- 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

# The Standard Metric

# The Standard Metric

- Chapter:
- (p.221) Chapter Twenty Six The Standard Metric
- Source:
- Descent in Buildings (AM-190)
- Author(s):
### Bernhard M¨uhlherr

### Holger P. Petersson

### Richard M. Weiss

- Publisher:
- Princeton University Press

This chapter presents some results about groups generated by reflections and the standard metric on a Bruhat-Tits building. It begins with definitions relating to an affine subspace, an affine hyperplane, an affine span, an affine map, and an affine transformation. It then considers a notation stating that the convex closure of a subset *a* of *X* is the intersection of all convex sets containing *a* and another notation that denotes by AGL(*X*) the group of all affine transformations of *X* and by Trans(*X*) the set of all translations of *X*. It also describes Euclidean spaces and assumes that the real vector space *X* is of finite dimension *n* and that *d* is a Euclidean metric on *X*. Finally, it discusses Euclidean representations and the standard metric.

*Keywords:*
reflection, standard metric, Bruhat-Tits building, affine subspace, affine hyperplane, affine transformation, convex closure, Euclidean space, Euclidean metric, Euclidean representation

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- 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