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

Abstract We present a novel proof for the fact that thick spherical buildings of types $$\mathsf {H_3}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>H</mml:mi> <mml:mn>3</mml:mn> </mml:msub> </mml:math> and $$\mathsf {H_4}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>H</mml:mi> <mml:mn>4</mml:mn> </mml:msub> </mml:math> do not exist. For that, we first provide an elementary axiom system for Lie incidence geometries associated with buildings of type $$\mathsf {H_3}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>H</mml:mi> <mml:mn>3</mml:mn> </mml:msub> </mml:math> . 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 $$\mathsf {H_3}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>H</mml:mi> <mml:mn>3</mml:mn> </mml:msub> </mml:math> , 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 $$\mathsf {H_3}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>H</mml:mi> <mml:mn>3</mml:mn> </mml:msub> </mml:math> and $$\mathsf {H_4}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>H</mml:mi> <mml:mn>4</mml:mn> </mml:msub> </mml:math> , without invoking Tits’s extension theorem. Together with similar geometric constructions of the author for root elations of buildings of types $$\mathsf {B_n}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>B</mml:mi> <mml:mi>n</mml:mi> </mml:msub> </mml:math> and $$\mathsf {C_n}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>C</mml:mi> <mml:mi>n</mml:mi> </mml:msub> </mml:math> , and known constructions for buildings of type $$\mathsf {A_n}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>A</mml:mi> <mml:mi>n</mml:mi> </mml:msub> </mml:math> , 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.