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The problem can be reduced further in the following manner. Much of the information about the structure of a building is contained in the different J-residues for |J| = 2. In fact, by including a labeling scheme on the chambers, it is possible in many cases to collate all of the important information about the building into a foundation consisting of these J-residues and the labeling information. Foundations allow us to reconstruct the building by gluing together the J-residues according to an equivalency determined by the labeling.

It turns out that to each affine building we can associate a spherical “building at infinity”. By using the classification of spherical buildings, we can again reduce the problem of classifying affine buildings to a more manageable question to one about root groups and labeling schemes. These classification theorems come in useful for working with Lie groups or manifolds: if we can construct a building based on the object in question, our classification theorem will help us determine its properties.

As the close link between buildings and BN-pairs suggests, the theory of buildings can be used to derive many more useful theorems in group theory. 30 References [1] Ronan, Mark Lectures on Buildings, Academic Press, Inc. (1989) [2] Brown, Kenneth Buildings, Springer-Verlag New York Inc.

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Buildings (Cambridge University Part III) by Brian Lehmann


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