the exceptional Lie groups." 4 Lizhen Ji intersection for arithmetic varieties, algebraic K-theories, combinatorial and geo- metric group theory, global differential geometry, geometric and algebraic topology, cohomology groups of arithmetic groups and S-arithmetic groups, Mostow strong rigidity and Margulis super-rigidity of lattice subgroups of semisimple Lie groups and of nonpositively curved manifolds, classification of isoparametric submani- folds in Rn of high codimension, and existence of hyperbolic structures on three dimensional manifolds in Thurston's geometrization program. Basically, there are two reasons why buildings are useful for questions and problems on semisimple (or more generally reductive) groups and their subgroups, and generalizations such as Kac-Moody groups: 1. For semisimple (or reductive) Lie groups and algebraic groups, buildings pro- vide effective ways to parametrize subgroups such as parabolic subgroups of reductive algebraic groups over any fields and parahoric subgroups of reduc- tive algebraic groups over locally compact, totally disconnected fields, and relations between these subgroups can also be expressed and reflected effec- tively by the geometry of buildings. Hence properties and structures of the groups can be described, understood, and proved via buildings. Rich geom- etry (for example existence of ample collections of subspaces) of buildings is a geometric reflection of rich structure of the associated (parahoric) groups. 2. Buildings also provide spaces on which the algebraic groups and their sub- groups act, and representation spaces of these groups can be constructed and studied. Furthermore, the automorphism groups of locally finite buildings provide natural locally compact topological groups which can be used to understand groups that act on the buildings, for example, the Kac-Moody groups. In this paper, several applications of buildings in differential geometry, geo- metric topology and group cohomology theory will be emphasized. There are four underlying themes in these applications: 1. Spherical buildings often describe the geometry at infinity of symmetric spaces and locally symmetric spaces, and hence are large scale geometric invariants of these spaces. The topology of spherical buildings often de- scribes precisely the topology at infinity of locally symmetric spaces. Both spherical buildings and R-buildings also appear as limiting objects under degeneration or scaling of Riemannian manifolds and more generally metric spaces. 2. Euclidean buildings are analogues of symmetric spaces for semisimple groups defined over local fields and their discrete subgroups, and they play similar roles in understanding properties and structures of algebraic groups and their subgroups, for example, generalization of the standard decompositions such as the Cartan decomposition and Iwasawa decomposition. 3. Euclidean buildings and symmetric spaces of noncompact type provide the most important examples of CAT(0)-spaces, which are simply connected and Buildings and their applications in geometry and topology 5 nonpositively curved geodesic metric spaces. This fact is used crucially in proving the super-rigidity and arithmeticity of co-finite discrete subgroups of the rank one simple Lie group Sp(1; n) and F4(−20), the isometry groups of quaternionic hyperbolic space and the Cayley plane. Furthermore, groups that act properly and isometrically on buildings provide some of the most in- teresting groups in geometric group theory, for example, Kac-Moody groups over finite fields. 4. Spherical buildings of higher rank