zbMATH — the first resource for mathematics

The structure of affine buildings. (English) Zbl 1166.51001
Annals of Mathematics Studies 168. Princeton, NJ: Princeton University Press (ISBN 978-0-691-13881-7/pbk; 978-0-691-13659-2/hbk). x, 368 p. (2009).
The monograph assumes some familiarity with Coxeter groups and buildings and is a sequel to the author’s book [The structure of spherical buildings. Princeton, NJ: Princeton University Press (2004; Zbl 1061.51011)], which is cited frequently in the text, and a brief review of its most important definitions and results is given in Appendix A together with some additional small results on Coxeter chamber systems and buildings. The main goal of the book under review is to present a complete proof of the classification of affine buildings whose buildings at infinity satisfy the Moufang property.
All spherical buildings of rank at least 3 as well as all their irreducible residues of rank at least 2 satisfy the Moufang property, and the author and J. Tits [Moufang polygons. Springer Monographs in Mathematics. Berlin: Springer (2002; Zbl 1010.20017)] classified all Moufang generalized polygons. A summary of the results on Moufang spherical buildings is given in Appendix B. The proof of the classification of Bruhat-Tits buildings largely follows the work of F. Bruhat and J. Tits [Publ. Math., Inst. Hautes Étud. Sci. No. 41, 5–251 (1972; Zbl 0254.14017)] and J. Tits [Buildings and the geometry of diagrams, Lect. 3rd 1984 Sess. C.I.M.E., Como/Italy 1984, Lect. Notes Math. 1181, 159–190 (1986; Zbl 0611.20026)] with a greater emphasis on the role of root data in the case of rank 2.
The first few chapters serve to introduce affine buildings, present their basic properties and substructures, construct the building at infinity and establish that this is a building of spherical type. Root data with valuations are already introduced in Chapter 3 although they are not used until much later in the text. In Chapter 9 affine buildings of rank 1, that is, thick trees, are investigated and the important connection between trees and discrete valuations of fields is established. The following two chapters introduce two families of trees, wall trees and panel trees, for affine buildings of arbitrary rank and Chapter 12 shows that an affine building is uniquely determined by its building at infinity together with its structure of wall and panel trees.
The next chapter uses the Moufang property at infinity to obtain a valuation of the associated root datum and Chapter 14 shows that every root datum with valuation arises in this manner from an affine building. This establishes that Bruhat-Tits pairs \((\Delta,{\mathcal A})\), where \({\mathcal A}\) is the system of apartments of the Bruhat-Tits building \(\Delta\), are classified by root data (of Moufang spherical buildings of rank at least 2) with valuation, up to equipollence of valuations. From the determination of all Moufang spherical buildings one knows all possible root data. The aim then is to determine when such a root datum has a valuation. Chapters 15 and 16 reduce this problem to the rank 2 case and develop general conditions for a valuation of the underlying field (or skew field or octonian division algebra) of a root datum to extend to a valuation of the root datum. The problem is then solved in chapters 19 to 25 by a case by case analysis for each the seven families of Moufang polygons.
Since the building at infinity also depends on the system of apartments, one has, for the classification of Bruhat-Tits buildings (rather than Bruhat-Tits pairs), to determine when two root data with valuations correspond to the same Bruhat-Tits building. This is done on Chapter 17 with the investigation of completions. The following chapter then examines more closely the structure of the residues of a Bruhat-Tits building. Chapters 26 and 27 add some comments on Bruhat-Tits pairs of type \(F_4\) and algebraic Bruhat-Tits buildings, and summarizes the classification. The last chapter before the appendices introduces locally finite Bruhat-Tits buildings and outlines some of their principal features.

51-02 Research exposition (monographs, survey articles) pertaining to geometry
20-02 Research exposition (monographs, survey articles) pertaining to group theory
51E24 Buildings and the geometry of diagrams
20E42 Groups with a \(BN\)-pair; buildings
Full Text: DOI