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The structure of spherical buildings. (English) Zbl 1061.51011
Princeton, NJ: Princeton University Press (ISBN 0-691-11733-0/hbk). xi, 135 p. $ 45.00; £ 29.95 (2004).
In this book, the author introduces the reader into the theory of Tits buildings. There are not so many books on that topic, the standard references being the book “Lectures on buildings” [Academic Press, Boston (1989; Zbl 0694.51001)] by {\it M. Ronan}, and “Buildings” [Springer-Verlag (1998; Zbl 0922.20034)] by {\it K. S. Brown}. Of course, there is also the research book “Spherical buildings and finite BN-pairs” [Springer Lect. Notes 386 (1974; Zbl 0295.20047)] by {\it J. Tits} himself, where he classifies all spherical buildings of rank $\geq 3$, but proves a lot more than that. In the Addendum to that book, Tits introduces the concept of a Moufang building, and in particular of a Moufang generalized polygon. He then claims that the classification of spherical buildings of rank at least 3 would be simpler if one would have at one’s disposal a list of all Moufang polygons (hence, a classification of Moufang buildings of rank 2). This very list was completed only in 2002 with the appearance of the book “Moufang polygons” [Springer-Verlag (2002; Zbl 1010.20017)] by {\it J. Tits} and {\it R. M. Weiss}, containing the first published proof of the complete classification of Moufang polygons. In that book, the authors also explain Tits’ claim in the Addendum mentioned above. The book under review introduces buildings from a graph theoretic point of view (using the concept of a {\it chamber system}), and takes the reader on a trip culminating in the heart of Tits’ claim, explaining and proving why the classification of Moufang polygons provides an alternative approach to the classification of rank $\geq 3$ spherical buildings. Hence basically, the book is about Theorem 4.1.2 of Tits’ Lecture Notes mentioned before. The proof of Tits was so far the only existing proof in the literature -- now there are two proofs. The proof in the book under review -- written more than 30 years after Jacques Tits produced his proof -- differs from the original one in that it uses modern insight and techniques that were developed after 1974. For instance, ingenuous arguments of Bernhard Mühlherr and Mark Ronan are used (these arguments mostly found there origin in the theory of twin buildings, and the latter were introduced only in the eighties!). The book ends with a sketch of the proof of the classification of rank $\geq 3$ buildings using the classification of Moufang polygons. The book is very well written, and it is very suitable for a course, or a seminar. So anybody interested in this topic might want to get a copy. On the back cover, I wrote the following lines: “This is the best currently available introduction to the theory of buildings. And it brings the reader to a very important theorem in the theory of spherical buildings. Moreover, it is very carefully written: obviously the author spent quite some time arranging the different results in the right order, which isn’t a straightforward task. As for explaining the really hard part of the classification of spherical buildings, this book is a perfect complement to the existing literature.” And I still stand behind these words.

51E24Buildings and the geometry of diagrams
20E42Groups with a $BN$-pair; buildings