A point-line approach to the nonexistence of buildings of types ${\mathsf{H}}_3$ and ${\mathsf{H}}_4$

Abstract

We present a novel proof for the fact that thick spherical buildings of types ${\mathsf{H}}_3$ and ${\mathsf{H}}_4$ do not exist. For that, we first provide an elementary axiom system for Lie incidence geometries associated with buildings of type ${\mathsf{H}}_3$. This way, we can write our arguments purely in the language of point-line geometries, not needing the theory of buildings. Assuming the existence of thick spherical buildings of type ${\mathsf{H}}_3$, we construct nontrivial root elations of generalized pentagons contained within them. This leads to a contradiction with Tits's result on the nonexistence of Moufang generalized pentagons. Consequently, we obtain a new, direct, and geometric proof for the nonexistence of thick spherical buildings of types ${\mathsf{H}}_3$ and ${\mathsf{H}}_4$, without invoking Tits's extension theorem. Together with similar geometric constructions of the author for root elations of buildings of types ${\mathsf{B}}_n$ and ${\mathsf{C}}_n$, and known constructions for buildings of type ${\mathsf{A}}_n$, this yields an alternative, elementary proof for the fact that all thick irreducible spherical buildings of rank $3$ have the Moufang property, not using Tits's extension theorem.

New definition for icosahedral geometries. Proofs for polar spaces are now in a separate article