Buildings

Covering all three approaches to Jacques Tits's beautiful theory of buildings, this accessible volume is suitable as a textbook introduction to the subject as well as a self-study text for researchers. It contains numerous exercises.

This text started out as a revised version of Buildings by the second-named author [53], but it has grown into a much more voluminous book. The earlier book was intended to give a short, friendly, elementary introduction to theory, accessible to readers with a minimal background.Moreover, it approached buildings from only one point of view, sometimes called the old-fashioned approach: A building is a simplicial complex with certain properties. The current book includes all the material of the earlier one, but we have added a lot. In particular, we have included the modern (or W-metric) approach to buildings, which looks quite different from the old-fashioned approach but is equivalent to it. This has become increasingly important in the theory and applications of buildings. We have also added a thorough treatment of the Moufang property, which occupies two chapters. And we have added many new exercises and illustrations. Some of the exercises have hints or solutions in the back of the book. A more extensive set of solutions is available in a separate solutions manual, which may be obtained from Springer's Mathematics Editorial Department. We have tried to add the new material in such a way that readers who are content with the old-fashioned approach can still get an elementary treatment of it by reading selected chapters or sections. In particular, many readers will want to omit the optional sections (marked with a star). The introduction below provides more detailed guidance to the reader.

Contains all of the material from the previous book, Buildings by K. S. Brown (a short, friendly, elementary introduction to the theory of buildings), and substantially revised, updated, new material Includes advanced content that is appropriate for more advanced students or for self-study, including two new chapters on the Moufang propert Introduces many new exercises and illustrations, as well as hints and solutions--including a separate, extensive solutions manual Thoroughly focuses on all three approachs to buildings, old-fashioned," combinatorial (chamber systems), and metric so that the reader can learn all three or focus on only one Includes appendices on cell complexes, root systems and algebraic groups Request lecturer material: sn.pub/lecturer-material

Autorentext

Kenneth S. Brown has been a professor at Cornell since 1971. He received his Ph.D. in 1971 from MIT. He has published many works, including Buildings with Springer-Verlag in 1989, reprinted in 1998.

Peter Abramenko received his Ph.D. in 1987 from the University of Frankfurt, Germany. He held various academic positions afterwards, including a Heisenberg fellowship from 1998 until 2001. Since 2001, he is Associate Professor at the University of Virginia in Charlottesville. He has previously published Twin Buildings and Applications to S-Arithmetic Groups for the Lecture Notes in Mathematics series for Springer (1996).

Klappentext

This book treats Jacques Tits's beautiful theory of buildings, making that theory accessible to readers with minimal background. It includes all the material of the earlier book Buildings by the second-named author, published by Springer-Verlag in 1989, which gave an introduction to buildings from the classical (simplicial) point of view. This new book also includes two other approaches to buildings, which nicely complement the simplicial approach: On the one hand, buildings may be viewed as abstract sets of chambers with a Weyl-group-valued distance function; this point of view has become increasingly important in the theory and applications of buildings. On the other hand, buildings may be viewed as metric spaces. Beginners can still use parts of the new book as a friendly introduction to buildings, but the book also contains valuable material for the active researcher.

There are several paths through the book, so that readers may choose to concentrate onone particular approach. The pace is gentle in the elementary parts of the book, and the style is friendly throughout. All concepts are well motivated. There are thorough treatments of advanced topics such as the Moufang property, with arguments that are much more detailed than those that have previously appeared in the literature.

This book is suitable as a textbook, with many exercises, and it may also be used for self-study.

Inhalt
Finite Reflection Groups.- Coxeter Groups.- Coxeter Complexes.- Buildings as Chamber Complexes ..- Buildings as W-Metric Spaces.- Buildings and Groups.- Root Groups and the Moufang Property ..- Moufang Twin Buildings and RGD Systems.- The Classification of Spherical Buildings.- Euclidean and Hyperbolic Reflection Groups.- Euclidean Buildings ..- Buildings as Metric Spaces.- Applications to the Cohomology of Groups.- Other Applications.