Lectures' notes on BruhatTits buildings

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Annexe: axiomatic definitions and bibliography


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First part: Action of a group on an Euclidean building

I. Motivations and definitions
Historical motivation
The spherical building of GL(V)
The general definition of buildings

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II. Euclidean buildings
The Euclidean distance
Topological properties of Euclidean buildings
The spherical building at infinity

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0. Interlude on Reductive groups over a field

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III. Action on buildings: combinatorial aspects
Tits systems
Action of a group on a general building
Root groups datum

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IV. Action on Euclidean buildings: topological aspects
The BruhatTits fixed point theorem (with a proof of Serre)
Bounded subsets
Application: maximal thingy subgroups
Some decompositions: Cartan, Iwasawa, Mixed Bruhat

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V. The building of SL(V)
Space of norms
Simplicial structure
Action of matrices
Fundamental domains

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Second part: Constructions in BruhatTits theory

VI. Presentation of the Étale descent and quasisplitting
Presentation of the descent method
The ∗action and root systems
Valuation of a root groups datum
Parametrization of root groups

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VII. The Euclidean BruhatTits building of a quasisplit reductive group
The standard apartment
Gluing of apartments

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VIII. Integral models
Integral models of root groups
Integral models of tori
Integral models of the whole group

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IX. Étale descent
The underlying set of points
The cellular structure
The building structure
Action of the group on the building
The special fiber

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X. Compactification of Euclidean buildings
Some basics in Berkovich geometry
Affinoid subgroup stabilizing a point
Realization of the building in flag varieties
Berkovich compactifications of the building
Properties of the action of the group of rational points on the compactifications

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