Buildings and Tits Systems

Introduction

A building is a simplicial complex assembled from copies of a Coxeter complex (simplicial complex in the standard sense, closed under faces; the reference is Munkres, Elements of Algebraic Topology, and no homology is used here): it is a union of apartments, each apartment being the Coxeter complex of a Coxeter group $(W,S)$, so that any two chambers lie in a common apartment. The building is thick when every panel — a codimension-one simplex, that is, a chamber with one vertex deleted — belongs to at least three chambers, and it is spherical or affine according to whether the Coxeter group $W$ is finite or affine. The fundamental examples are the complex of flags of a vector space over a field, whose apartments are the flags of a fixed basis and whose Coxeter group is the symmetric group; the generalised polygons, which are the rank-two buildings and include the incidence structures of projective planes and the incidence graphs of quadrics; and the trees of degree at least three, which are the rank-one buildings and the simplest case of all.

The group-theoretic companion of a building is a Tits system, or $BN$-pair: a group $G$ with subgroups $B$ and $N$ such that $G = BNB$, the intersection $B\cap N$ is normal in $N$, the quotient $W = N/(B\cap N)$ is a Coxeter group generated by a set $S$ of involutions, and the multiplication by the elements of $S$ satisfies the exchange condition $sBwB \subseteq BwB \cup BswB$. The group then decomposes as the disjoint union of the double cosets $BwB$ — the Bruhat decomposition — and it acts on a building whose chambers are the cosets $G/B$, whose apartments are the cosets of the subgroup generated by $B$ and $N$, and whose parabolics are the conjugates of the subgroups containing $B$. The building records the group's structure, and the Tits system is the algebraic form of the building; the correspondence is exact for the groups of Lie type, where $B$ is a Borel subgroup, $N$ the normaliser of a maximal torus, and $W$ the Weyl group.

The theory is the combinatorial and topological background of the structure theory of the groups of Lie type over arbitrary fields and of the reductive groups over local fields, and it is the setting in which the rigidity theorems of lattices are proved. The article develops the Coxeter complex and the chamber systems, the definition of a building with the apartments and the thickness, the Tits systems and the Bruhat decomposition, the parabolic subgroups, the examples — the flag complexes of vector spaces, the generalised polygons, the trees, the buildings at infinity of symmetric spaces — and the classification theorems, with the Solomon–Tits theorem on the homology and the Moufang condition. The input from above is the theory of Coxeter groups and their word problems of Coxeter Groups in Part I, the trees and ends of Geometric Group Theory and Bass–Serre Theory, and the classical groups and their forms. The simplicial complex, the flag complex and the building are defined in line, since no earlier article introduces them.

The boundary with Part III is the one fixed for this block. What is developed here is the combinatorial incidence and the group-theoretic structure: chambers, apartments, types, Coxeter complexes, Tits systems, Bruhat decompositions, parabolic subgroups and the classification. What is deferred is the analytic theory: the harmonic analysis on the building, the Hecke algebras and the spherical functions (which use the convolution algebra $L^1$ and the measure), the $L^2$-cohomology and the spectra of the buildings, and the ergodic theory of the group actions on the buildings and their boundaries, all of which belong to Analysis on Groups and to the automorphic and ergodic theory of Part III. The affine buildings over a local field are the subject, and the groups that carry the Tits systems over a local field are the $p$-adic Lie groups of $p$-adic Lie Groups. No physics is invoked.

Coxeter Groups and Coxeter Complexes

Coxeter Systems

Definition. A Coxeter system is a pair $(W,S)$ consisting of a group $W$ and a set $S \subseteq W$ together with a symmetric matrix of integers $m : S\times S \to \{1,2,\dots,\infty\}$ with $m(s,s) = 1$ and $m(s,s') \geq 2$ for $s \neq s'$, such that $(W,S)$ has the presentation

$$ W = \bigl\langle S \ \big|\ (ss')^{m(s,s')} = 1 \ \text{ for all } s,s' \in S \ \text{ with } m(s,s') < \infty \bigr\rangle . $$

The group $W$ is a Coxeter group; it is spherical if it is finite and affine if it is infinite and acts properly discontinuously with compact quotient on an affine space of dimension $|S|-1$. For a subset $J \subseteq S$ the subgroup $W_J$ generated by $J$ is the parabolic subgroup of type $J$, and it is a Coxeter group with system $(W_J,J)$. The properties of the Coxeter system — the word problem, the Bruhat order, the classification of the spherical and affine ones — are those of Coxeter Groups.

Definition (the Coxeter complex). The Coxeter complex $\Sigma(W,S)$ is the simplicial complex whose simplices are the cosets $wW_J$ with $J \subseteq S$ and $W_J$ finite, ordered by inclusion of cosets: the simplex $wW_J$ is a face of $w'W_{J'}$ when $w'W_{J'}\subseteq wW_J$. The complex is realised with the type of a simplex $wW_J$ given by the complement $S\smallsetminus J$, so that the vertices have types in $S$. The chambers are the cosets $wW_\emptyset = \{w\}$, and the panels are the cosets $wW_{\{s\}}$, of type $S\smallsetminus\{s\}$. Every chamber has exactly one vertex of each type, and every panel lies in exactly two chambers of $\Sigma(W,S)$.

Example (the rank-one and rank-two cases). For $W = \mathbb{Z}/2$ with $S = \{s\}$ the Coxeter complex is a single edge, with two chambers; for $W$ the dihedral group of order $2m$ with $S = \{s,t\}$ it is the boundary of a regular $2m$-gon subdivided into $2m$ edges, so the rank-two Coxeter complexes are the triangulated circles. For $W$ spherical of type $A_{n-1}$ the Coxeter complex is the flag complex of the standard $\mathbb{R}^n$ with the chambers being the complete flags of the standard basis.

Chamber Systems

Definition. A chamber system over $S$ is a set $\mathcal{C}$ of chambers with an equivalence relation $\sim_s$ on $\mathcal{C}$ for each $s \in S$, such that two chambers related by $\sim_s$ are distinct unless equal, the classes of $\sim_s$ are the panels of type $S\smallsetminus\{s\}$, and the Weyl distance $d : \mathcal{C}\times\mathcal{C} \to W$ defined by $d(x,y) = w$ when the chambers $x,y$ can be joined by a gallery whose sequence of types spells a reduced word for $w$, is well defined. The chamber system is thick if every panel contains at least three chambers, and a building of type $(W,S)$ is a thick chamber system in which the residues — the connected components of the sub-chamber-system obtained by deleting one type — are buildings of the smaller type, together with the requirement that any two chambers be joined by a gallery. Equivalently, a building is a simplicial complex with a type map to $S$ that is the union of a family of subcomplexes each isomorphic to $\Sigma(W,S)$, such that any two chambers lie in a common member of the family.

Theorem (equivalence of the definitions). Let $(W,S)$ be a Coxeter system. A thick chamber system over $S$ with the Weyl distance is a building if and only if it is the chamber system of a simplicial complex that is the union of apartments each isomorphic to the Coxeter complex $\Sigma(W,S)$ with the property that any two chambers lie in a common apartment, the apartment system definition. The simplicial complex of a building is determined by its chamber system, and conversely.

Proof sketch. From a chamber system one recovers the simplicial complex whose simplices are the residues, with the type map; the residues of type $S\smallsetminus J$ correspond to the cosets of $W_J$. The apartment condition is proved by constructing, for two chambers, a sub-chamber-system isomorphic to $\Sigma(W,S)$ containing both, using the Weyl distance and the exchange condition in $W$; the details are the standard equivalence of Tits and are quoted from the literature.

Example (the thin and thick cases). The Coxeter complex $\Sigma(W,S)$ itself is a building that is not thick, the number of chambers per panel being two; a building is thick when every panel has at least three chambers. The thick building of type $A_1$ is a set of at least three points, the projective line over a field with $q+1$ points, with the single empty panel the only panel and the apartments the pairs of points. The thick affine building of type $\tilde{A}_1$ is a thick tree: the chambers are the edges, the panels are the vertices, the Coxeter system is the infinite dihedral group and the Coxeter complex is a line subdivided into edges, so the apartments are the bi-infinite geodesic lines, and thickness means that every vertex has degree at least three. The regular tree of degree $q+1$ is the building of $SL_2(\mathbb{Q}_p)$ with $q$ the cardinality of the residue field.

Buildings and Tits Systems

Tits Systems

Definition. A Tits system, or $BN$-pair, in a group $G$ is a pair of subgroups $B, N$ such that

(BN1) $G = BNB$ and $B\cap N$ is normal in $N$;

(BN2) $W := N/(B\cap N)$ is generated by a set $S$ of involutions;

(BN3) $sBwB \subseteq BwB \cup BswB$ for all $s \in S$ and $w \in W$;

(BN4) $sBs \neq B$ for all $s \in S$.

The group $W$ with the generating set $S$ is the Weyl group of the Tits system, and it is a Coxeter group with Coxeter matrix determined by the orders of the products of the elements of $S$; the subgroup $B$ is the Borel subgroup, the conjugates of $B$ are the Borel subgroups of the system, and a parabolic subgroup is a subgroup containing a conjugate of $B$.

Theorem (Bruhat decomposition). Let $(G,B,N)$ be a Tits system with Weyl group $W$ and generating set $S$. Then

$$ G = \bigsqcup_{w\in W} BwB , $$

the union being disjoint; every parabolic subgroup of $G$ is conjugate to a subgroup of the form $P_J = B W_J B$ for a unique subset $J \subseteq S$, and $P_J$ is the disjoint union of the double cosets $BwB$ with $w \in W_J$. Consequently the map $J \mapsto P_J$ is a bijection between the subsets of $S$ and the conjugacy classes of parabolic subgroups, and $B = P_\emptyset$ is a maximal proper parabolic subgroup only when $|S| = 1$.

Proof sketch. The exchange condition (BN3) is the combinatorial heart: it says that multiplication by $s$ moves a chamber at Weyl distance $w$ to a chamber at distance $w$ or $sw$, which forces the double cosets to be the $BwB$ and to be disjoint, and gives the Bruhat decomposition by an induction on the length of $w$ in the Coxeter group. The parabolic statement follows by applying the same argument inside the parabolic groups, which are themselves Tits systems with Weyl group $W_J$. The theorem is Tits's and is quoted from the literature.

The Building of a Tits System

Definition. Let $(G,B,N)$ be a Tits system with Weyl group $(W,S)$. The building $\Delta(G,B)$ is the simplicial complex whose simplices are the cosets $gP_J$ for $g\in G$ and $J \subseteq S$, with incidence by inclusion, with type $S\smallsetminus J$, and with the chambers the cosets $gB$; the group $G$ acts on $\Delta(G,B)$ by left translation, the stabiliser of the chamber $gB$ is $gBg^{-1}$, and the building is thick exactly when the parabolic subgroups $P_{S\smallsetminus\{s\}}$ are not equal to $B$ for any $s$, which is the content of (BN4).

Theorem (the building of a Tits system). The complex $\Delta(G,B)$ is a thick building of type $(W,S)$; its apartments are the translates by the elements of $N$ of the standard apartment, which is the set of cosets $nP_J$ with $n \in N$ and $J \subseteq S$, and every apartment is isomorphic to the Coxeter complex $\Sigma(W,S)$. The group $G$ acts on $\Delta(G,B)$ by type-preserving simplicial automorphisms, transitively on the chambers and on the pairs (chamber, apartment containing it), and the building is the unique building of type $(W,S)$ on which $G$ acts with the given chamber stabiliser.

Proof sketch. The Bruhat decomposition gives the chamber system: the chambers are the cosets $G/B$, and the Weyl distance is defined by the Bruhat cells $BwB$, which are well defined by the decomposition and satisfy the exchange condition (BN3); the residue of type $S\smallsetminus\{s\}$ at a chamber is the set of chambers in the coset of $P_{\{s\}} = B\cup BsB$, which has at least three elements by (BN4). The apartments are the translates of the standard apartment by the elements of $N$, and the axiom that two chambers lie in an apartment follows from the decomposition of an element of $G$ into a product of elements of $B$ and $N$. The theorem is Tits's and is quoted from the literature.

Parabolic Subgroups and Residues

Definition. Let $\Delta$ be a building of type $(W,S)$. For a chamber $c$ and a subset $J \subseteq S$, the residue of type $J$ at $c$ is the set of chambers connected to $c$ by galleries whose types lie in $J$; it is itself a building of type $(W_J,J)$. A parabolic subgroup of a group of automorphisms $G$ of $\Delta$ is the stabiliser of a residue; when $G$ has a Tits system, the parabolic subgroups are the stabilisers of the residues of the building $\Delta(G,B)$ and they are the conjugates of the $P_J$.

Theorem (residues and parabolics). In a thick building of type $(W,S)$, the residue of type $J$ at a chamber $c$ is a building of type $(W_J,J)$ with the same Coxeter complex as its apartments; a residue of type $S\smallsetminus\{s\}$ is a panel, a residue of type $\emptyset$ is a chamber, and the automorphism group of the building acts on the residues by type-preserving automorphisms. In the Tits-system case the residues of type $J$ at $c = gB$ are the cosets $gP_J$, so the residues of the building are exactly the cosets of the parabolics.

Proof sketch. The residue of a chamber is the connected component of the chamber system after the types outside $J$ are forgotten, and the building axioms restrict to give a building of the smaller type with the same apartment system restricted; in the Tits-system case the identification of the residues with the cosets of the parabolic subgroups is the definition of the $P_J$ and the Bruhat decomposition. The statements are standard and are quoted from the literature.

Examples

The Flag Complex of a Vector Space

Example (the projective building). Let $V$ be a vector space of dimension $n$ over a field $K$ and let $\Delta(V)$ be the flag complex whose simplices are the nontrivial proper subspaces of $V$ ordered by inclusion and whose chambers are the complete flags $0 \subsetneq V_1 \subsetneq \cdots \subsetneq V_{n-1} \subsetneq V$. Then $\Delta(V)$ is a building of type $A_{n-1}$ with Weyl group the symmetric group $S_n$: the type of a subspace is its dimension, the apartments are the flags of a fixed basis, and a panel is the set of chambers through a fixed flag with one entry deleted, which is the set of the $q+1$ points of a projective line over $K$; hence the building is thick for every field $K$, the panel having at least three chambers as soon as $|K|\geq2$. The group $GL_n(K)$ acts transitively on the chambers, the stabiliser of a chamber is the group of upper triangular matrices — the Borel subgroup — and the Tits system is $(GL_n(K), B, N)$ with $N$ the group of monomial matrices; the Weyl group is $S_n$ and the Bruhat decomposition is the classical decomposition of $GL_n(K)$ into the cells of the Bruhat decomposition of the flag variety. The parabolic subgroups are the stabilisers of partial flags, and the residues are the complexes of flags of a fixed type.

Example (the symplectic and orthogonal buildings). Let $V$ carry a nondegenerate alternating form or a quadratic form. The polar space of the form is the incidence structure of the totally isotropic subspaces; its flag complex is a building of type $C_n$ for a symplectic form and of type $D_n$ or $B_n$ for a quadratic form, and the group preserving the form acts on it with a Tits system whose Borel subgroup is the stabiliser of a maximal totally isotropic flag. The forms themselves and the classical groups defined by them are those,; the building of the polar space is the geometric object attached to the group from the form side.

Example (the generalised polygons). A generalised $m$-gon is a bipartite incidence structure in which the incidence graph has diameter $m$ and girth $2m$; equivalently it is a rank-two building of type $I_2(m)$, the dihedral group of order $2m$. A projective plane is a generalised $3$-gon, a generalised quadrangle is a generalised $4$-gon, and the incidence graph of a generalised $m$-gon is the building itself. The classification of the Moufang generalised polygons by Tits and Weiss gives the algebraic families: the polygons arising from the projective planes over a field or a division ring, from the quadrics, from the hermitian forms, and the two exceptional families of the Ree groups. The rank-two buildings are not classified in general, and this is the reason the general theory of buildings concentrates on rank at least three.

Trees and the Buildings at Infinity

Example (the tree as a building). A thick tree of degree at least three, with its edges as chambers and its vertices as panels, is an affine building of type $\tilde{A}_1$, the simplest affine building: the Coxeter system is the infinite dihedral group with two generators, the Coxeter complex is a line subdivided into edges, and the apartments are the bi-infinite geodesic lines. When the tree is the regular tree of degree $q+1$, the residues are the sets of $q+1$ edges at a vertex, and the building is the Bruhat–Tits building of $SL_2(\mathbb{Q}_p)$ with $q$ the cardinality of the residue field. The theory of Bass–Serre Theory, which is exactly the theory of the groups acting on the trees, is the rank-one model for the whole theory, and the affine buildings of higher rank are the buildings of the reductive groups over a local field, treated in Bruhat–Tits Theory.

Example (the building at infinity of a symmetric space). Let $X = G/K$ be a symmetric space of noncompact type with $G$ a connected semisimple Lie group without compact factors and $K$ a maximal compact subgroup. The spherical building at infinity $\Delta_\infty(X)$ has as chambers the Weyl chambers of the tangent space at infinity of $X$, and it is a spherical building of type equal to the restricted root system of $G$; the group $G$ acts on it with a Tits system whose parabolic subgroups are the stabilisers of the faces of the Weyl chamber at infinity, that is, the parabolics of $G$. This building is the boundary on which the rigidity theory of Lattices in Lie Groups and the superrigidity of Arithmetic Groups operate: the Furstenberg boundary of $G$ is a flag manifold, a residue of the building at infinity, and the rigidity proofs use the action of a lattice on this spherical building.

Classification and Structure

The Classification of Spherical Buildings

Theorem (Tits's classification). Let $\Delta$ be a thick, irreducible spherical building of rank at least three. Then $\Delta$ is the building of a Tits system of the following type:

(a) the flag complex of a finite-dimensional vector space over a division ring (type $A_n$);

(b) the flag complex of a polar space, that is, of a vector space with a nondegenerate sesquilinear or quadratic form (types $B_n$, $C_n$, $D_n$, $^2A_n$, $^2D_n$);

(c) the building of one of the exceptional types $E_6, E_7, E_8, F_4$ or $G_2$, associated with a simple algebraic group of the corresponding type over a division ring, with the appropriate twisting.

Consequently every thick irreducible spherical building of rank at least three arises from a division ring with a form or a division algebra with an involution, and its automorphism group is a group of Lie type in the appropriate sense. The rank-two cases are the generalised polygons, and the Moufang ones among them are classified by the theorem of Tits and Weiss: they are the buildings of the groups of relative rank two over a field, together with the two exceptional families.

Proof sketch. The proof is Tits's classification of spherical buildings, which proceeds by showing first that a thick irreducible spherical building of rank at least three carries a root group structure from the geometry of the panels, that the root groups satisfy the axioms of a group with a Tits system, and that the building is then the building of that group; the reconstruction of the group from the building is the essential content. In the rank-two case the analogous reconstruction requires the Moufang condition, which the general polygon need not satisfy, and the classification is the theorem of Tits and Weiss. The results are quoted from the literature.

Remark (the Moufang condition). A building satisfies the Moufang condition if for every panel the group generated by the root groups of the apartments through the panel acts transitively on the set of apartments containing the panel. Buildings of rank at least three are automatically Moufang, which is why the classification is complete in that range; in rank two the Moufang condition is an extra hypothesis, satisfied by the classical and exceptional polygons and encoding the presence of a large automorphism group.

The Solomon–Tits Theorem

Theorem (Solomon–Tits). Let $\Delta$ be a spherical building of rank $n \geq 2$, so that its dimension is $n-1$. Then the reduced homology of the simplicial complex $\Delta$ vanishes in all degrees except $n-1$, where it is a free abelian group; equivalently, $\Delta$ is $(n-2)$-connected and has the homotopy type of a wedge of $(n-1)$-dimensional spheres. The rank of the top homology is computed from the parameters of the building: for the flag complex of an $N$-dimensional vector space over $\mathbb{F}_q$, which has rank $n = N-1$, the top reduced homology is free of rank

$$ q^{\,n(n+1)/2} = q^{\,N(N-1)/2}, $$

the order of a Sylow $p$-subgroup of $GL_N(\mathbb{F}_q)$, where $q$ is a power of the prime $p$, and the dimension of the Steinberg representation.

Proof sketch. The proof uses the summation over the apartments of the building together with the analogue for a Coxeter group of the Euler characteristic: the alternating sum of the ranks of the reduced homology of the link of a simplex is computed by the Möbius function of the Coxeter group, and the Solomon–Tits theorem states that the building has the homology of a wedge of spheres in the top degree only. For the flag complex of $\mathbb{F}_q^N$ the rank is verified on the small cases: for $N = 2$ the building is the projective line with $q+1$ points, so the reduced homology is free of rank $q$ in degree $0$; for $N = 3$ the building is the incidence graph of the projective plane of order $q$, with $2(q^2+q+1)$ vertices and $(q+1)(q^2+q+1)$ edges, so the reduced homology is free of rank $(q+1)(q^2+q+1)-2(q^2+q+1)+1 = q^3$ in degree $1$; both agree with $q^{N(N-1)/2}$. The homological language is that of the section Algebraic Topology; the result is quoted from the literature.

Corollary (applications). The Solomon–Tits theorem is the input into the computation of the cohomology of the arithmetic groups and of the groups of Lie type: the Steinberg module of a group of Lie type is the top homology of its building, a projective module whose character is the Steinberg character — finite-dimensional of dimension equal to the order of a Sylow $p$-subgroup for a finite group of Lie type, and of infinite rank over the group ring for an arithmetic group; the Borel–Serre compactification of a locally symmetric space uses the spherical building at infinity, and the cohomology of the arithmetic lattice is computed from the action on the building, as in Arithmetic Groups. The building therefore supplies the combinatorial-topological input for the arithmetic and rigidity theory of this Part, with the analytic and cohomological consequences developed in the section Algebraic Topology and in Part III.

The Boundary with Analysis

The buildings and the Tits systems are combinatorial and topological objects; the analysis of the buildings is Part III.

• The Hecke algebra $\mathcal{H}(G,B)$ of a group with a Tits system, the spherical functions and the Satake isomorphism use the convolution algebra of functions on $G$ and therefore the measure and the integration of Part III.
• The harmonic analysis on a building, the spectral theory of the Laplacian of a building, the Poincaré series and the growth of the chambers are Analysis on Groups in Part III; the growth of the building, in contrast, is a graph-theoretic invariant treated in Geometric Group Theory.
• The $L^2$-cohomology of the arithmetic quotients, the automorphic forms on the groups of Lie type and the Steinberg representation as a representation-theoretic object are Part III; the module-theoretic statement of the Steinberg module is stated above as a corollary.
• The buildings over a local field, their apartments and their residues, are not covered here; the groups acting on them are the $p$-adic Lie groups of $p$-adic Lie Groups.
• What is not deferred: the Coxeter complex and the chamber systems; the definition of a building by apartments and by the Weyl distance; thick, spherical and affine buildings; Tits systems and the Bruhat decomposition; parabolic subgroups and residues; the examples of the flag complexes, the polar spaces, the generalised polygons, the trees and the building at infinity; and the classification theorems with the Solomon–Tits theorem at the level of statement.

Summary

A Coxeter system $(W,S)$ has a Coxeter complex $\Sigma(W,S)$ whose simplices are the cosets of the finite parabolic subgroups, whose chambers are the elements of $W$ and whose panels are the cosets of the $W_{\{s\}}$; a building of type $(W,S)$ is a thick simplicial complex with a type map in which any two chambers lie in a common apartment, a subcomplex isomorphic to $\Sigma(W,S)$. Equivalently a building is a thick chamber system with a Weyl distance in $W$, and the two descriptions determine each other. The thick building of type $A_1$ is a projective line, a set of at least three points; the thick affine building of type $\tilde{A}_1$ is a thick tree; the rank-two spherical buildings are the generalised polygons; and the flag complex of a vector space of dimension $N$ is a spherical building of type $A_{N-1}$ with the symmetric group as Weyl group.

A Tits system, or $BN$-pair, in a group $G$ is a pair of subgroups with $G = BNB$, $B\cap N$ normal in $N$, $W = N/(B\cap N)$ a Coxeter group generated by involutions, and the exchange condition $sBwB\subseteq BwB\cup BswB$. It gives the Bruhat decomposition $G = \bigsqcup_{w\in W}BwB$ and the building $\Delta(G,B)$ whose chambers are the cosets $G/B$, whose apartments are the translates by $N$ of the standard apartment, and whose parabolic subgroups are the stabilisers of the residues, conjugate to the groups $P_J = BW_JB$. The classical examples are the group $GL_n(K)$ with the upper triangular Borel subgroup and the monomial $N$, the groups of a form acting on the polar spaces, and the group of a reductive group over a local field acting on its affine building.

Tits's classification states that every thick irreducible spherical building of rank at least three is the building of a division ring with a form or of an exceptional algebraic group, and the Moufang generalised polygons of rank two are classified by Tits and Weiss. The Solomon–Tits theorem states that a spherical building of rank $n \geq 2$ has the homology of a wedge of $(n-1)$-spheres and is $(n-2)$-connected; it gives the Steinberg module and is the combinatorial input into the cohomology of the arithmetic groups. The measure-theoretic, spectral and automorphic theory of the buildings belongs to Part III, and the affine buildings over a local field.

Summary of Notation

Further Reading

• Jacques Tits, Buildings of Spherical Type and Finite $BN$-Pairs (Springer Lecture Notes 386, 1974), for the definition, the Tits systems and the classification.
• Jacques Tits, A local approach to buildings, in The Geometric Vein (Springer, 1981), 519–547, for the chamber-system approach and the local-to-global theory.
• Kenneth S. Brown, Buildings (Springer, 1989), for a textbook treatment of the Coxeter complexes, the buildings and the Solomon–Tits theorem.
• Peter Abramenko and Kenneth S. Brown, Buildings: Theory and Applications (Springer, 2008), for the modern account with the applications to groups of Lie type.
• Armand Borel and Jacques Tits, Groupes réductifs, Publications Mathématiques de l'IHÉS 27 (1965), 55–150, for the Tits systems of the reductive groups.
• Armand Borel, Linear Algebraic Groups (Springer, 2nd ed. 1991), for the Borel subgroups, the Bruhat decomposition and the parabolics.
• Jacques Tits and Richard M. Weiss, Moufang Polygons (Springer, 2002), for the classification of the Moufang generalised polygons.
• Louis Solomon, The Steinberg character of a finite group with a $BN$-pair, in Theory of Finite Groups (Harvard, 1969), for the Steinberg module and the homology of the building.
• Bernhard Mühlherr and Richard M. Weiss, Generalized polygons and the classification of buildings, in Handbook of Incidence Geometry (North-Holland, 1995), for the rank-two theory and the classification programme.
• James R. Munkres, Elements of Algebraic Topology (Addison-Wesley, 1984), for the standard notion of a simplicial complex used in the definition.
• Jean-Pierre Serre, Trees (Springer, 1980), for the rank-one buildings and the model case of the whole theory.