Accueil - Recherche - Mini-cours - Enseignement - Diffusion et communication


Bruhat-Tits theory

Santiago de Chile, June,25 to July,6

Lectures' notes on Bruhat-Tits buildings

Full text

Handwritten PDF

Annexe: axiomatic definitions and bibliography


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
Handwritten PDF

    II. Euclidean buildings

  • The Euclidean distance
  • Topological properties of Euclidean buildings
  • The spherical building at infinity
Handwritten PDF

    0. Interlude on Reductive groups over a field

Handwritten PDF

    III. Action on buildings: combinatorial aspects

  • Tits systems
  • Action of a group on a general building
  • Root groups datum
Handwritten PDF

    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
Handwritten PDF

    V. The building of SL(V)

  • Space of norms
  • Simplicial structure
  • Action of matrices
  • Fundamental domains
Handwritten PDF

Second part: Constructions in Bruhat-Tits theory

    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
Handwritten PDF

    VII. The Euclidean Bruhat-Tits building of a quasi-split reductive group

  • The standard apartment
  • Gluing of apartments
Handwritten PDF

    VIII. Integral models

  • Integral models of root groups
  • Integral models of tori
  • Integral models of the whole group
Handwritten PDF

    IX. Étale descent

  • The underlying set of points
  • The cellular structure
  • The building structure
  • Action of the group on the building
  • The special fiber
Handwritten PDF

    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
Handwritten PDF

Dernière mise à jour : 19 juin 2018