- Title Pages
- Dedication
- Epigraph
- Preface
- Chapter One Buildings
- Chapter Two Quadratic Forms
- Chapter Three Moufang Polygons
- Chapter Four Moufang Quadrangles
- Chapter Five Linked Tori, I
- Chapter Six Linked Tori, II
- Chapter Seven Quadratic Forms over a ∈ Local Field
- Chapter Eight Quadratic Forms of Type E6, E7 and E8
- Chapter Nine Quadratic Forms of Type F4
- Chapter Ten Residues
- Chapter Eleven Unramified Quadrangles of Type E6, E7 and E8
- Chapter Twelve Semi-ramified Quadrangles of Type E6, E7 and E8
- Chapter Thirteen Ramified Quadrangles of Type E6, E7 and E8
- Chapter Fourteen Quadrangles of Type E6, E7 and E8: Summary
- Chapter Fifteen Totally Wild Quadratic Forms of Type E7
- Chapter Sixteen Existence
- Chapter Seventeen Quadrangles of Type F4
- Chapter Eighteen The Other Bruhat-Tits Buildings
- Chapter Nineteen Coxeter Groups
- Chapter Twenty Tits Indices
- Chapter Twenty One Parallel Residues
- Chapter Twenty Two Fixed Point Buildings
- Chapter Twenty Three Subbuildings
- Chapter Twenty Four Moufang Structures
- Chapter Twenty Five Fixed Apartments
- Chapter Twenty Six The Standard Metric
- Chapter Twenty Seven Affine Fixed Point Buildings
- Chapter Twenty Eight Pseudo-Split Buildings
- Chapter Twenty Nine Linear Automorphisms
- Chapter Thirty Strictly Semi-linear Automorphisms
- Chapter Thirty One Galois Involutions
- Chapter Thirty Two Unramified Galois Involutions
- Chapter Thirty Three Residually Pseudo-Split Buildings
- Chapter Thirty Four Forms of Residually Pseudo-Split Buildings
- Chapter Thirty Five Orthogonal Buildings
- Chapter Thirty Six Indices for the Exceptional Bruhat-Tits Buildings
- Bibliography
- Index

# Affine Fixed Point Buildings

# Affine Fixed Point Buildings

- Chapter:
- (p.233) Chapter Twenty Seven Affine Fixed Point Buildings
- Source:
- Descent in Buildings (AM-190)
- Author(s):
### Bernhard M¨uhlherr

### Holger P. Petersson

### Richard M. Weiss

- Publisher:
- Princeton University Press

This chapter shows that if Ξ is an affine building and Γ is a finite descent group of Ξ, then Γ is a descent group of Ξ^{∞} and (Ξ^{∞}) is congruent to (Ξ^{∞}). Ξ^{Γ} and Ξ can be viewed as metric spaces. The chapter first considers the assumptions that Π is an irreducible affine Coxeter diagram, Ξ is a thick building of type Ξ, Γis a finite descent group of Ξ, and Tits index �� = (Π, Θ, *A*). It then describes apartments that are endowed with reflection hyperplanes and reflection half-spaces before concluding with a theorem about a canonical isomorphism from the fixed point building Ξ^{Γ} to (Ξ^{Γ}).

*Keywords:*
affine building, metric space, affine Coxeter diagram, thick building, finite descent group, Tits index, apartment, reflection hyperplane, reflection half-space, fixed point building

Princeton Scholarship Online requires a subscription or purchase to access the full text of books within the service. Public users can however freely search the site and view the abstracts and keywords for each book and chapter.

Please, subscribe or login to access full text content.

If you think you should have access to this title, please contact your librarian.

To troubleshoot, please check our FAQs , and if you can't find the answer there, please contact us.

- Title Pages
- Dedication
- Epigraph
- Preface
- Chapter One Buildings
- Chapter Two Quadratic Forms
- Chapter Three Moufang Polygons
- Chapter Four Moufang Quadrangles
- Chapter Five Linked Tori, I
- Chapter Six Linked Tori, II
- Chapter Seven Quadratic Forms over a ∈ Local Field
- Chapter Eight Quadratic Forms of Type E6, E7 and E8
- Chapter Nine Quadratic Forms of Type F4
- Chapter Ten Residues
- Chapter Eleven Unramified Quadrangles of Type E6, E7 and E8
- Chapter Twelve Semi-ramified Quadrangles of Type E6, E7 and E8
- Chapter Thirteen Ramified Quadrangles of Type E6, E7 and E8
- Chapter Fourteen Quadrangles of Type E6, E7 and E8: Summary
- Chapter Fifteen Totally Wild Quadratic Forms of Type E7
- Chapter Sixteen Existence
- Chapter Seventeen Quadrangles of Type F4
- Chapter Eighteen The Other Bruhat-Tits Buildings
- Chapter Nineteen Coxeter Groups
- Chapter Twenty Tits Indices
- Chapter Twenty One Parallel Residues
- Chapter Twenty Two Fixed Point Buildings
- Chapter Twenty Three Subbuildings
- Chapter Twenty Four Moufang Structures
- Chapter Twenty Five Fixed Apartments
- Chapter Twenty Six The Standard Metric
- Chapter Twenty Seven Affine Fixed Point Buildings
- Chapter Twenty Eight Pseudo-Split Buildings
- Chapter Twenty Nine Linear Automorphisms
- Chapter Thirty Strictly Semi-linear Automorphisms
- Chapter Thirty One Galois Involutions
- Chapter Thirty Two Unramified Galois Involutions
- Chapter Thirty Three Residually Pseudo-Split Buildings
- Chapter Thirty Four Forms of Residually Pseudo-Split Buildings
- Chapter Thirty Five Orthogonal Buildings
- Chapter Thirty Six Indices for the Exceptional Bruhat-Tits Buildings
- Bibliography
- Index