By Brian Lehmann

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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.

For such buildings, each apartment will correspond to a tiling of a Euclidean vector space. Thus, we can impose a natural metric on affine buildings that is derived from this local Euclidean structure. 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.

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 every spherical building yields a foundation. Furthermore, these foundations each yield a unique building provided that they satisfy some conditions on their restriction to spherical buildings of rank 3.

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Buildings (Cambridge University Part III) by Brian Lehmann
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