Non-discrete Euclidean Buildings for the Ree and Suzuki groups

TL;DR

This paper classifies non-discrete Euclidean buildings, called Bruhat-Tits spaces, associated with Ree and Suzuki groups, revealing their structure and embeddings within ambient buildings.

Contribution

It provides a complete classification of Bruhat-Tits spaces linked to Ree and Suzuki groups, extending the understanding of their geometric and algebraic properties.

Findings

Classified Bruhat-Tits spaces for Ree and Suzuki groups.

Identified fixed point sets as buildings or generalized octagons.

Established natural embeddings into ambient Bruhat-Tits spaces.

Abstract

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 B_2, F_4 or G_2 associated with a Ree or Suzuki group endowed with the usual root datum. (In the B_2 and G_2 cases, this fixed point set is a building of rank one; in the F_4 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.