Descent in Buildings (AM-190)

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 nece...

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Bibliographic Details
Main Authors:	Mühlherr, Bernhard, Petersson, Holger P, Weiss, Richard M
Format:	eBook Book
Language:	English
Published:	Princeton Princeton University Press 2015
Edition:	1
Series:	Annals of Mathematics Studies
Subjects:	Addition Additive group Affine space Affine transformation Algebra & number theory Algebraic Algebraic geometry Algebraic group Algebraic structure Algebraically closed field Automorphism Bijection Biquaternion Buildings (Group theory) Cardinality Combinatorial geometry Combinatorics Composition algebra Convex hull Coset Coxeter group Diagram (category theory) Dimension (vector space) Discrete valuation Division algebra Dynkin diagram Embedding Empty set Equivalence class Euclidean space Existential quantification Field extension Finite set Galois group General Topics for Engineers Geometry Half-space (geometry) Homomorphism Hyperbolic geometry Hyperplane Linear combination Linear map Linear space (geometry) Linear subspace Local field Mathematical induction MATHEMATICS MATHEMATICS / General MATHEMATICS / Geometry / General Metric space Module (mathematics) Moufang polygon Moufang set Non-abelian Octonion Octonion algebra Permutation Power set Projective space Purely inseparable extension Quadratic form Quaternion Quaternion algebra Residue field Root system Scientific notation Separable extension Set (mathematics) Simplicial complex Special case Splitting field Subgroup Subring Subset Substructure Summation Surjective function Theorem Unit sphere Vector space quaternion division algebra discrete valuation anti-isomorphism abelian group building isomorphism separable quadratic extension trace map biquaternion division algebra pseudo-split building wild quadratic space split quadratic space ramified quadrangle relative Coxeter diagram unramified separable quadratic extension arctic region weak isomorphism scalar multiplication compatible representation vector space absolute Coxeter system Coxeter group Structure Theorem pseudo-quadratic space Moufang polygon exceptional quadrangle automorphism Moufang quadrangle affine building Coxeter diagram gem quadratic module semi-ramified quadrangle Clifford invariant tamely ramified division algebra involutory set polar space descent group Moufang building descent relative Coxeter group fixed point theory isotropic quadratic space generalized quadrangle proper involutory set absolute Coxeter diagram Moufang structure projection map Fundamental Theorem of Descent vertex non-abelian group round quadratic space quadratic space Coxeter system anisotropic quadratic space residual quadratic spaces relative rank unramified quadratic space unramified quadrangle chamber special vertex absolute rank Moufang set quadratic form standard involution Euclidean plane bilinear form algebraic group subbuilding of split type Moufang condition Tits index hyperbolic quadratic space anisotropic pseudo-quadratic space ramified separable quadratic extension fixed point building hyperbolic plane simplicial complex subbuilding root group sequence trace relative Coxeter system unramified quaternion division algebra thick building root spherical building Bruhat-Tits building parallel residues proper indifferent set ramified quaternion division algebra residue exceptional Moufang quadrangle hyperbolic quadratic module length function Pfister form canonical isomorphism thin T-building apartment finite dimension Hyperbolic geometry Composition algebra Bijection Subset Biquaternion Quaternion algebra Separable extension Subgroup Substructure Octonion Non-abelian Vector space Mathematical induction Simplicial complex Dynkin diagram Power set Addition Cardinality Algebraic group Permutation Homomorphism Affine transformation Hyperplane Existential quantification Dimension (vector space) Coset Convex hull Subring Special case Root system Summation Unit sphere Projective space Algebraic structure Equivalence class Algebraically closed field Metric space Quaternion Octonion algebra Residue field Set (mathematics) Embedding Diagram (category theory) Division algebra Linear subspace Linear map Linear combination Splitting field Empty set Theorem Euclidean space Quadratic form Galois group Automorphism Field extension Linear space (geometry) Discrete valuation Affine space Half-space (geometry) Additive group Purely inseparable extension Local field Finite set Module (mathematics) Scientific notation Surjective function Algebraic geometry
ISBN:	9781400874019, 1400874017, 9780691166902, 0691166900, 0691166919, 9780691166919
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Summary:	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.
Bibliography:	Includes bibliographical references (p. [327]-331) and index
ISBN:	9781400874019 1400874017 9780691166902 0691166900 0691166919 9780691166919
DOI:	10.1515/9781400874019