On semi-finite hexagons of order $(2, t)$ containing a subhexagon

TL;DR

This paper proves that semi-finite generalized hexagons of order (2,t) cannot contain certain subhexagons, and any near hexagon with such a subhexagon must be finite, with specific isomorphism conditions.

Contribution

It establishes non-existence results for semi-finite hexagons containing a subhexagon of order 2 and characterizes the structure of near hexagons with such subgeometries.

Findings

No semi-finite generalized hexagon of order (2,t) contains a subhexagon of order 2.

Any near hexagon of order (2,t) with a subhexagon of order 2 must be finite.

If the subhexagon is isomorphic to the dual of H(2), the near hexagon is either H^D(2) or T(2,8).

Abstract

The research in this paper was motivated by one of the most important open problems in the theory of generalized polygons, namely the existence problem for semi-finite thick generalized polygons. We show here that no semi-finite generalized hexagon of order $(2,t)$ can have a subhexagon $H$ of order $2$. Such a subhexagon is necessarily isomorphic to the split Cayley generalized hexagon $H(2)$ or its point-line dual $H^D(2)$. In fact, the employed techniques allow us to prove a stronger result. We show that every near hexagon $\mathcal{S}$ of order $(2,t)$ which contains a generalized hexagon $H$ of order $2$ as an isometrically embedded subgeometry must be finite. Moreover, if $H \cong H^D(2)$ then $\mathcal{S}$ must also be a generalized hexagon, and consequently isomorphic to either $H^D(2)$ or the dual twisted triality hexagon $T(2,8)$.