##
**Lectures on buildings.**
*(English)*
Zbl 0694.51001

Perspectives in Mathematics, 7. Boston, MA: Academic Press, Inc. ix, 201 p. $ 27.95 (1989).

Lie (type) groups are best studied through their action on the associated building. This fact stood basically at the beginning of Tits’ work on those geometric structures - which he discovered, investigated and later on classified as far as one can only hope for, and which are called “buildings”. But this fact also makes the study of buildings a must for everybody who is interested in groups (say finite, simple), the Lie type groups being the most important and interesting groups to be found. Buildings, moreover, show themselves so many fascinating aspects that the above motivation for the study of buildings is actually no longer needed. Nevertheless, up to 1988, there was no introductory book, no standard text book on the subject, that could serve as a basis for courses or seminars on buildings. Hence both, group theorists and geometers, are happy to see the author’s book appear. In fact, this is a book that has grown out of a series of lectures given by the author and is therefore readable for anyone who wants to learn about buildings, but is also careful enough in the choice of material to reach theorems (mainly by Tits) that are of great depth and enormous importance.

Instead of commenting on certain aspects of the book, let me give a short survey on it.

The author starts off (chapter 1) defining chamber systems - the convenient language for talking about buildings and more general geometries or complexes - which have been used now ever since the Local Approach. Chapter 2 is devoted to (the chamber systems of) Coxeter complexes. Here chambers correspond to elements of the underlying Coxeter group. Coxeter complexes provide not only a most interesting class of examples for complexes (chamber systems,...), so deserve to be studied in their own right, but are most important serving as the apartments in the buildings to be defined. And, of course, crucial for the understanding of buildings is to know what is happening in their apartments.

The definition of buildings (as chamber systems) given (in chapter 3) is one more recent definition by Tits, and not the original one (which is given as Theorem (3.11)). The main axiom is the existence of a “Coxeter group valued metric” on the chamber system, the distance between two chambers being the common type of all the galleries joining them in the building (in every apartment they lie in).

Chapter 4 gives the local approach to buildings. It is here that the simple (2-)connectedness of buildings is proved and Tits’ Universal 2- cover Theorem can be found.

Tits’ classification (list) of buildings of spherical, irreducible type, rank at least 3, is given in chapter 8: here Tits diagrams are described and explained and material is brought together in a very neat way. The classification consists mainly of uniqueness theorems for certain types, given certain parameters (fields, small rank residues). In the form presented, the classification turns out to be a very abstract and local one - the approach taken (going back to joint work of Ronan and Tits) is through labellings and foundations. This rather constructive point of view is developed in chapter 7. Its origin - or at least its natural field of application - lies in Moufang buildings. The Moufang property assures the existence of “root subgroups”, i.e. subgroups of the automorphism group of the building I acting locally in a certain way and being isomorphic to the additive group of some field k, which in the end will be characterized for the whole of I. Such Moufang buildings are studied in chapter 6 for the spherical, irreducible type and rank at least 3 (here the buildings all have the Moufang property, this is part of Tits’ classification theorem), and in the Appendices 1 and 2 for the rank 2 case.

But the reader last not least finds also much of what he wants to know about affine buildings. In chapters 9 and 10 the relations between an affine building and its (spherical) building at infinity is described, and even Tits’ classification of affine buildings of rank at least 4 is given in chapter 10. In this situation the building at infinity does have the Moufang property, and the problem is to come back from the root groups on the building at infinity to the original building which wants to be classified. One has to look at “root data with valuation” and in the end construct a Tits system (BN-pair), which gives back the (affine) building. Tits systems, this concept played a central role in the recognition of finite simple groups and is somehow equivalent to the building in the finite case, are introduced and described in chapter 5.

The “lectures” supply material on buildings that gathers around the theorems from J. Tits [‘Buildings of spherical type and finite BN- pairs (1974; Zbl 0295.20047), The geometric vein, The Coxeter Festschr., 519-547 (1982; Zbl 0496.51001), Lect. Notes Math. 1181, 159-190 (1986; Zbl 0611.20026), and Lond. Math. Soc. Lect. Note Ser. 121, 110-127 (1986; Zbl 0631.20019)], but also from F. Bruhat and J. Tits [Inst. Haut. Étud. Sci., Publ. Math. 41, 5-251 (1972; Zbl 0254.14017)] on affine buildings. Moreover, much can be found that one had to look for in Bourbaki: Groupes et algèbres de Lie IV, V, VI so far.

The appendices contain a proof of the Tits-Weiss non-existence theorem on Moufang generalized polygons, but also contain tables on finite Coxeter groups and Lie type groups.

To sum up the remarks made, and to make clear the impression the reviewer has got from the text, the author’s book is

- important, as a text book for courses on the subject but also as an introduction to the area that goes as far as to the main theorems (by Tits) on buildings that have appeared in a period of now more than 15 years.

- written very clearly and is enjoyable to read,

- containing exercises which are helpful and provide necessary examples in great numbers.

Instead of commenting on certain aspects of the book, let me give a short survey on it.

The author starts off (chapter 1) defining chamber systems - the convenient language for talking about buildings and more general geometries or complexes - which have been used now ever since the Local Approach. Chapter 2 is devoted to (the chamber systems of) Coxeter complexes. Here chambers correspond to elements of the underlying Coxeter group. Coxeter complexes provide not only a most interesting class of examples for complexes (chamber systems,...), so deserve to be studied in their own right, but are most important serving as the apartments in the buildings to be defined. And, of course, crucial for the understanding of buildings is to know what is happening in their apartments.

The definition of buildings (as chamber systems) given (in chapter 3) is one more recent definition by Tits, and not the original one (which is given as Theorem (3.11)). The main axiom is the existence of a “Coxeter group valued metric” on the chamber system, the distance between two chambers being the common type of all the galleries joining them in the building (in every apartment they lie in).

Chapter 4 gives the local approach to buildings. It is here that the simple (2-)connectedness of buildings is proved and Tits’ Universal 2- cover Theorem can be found.

Tits’ classification (list) of buildings of spherical, irreducible type, rank at least 3, is given in chapter 8: here Tits diagrams are described and explained and material is brought together in a very neat way. The classification consists mainly of uniqueness theorems for certain types, given certain parameters (fields, small rank residues). In the form presented, the classification turns out to be a very abstract and local one - the approach taken (going back to joint work of Ronan and Tits) is through labellings and foundations. This rather constructive point of view is developed in chapter 7. Its origin - or at least its natural field of application - lies in Moufang buildings. The Moufang property assures the existence of “root subgroups”, i.e. subgroups of the automorphism group of the building I acting locally in a certain way and being isomorphic to the additive group of some field k, which in the end will be characterized for the whole of I. Such Moufang buildings are studied in chapter 6 for the spherical, irreducible type and rank at least 3 (here the buildings all have the Moufang property, this is part of Tits’ classification theorem), and in the Appendices 1 and 2 for the rank 2 case.

But the reader last not least finds also much of what he wants to know about affine buildings. In chapters 9 and 10 the relations between an affine building and its (spherical) building at infinity is described, and even Tits’ classification of affine buildings of rank at least 4 is given in chapter 10. In this situation the building at infinity does have the Moufang property, and the problem is to come back from the root groups on the building at infinity to the original building which wants to be classified. One has to look at “root data with valuation” and in the end construct a Tits system (BN-pair), which gives back the (affine) building. Tits systems, this concept played a central role in the recognition of finite simple groups and is somehow equivalent to the building in the finite case, are introduced and described in chapter 5.

The “lectures” supply material on buildings that gathers around the theorems from J. Tits [‘Buildings of spherical type and finite BN- pairs (1974; Zbl 0295.20047), The geometric vein, The Coxeter Festschr., 519-547 (1982; Zbl 0496.51001), Lect. Notes Math. 1181, 159-190 (1986; Zbl 0611.20026), and Lond. Math. Soc. Lect. Note Ser. 121, 110-127 (1986; Zbl 0631.20019)], but also from F. Bruhat and J. Tits [Inst. Haut. Étud. Sci., Publ. Math. 41, 5-251 (1972; Zbl 0254.14017)] on affine buildings. Moreover, much can be found that one had to look for in Bourbaki: Groupes et algèbres de Lie IV, V, VI so far.

The appendices contain a proof of the Tits-Weiss non-existence theorem on Moufang generalized polygons, but also contain tables on finite Coxeter groups and Lie type groups.

To sum up the remarks made, and to make clear the impression the reviewer has got from the text, the author’s book is

- important, as a text book for courses on the subject but also as an introduction to the area that goes as far as to the main theorems (by Tits) on buildings that have appeared in a period of now more than 15 years.

- written very clearly and is enjoyable to read,

- containing exercises which are helpful and provide necessary examples in great numbers.

Reviewer: Th.Meixner

### MSC:

51-02 | Research exposition (monographs, survey articles) pertaining to geometry |

51E25 | Other finite nonlinear geometries |

20-02 | Research exposition (monographs, survey articles) pertaining to group theory |

20F65 | Geometric group theory |

20G15 | Linear algebraic groups over arbitrary fields |

20D06 | Simple groups: alternating groups and groups of Lie type |