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Mathematics - Algebra | Buildings

Buildings

Theory and Applications

Series: Graduate Texts in Mathematics, Vol. 248

Abramenko, Peter, Brown, Kenneth S.

2008, XXII, 754p. 100 illus..

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  • 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 property
  • 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

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 on one 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.

Content Level » Research

Keywords » Moufang property - cohomology - combinatorial approach - coxeter - euclidean buildings - homology - metric approach - simplicial approach

Related subjects » Algebra

Table of contents 

Preface.- Introduction.- 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.- Cell Complexes.- Root Systems.- Algebraic Groups.

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