Tits building

Tits building
Name	Tits building
Named after	Jacques Tits
Field	Algebraic group theory, Incidence geometry
Introduced	1950s

Tits building

The Tits building is a combinatorial and geometric structure introduced by Jacques Tits that encodes the parabolic subgroup structure of Chevalley groups, Lie groups, and algebraic groups; it unites concepts from projective geometry, spherical geometry, and incidence geometry. The theory of buildings provides a unifying framework linking the classification of semisimple algebraic groups, the study of Coxeter groups, and the geometry of symmetric spaces such as Riemannian symmetric spaces and Bruhat–Tits trees.

Introduction

A Tits building associates to a semisimple algebraic group G over a field or local field a simplicial complex (or a chamber system) whose simplices correspond to flags of parabolic subgroups, reflecting incidence relations analogous to those in projective planes and flag varietys. Jacques Tits developed the concept while studying the geometric underpinnings of the Weyl group and the BN-pair formalism for groups like SL_n, Sp_{2n}, SO_n, and exceptional groups such as E_8, E_7, E_6, F_4, and G_2. Buildings come in spherical, affine, and hyperbolic types tied to Coxeter diagrams and Dynkin diagrams, and they play a central role in the classification theorems of Tits' classification and in rigidity results such as those of Mostow and Margulis.

Definition and Construction

Formally, a building is constructed from a Coxeter system (W, S) where W is a Coxeter group with generating set S and relations encoded by a Coxeter diagram. Chambers of the building correspond to cosets of a Borel subgroup in groups with a BN-pair; adjacency is determined by simple reflections in S, yielding a chamber system with gallery distances measured via W. For algebraic groups over local fields like Q_p or R, one obtains affine buildings modeled on Euclidean Coxeter complexes; for groups over finite fields one obtains spherical buildings modeled on finite spherical Coxeter groups such as A_n, B_n, C_n, and D_n. The Bruhat decomposition and the Iwahori subgroup appear naturally in this construction, connecting to the Iwahori–Hecke algebra and the representation theory of p-adic groups such as GL_n(Q_p).

Examples and Types

Classical examples include the spherical building of type A_{n-1} associated to SL_n(k) whose simplices correspond to chains of subspaces of an n-dimensional vector space over a field k, reproducing the structure of the Grassmannian and the full flag variety. The affine building for SL_2(Q_p) is the Bruhat–Tits tree, a one-dimensional simplicial complex used in the study of modular curves and p-adic analytic spaces. Exceptional buildings arise from exceptional algebraic groups such as E_8 and F_4 and link to geometries like the Moufang polygons and hermitian symmetric space analogues. Twin buildings appear in the theory of Kac–Moody groups, tying to Kac–Moody algebras and infinite-dimensional generalizations of finite-type classifications.

Geometric and Combinatorial Properties

Buildings admit a metric and curvature interpretation: spherical buildings are CAT(1) spaces while affine buildings are CAT(0) spaces, providing synthetic curvature bounds used in rigidity theorems of Gromov and in geometric group theory of nonpositively curved spaces. The gallery metric and the Tits metric give rise to a spherical building boundary at infinity for Euclidean buildings, important in the study of discrete subgroups like arithmetic subgroups and lattices. Combinatorially, buildings satisfy the Moufang property in many cases, reflecting transitivity properties of root groups and linking to the classification of Moufang loops and Jordan algebra constructions related to groups like Spin_n and E_6.

Connections to Group Theory and Buildings

The intimate connection between buildings and group theory manifests via BN-pairs, where the Weyl group W arises as N/T in a BN-pair (B, N), and parabolic subgroup incidence is recovered as stabilizers of faces. Buildings provide geometric models for the action of reductive groups over local and global fields, underpinning strong approximation results, the structure of Hecke algebras, and representation-theoretic phenomena studied by Harish-Chandra, Langlands, and others. In the arithmetic context, the action of p-adic groups on affine buildings facilitates the study of automorphic forms, Shimura varieties, and the reduction theory for arithmetic groups like SL_n(Z) and Sp_{2n}(Z).

Applications and Further Developments

Applications of building theory include proofs of rigidity and classification theorems such as Tits' classification of spherical buildings, superrigidity results of Margulis and Mostow for lattices in higher-rank groups, and constructions in combinatorics and finite geometry like generalized polygons used by Feit and Higman. Buildings also appear in algebraic topology via Brown's criterion for finiteness properties of groups, in noncommutative geometry through actions of C*-algebra-related groups on boundaries, and in mathematical physics via connections between Kac–Moody groups and string-theoretic symmetry groups like E_8 × E_8. Contemporary research explores interactions with tropical geometry, Berkovich spaces, and computational aspects in the study of expander graphs arising from arithmetic quotients of buildings, connecting to work by Lubotzky and Margulis.

Category:Buildings (mathematics)