A point-line approach to the nonexistence of buildings of types $${{\,\mathrm{\textsf{H}_3}\,}}$$ and $${{\,\mathrm{\textsf{H}_4}\,}}$$
Year: 2026
Type: article
Abstract: We present a novel proof for the fact that thick spherical buildings of types $$\mathsf {H_3}$$ H 3 and $$\mathsf {H_4}$$ H 4 do not exist. For that, we first provide an elementary axiom system for Lie incidence geometries associated with buildings of type $$\mathsf {H_3}$$ 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 $$\mathsf {H_3}$$ 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 $$\mathsf {H_3}$$ H 3 and $$\mathsf {H_4}$$ H 4 , without invoking Tits’s extension theorem. Together with similar geometric constructions of the author for root elations of buildings of types $$\mathsf {B_n}$$ B n and $$\mathsf {C_n}$$ C n , and known constructions for buildings of type $$\mathsf {A_n}$$ 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.
Source: Journal of Geometry
Author: Sira Busch
Institution: University of Münster
Language: English
Topic: Mathematics and Applications
Subfield: Geometry and Topology
Field: Mathematics
Domain: Physical Sciences
Open Access status: hybrid
Estimated APC paid: $3,190
OpenAlex ID: W7135227197