Buildings
The most important geometries of this book are of Coxeter type (cf. Definition 2.4.2). Their ‘building blocks’, that is, their rank two residues, are the generalized polygons (cf. Definition 2.2.7), which are precisely the rank two geometries of Coxeter t
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Buildings
The most important geometries of this book are of Coxeter type (cf. Definition 2.4.2). Their ‘building blocks’, that is, their rank two residues, are the generalized polygons (cf. Definition 2.2.7), which are precisely the rank two geometries of Coxeter type. In Chap. 4, we studied thin chamber systems of Coxeter type M and found that these are quotients of the very nice and regular universal chamber system C(M) for Coxeter systems of type M. In this chapter, we study special chamber systems of Coxeter type, called buildings, in which C(M) frequently occurs as a subsystem, which is called apartment. There are many possible definitions of buildings. In Sect. 11.1 we take a topological approach: closed galleries cannot be of types that are minimal expressions of nontrivial Coxeter group elements. In Corollary 11.2.6, we give the more classical definition in terms of apartments (but still in terms of chamber systems rather than geometries). In Sect. 11.2 we explore several useful properties of buildings, such as, in Corollary 11.2.12, their residual connectedness. By Theorem 3.4.6, this implies that buildings correspond to certain residually connected geometries over a Coxeter diagram. Accordingly, the term building will also be applied to geometries. In Proposition 11.1.9, we find that all of the projective geometries (Coxeter type An ), studied in Chaps. 5 and 6, and all of the polar geometries (Coxeter type Bn ), studied in Chaps. 7–10, are buildings. The geometries of type An−1 of Theorem 2.7.14 are also buildings (cf. Exercise 11.8.2). We will be mainly concerned with spherical buildings, which means that these have a spherical Coxeter type (cf. Definition 4.6.8). In Sect. 11.3, we study groups acting highly transitively (the technical term is ‘strongly transitively’) on buildings and arrive at the famous Tits systems in groups. In Sect. 11.4, we focus on shadow spaces of buildings. We derive various properties of these line spaces that have been used to characterize some of these spaces as shadow spaces of buildings. Often, these spaces have a family of convex subspaces isomorphic to polar spaces. Such spaces are called parapolar spaces and are studied in Sect. 11.5. In Sect. 11.6, we pay special attention to a particular kind of shadow spaces of buildings, called root shadow spaces, and show that these are root filtration spaces (cf. Definition 6.7.2). Root shadow spaces exist for each Weyl type, that is, F. Buekenhout, A.M. Cohen, Diagram Geometry, Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge / A Series of Modern Surveys in Mathematics 57, DOI 10.1007/978-3-642-34453-4_11, © Springer-Verlag Berlin Heidelberg 2013
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Buildings (m)
a spherical irreducible Coxeter type not isomorphic to one of H3 , H4 , I2 (m ∈ {5, 7, 8, 9, . . .}). We are motivated by the fact (not proved in this book) that each finite simple group of Lie type and rank at least three acts faithfully on a building of Weyl type. Finally, in Sect. 11.7 we explore to what extent line spaces can be recognized
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