NON-DISCRETE EUCLIDEAN BUILDINGS FOR THE REE AND SUZUKI GROUPS

We call a non-discrete Euclidean building a Bruhat-Tits space if its automorphism group contains a subgroup that induces the subgroup generated by all the root groups of a root datum of the building at infinity. This is the class of non-discrete Euclidean buildings introduced and studied by Bruhat a...

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Bibliographic Details
Published in:American journal of mathematics 2010-08, Vol.132 (4), p.1113-1152
Main Authors: Hitzelberger, Petra, Kramer, Linus, Weiss, Richard M.
Format: Article
Language:English
Subjects:
Automorphisms
Differential geometry
Educational buildings
Equipollence
Euclidean geometry
Euclidean space
Exact sciences and technology
General mathematics
General, history and biography
Geometry
Infinity
Mathematical analysis
Mathematical functions
Mathematical problems
Mathematics
Multivariate analysis
Probability and statistics
Real numbers
Residential buildings
Sciences and techniques of general use
Statistics
Vertices
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Summary:We call a non-discrete Euclidean building a Bruhat-Tits space if its automorphism group contains a subgroup that induces the subgroup generated by all the root groups of a root datum of the building at infinity. This is the class of non-discrete Euclidean buildings introduced and studied by Bruhat and Tits. We give the complete classification of Bruhat-Tits spaces whose building at infinity is the fixed point set of a polarity of an ambient building of type B₂, F₄ or G₂ associated with a Ree or Suzuki group endowed with the usual root datum. (In the B₂ and G₂ cases, this fixed point set is a building of rank one; in the F₄ case, it is a generalized octagon whose Weyl group is not crystallographic.) We also show that each of these Bruhat-Tits spaces has a natural embedding in the unique Bruhat-Tits space whose building at infinity is the corresponding ambient building.
ISSN:0002-9327
1080-6377
1080-6377
DOI:10.1353/ajm.0.0133