NON-DISCRETE EUCLIDEAN BUILDINGS FOR THE REE AND SUZUKI GROUPS

• 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
• ISSN: 0002-9327; 1080-6377; 1080-6377
• DOI: 10.1353/ajm.0.0133

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.