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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.

Buildings (Group theory) --- Combinatorial geometry. --- Geometric combinatorics --- Geometrical combinatorics --- Combinatorial analysis --- Discrete geometry --- Theory of buildings (Group theory) --- Tits's theory of buildings (Group theory) --- Linear algebraic groups --- Bruhat-Tits building. --- Clifford invariant. --- Coxeter diagram. --- Coxeter group. --- Coxeter system. --- Euclidean plane. --- Fundamental Theorem of Descent. --- Moufang building. --- Moufang condition. --- Moufang polygon. --- Moufang quadrangle. --- Moufang set. --- Moufang structure. --- Pfister form. --- Structure Theorem. --- Tits index. --- abelian group. --- absolute Coxeter diagram. --- absolute Coxeter system. --- absolute rank. --- affine building. --- algebraic group. --- anisotropic pseudo-quadratic space. --- anisotropic quadratic space. --- anti-isomorphism. --- apartment. --- arctic region. --- automorphism. --- bilinear form. --- biquaternion division algebra. --- building. --- canonical isomorphism. --- chamber. --- compatible representation. --- descent group. --- descent. --- discrete valuation. --- exceptional Moufang quadrangle. --- exceptional quadrangle. --- finite dimension. --- fixed point building. --- fixed point theory. --- gem. --- generalized quadrangle. --- hyperbolic plane. --- hyperbolic quadratic module. --- hyperbolic quadratic space. --- involutory set. --- isomorphism. --- isotropic quadratic space. --- length function. --- non-abelian group. --- parallel residues. --- polar space. --- projection map. --- proper indifferent set. --- proper involutory set. --- pseudo-quadratic space. --- pseudo-split building. --- quadratic form. --- quadratic module. --- quadratic space. --- quaternion division algebra. --- ramified quadrangle. --- ramified quaternion division algebra. --- ramified separable quadratic extension. --- relative Coxeter diagram. --- relative Coxeter group. --- relative Coxeter system. --- relative rank. --- residual quadratic spaces. --- residue. --- root group sequence. --- root. --- round quadratic space. --- scalar multiplication. --- semi-ramified quadrangle. --- separable quadratic extension. --- simplicial complex. --- special vertex. --- spherical building. --- split quadratic space. --- standard involution. --- subbuilding of split type. --- subbuilding. --- tamely ramified division algebra. --- thick building. --- thin T-building. --- trace map. --- trace. --- unramified quadrangle. --- unramified quadratic space. --- unramified quaternion division algebra. --- unramified separable quadratic extension. --- vector space. --- vertex. --- weak isomorphism. --- wild quadratic space.