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| spherical buildings | |
|---|---|
| Name | Spherical building |
| Type | Simplicial complex |
| Introduced | 1950s–1970s |
| Founder | Jacques Tits |
| Related | Coxeter group, BN-pair, Tits system |
spherical buildings Spherical buildings are simplicial complexes introduced to encode the algebraic and geometric structure of groups and incidence geometries; they arise in the work of Jacques Tits, connect to Coxeter groups and BN-pairs, and provide a unifying framework linking groups such as Chevalley groups, Steinberg groups, and classical groups like SL_n and Sp_{2n}. They serve as combinatorial and topological models for flag varieties associated to algebraic groups over fields and division rings, and they underpin rigidity theorems and classification results in algebraic group theory and geometric topology.
A spherical building is a simplicial complex whose geometric realizations are homeomorphic to spheres and whose apartments are Coxeter complexes associated to finite Coxeter groups such as A_n, B_n, D_n, and exceptional types E_6, E_7, E_8, F_4, H_3, H_4. The local structure is governed by Moufang conditions related to Tits systems and root groups occurring in algebraic groups like Chevalley groups and Ree groups. Key axioms require that any two simplices lie in a common apartment and that apartment intersections are controlled by the action of the underlying Coxeter group; these properties appear in the study of Bruhat decompositions and double coset decompositions in groups such as GL_n and PGL_n.
Prominent examples include spherical buildings of type A_{n-1} realized by flag complexes of projective spaces over fields or division algebras, giving connections to Projective spaces and groups like PGL_n and PSL_n. Buildings of type B_n and C_n correspond to polar spaces related to Orthogonal groups and Symplectic groups, while type D_n buildings model the geometry of even-dimensional quadratic forms and groups like SO_{2n}. Exceptional buildings arise from groups of types E_6, E_7, E_8, F_4 and the twisted groups like ^2E_6; classical constructions include buildings associated to SL_2(F) (a spherical building of rank 1) and higher-rank analogues tied to Chevalley group constructions and finite groups of Lie type such as Suzuki groups and Ree groups.
The combinatorics of a spherical building are encoded by its chamber system, galleries, and the Weyl distance function valued in a finite Coxeter group; chambers correspond to maximal flags in geometries like Grassmannians and apartment systems mirror Coxeter complexes such as those for A_n and B_n. The building's geometry supports notions of convexity, links of simplices isomorphic to lower-rank spherical buildings, and opposition relations central in the theory of polarities and dualities appearing in settings like Hermitian form geometries and Quadratic form geometries linked to Orthogonal groups.
Spherical buildings are naturally associated to finite Coxeter groups: each apartment is a Coxeter complex for a Coxeter system (W,S), and buildings of algebraic and Kac–Moody origin arise from groups endowed with BN-pairs (Tits systems). For reductive algebraic groups such as GL_n, SL_n, Sp_{2n}, and groups of exceptional type, the BN-pair yields a Weyl group isomorphic to a finite Coxeter group and a corresponding spherical building whose chambers reflect the parabolic subgroup lattice and the Bruhat decomposition of the group.
Spherical buildings furnish tools for proving simplicity, generation, and rigidity properties of groups like Chevalley groups, Steinberg groups, and finite groups of Lie type; they underlie Margulis superrigidity and Mostow rigidity phenomena for arithmetic lattices in Lie groups such as SL_n(R) and Sp_{2n}(R). Topologically, realizations of spherical buildings yield spaces homotopy equivalent to wedges of spheres that inform calculations in algebraic K-theory and cohomology of groups like GL_n(F); they also provide combinatorial models used in the study of manifold structures and exotic decomposition problems related to Thurston-type rigidity contexts.
Tits' classification links spherical buildings with irreducible finite Coxeter systems and corresponding simple algebraic groups over fields or skew fields; buildings of rank at least 3 satisfying Moufang conditions are classified by simple algebraic groups or finite groups of Lie type, with exceptional families corresponding to groups of types E_6, E_7, E_8, F_4, G_2 and twisted analogues like ^2F_4. Rank-2 spherical buildings coincide with generalized polygons classified under results by Feit and Higman and later work by Tits and Weiss that handle Moufang polygons; classification theorems often require hypotheses such as the Moufang property or presence of root group data tied to groups like Suzuki groups and Ree groups.
The theory was initiated by Jacques Tits in the 1950s and 1960s, who formulated buildings and Tits systems to study algebraic groups and projective geometries; further foundations were developed by Armand Borel, Claude Chevalley, François Bruhat, and Jacques Tits himself in the context of Bruhat decomposition and BN-pairs. Contributions to classification, Moufang theory, and polarity structures were made by Richard Weiss, J. Tits (continued), Donald G. Higman, Walter Feit, and researchers in finite group theory such as Robert Steinberg, Gerrit van de Geer (contextual), and later work by Peter Abramenko and Kenneth S. Brown on combinatorial and topological aspects. Modern developments link buildings to Arithmetic group actions, rigidity theorems of G. A. Margulis, and geometric group theory initiated by figures like Mikhael Gromov.
Category:Buildings (mathematics)