Lectures' notes on Bruhat-Tits buildings
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Annexe: axiomatic definitions and bibliography
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First part: Action of a group on an Euclidean building
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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 Bruhat-Tits 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 Bruhat-Tits theory
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VI. Presentation of the Étale descent and quasi-splitting
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 Bruhat-Tits building of a quasi-split 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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