Descent in Buildings begins with the resolution of a major open
question about the local structure of Bruhat-Tits buildings. The
authors then put their algebraic solution into a geometric context
by developing a general fixed point theory for groups acting on
buildings of arbitrary type, giving necessary and sufficient
conditions for the residues fixed by a group to form a kind of
subbuilding or "form" of the original building. At the center of
this theory is the notion of a Tits index, a combinatorial version
of the notion of an index in the relative theory of algebraic
groups. These results are combined at the end to show that every
exceptional Bruhat-Tits building arises as a form of a "residually
pseudo-split" Bruhat-Tits building. The book concludes with a
display of the Tits indices associated with each of these
exceptional forms. This is the third and final volume of a trilogy
that began with Richard Weiss' The Structure of Spherical Buildings
and The Structure of Affine Buildings.
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