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

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 $T(2,8)$.