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Buildings. Theory and applications. (English) Zbl 1214.20033
Graduate Texts in Mathematics 248. Berlin: Springer (ISBN 978-0-387-78834-0/hbk). xxii, 747 p. (2008).
Although the theory of buildings has many beautiful applications in mathematics, especially in group theory, it is often difficult for a newcomer to the topic to get started. This is in part due to the fact that there are several different approaches to the definition and study of buildings, and despite the wide range of introductory texts (for example [K. Brown, Buildings. New York: Springer-Verlag (1989; Zbl 0715.20017)] or [M. Ronan, Lectures on buildings. Chicago: University of Chicago Press (2009; Zbl 1190.51008)]) it is not always apparent which approach is best suited to which occasion, or even that these approaches are equivalent. To the reviewer’s knowledge, this book is the first one which presents all of the different approaches to buildings alongside one another, and it does so in an accessible yet thorough manner which should prove extremely useful to anyone wishing to learn more about this interesting area of mathematics.
The goal of presenting this introductory material in one place would be laudable enough, but the authors also succeed in exposing many of the diverse ramifications of the theory of buildings in group theory, ranging from groups of Lie type and Chevalley groups to Kac-Moody groups and $$S$$-arithmetic groups, to name a few examples. This material brings the reader to the cutting edge of current research and shows just how far the theory of buildings has come in the 50 years or so since Tits first started formalising the notion of a building.
The first few chapters of the book present basic material on finite reflection groups and their associated simplicial complexes (Chapter 1), moving onto Coxeter groups and complexes (Chapters 2 and 3), before presenting the first definition of a building as a simplicial complex (the “simplicial approach”) in Chapter 4. Chapter 5 develops the equivalent “combinatorial approach” to buildings, and we see here the advantage of developing both these approaches in the same book; the authors are able to make explicit all the connections between the two approaches, and show concretely how they are equivalent in Theorem 5.91.
Although one of the strengths of this book is the way that the different approaches to buildings are presented alongside each other, it is worth mentioning that the book will also excellently serve the reader who wishes to learn primarily about one of these approaches. The authors have provided different “pathways” through the book, which would make it an ideal text on which to base a reading group or series of graduate seminars.
From Chapter 6 onwards, the authors develop the group theory which goes hand in hand with the theory of buildings, looking first at the general construction of a BN-pair for a group acting nicely on a building (Chapter 6), and then RGD systems and the Moufang property (Chapters 7 and 8). In Chapter 9 we see the classification of spherical buildings, and then move on to Euclidean buildings in Chapters 10 and 11. Chapter 12 deals with the third “metric approach” to buildings. The final Chapters 13 and 14 are dedicated to describing some of the many applications of the theory of buildings to group cohomology, presentations of groups, finite groups, differential geometry, representation theory and harmonic analysis.
To sum up, this book is an extremely valuable addition to the mathematical literature. It provides a well written and accessible introduction to the theory of buildings, uniting all the different approaches in one place, and takes the reader from the basics all the way up to the present day. As well as this, it should also be an essential resource for experts in the area. In the reviewer’s opinion, the authors have succeeded in the goal they set themselves in the introduction, that “our exposition helps make Tits’s beautiful ideas accessible to a broad mathematical audience”.

##### MSC:
 20E42 Groups with a $$BN$$-pair; buildings 51E24 Buildings and the geometry of diagrams 20-02 Research exposition (monographs, survey articles) pertaining to group theory 51-02 Research exposition (monographs, survey articles) pertaining to geometry 20F55 Reflection and Coxeter groups (group-theoretic aspects) 51F15 Reflection groups, reflection geometries
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