The Moufang Condition and Root Automorphisms for Spherical Buildings of Rank 3

TL;DR

This paper provides a geometric approach to understanding root automorphisms in spherical buildings of rank 3, offering new insights and proofs for their Moufang properties and non-existence of certain types.

Contribution

It introduces a geometric construction of root elations, offers a new proof of Moufang properties for types Bn and Cn, and explains why thick Hm buildings cannot exist.

Findings

Root elations can be extended to ambient buildings in certain types.

Buildings of type Bn and Cn are Moufang, with new geometric proofs.

Thick, spherical buildings of type Hm cannot exist.

Abstract

We give direct, geometric constructions for nontrivial root elations for rank $2$ residues of higher rank buildings $\Delta$ of type $\mathsf{B_n}, \mathsf{C_n}$ and $\mathsf{H_m}$ for $n \in \mathbb{N}$ and $m \in \{3,4\}$. We show that we can extend these to the ambient building in the case that $\Delta$ has type $\mathsf{B_n}$ or $\mathsf{C_n}$. With that, we obtain a different proof for the fact that buildings of type $\mathsf{B_n}$ and $\mathsf{C_n}$ are Moufang. This geometric approach enables us to gain more insight into the root groups associated to these buildings and we obtain new results; Namely, that certain root elations generically fix more points than we previously knew and that every root elation in each point residual can be written as an even self-projectivity. Concerning $\mathsf{H_m}$, we will be able to see in a novel way why thick, spherical buildings of type $\mathsf{H_m}$ cannot exist. Altogther, this provides an alternative proof for the fact that all thick, irreducible, spherical buildings $\Delta$ of rank 3 have the Moufang property.