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 HD(2). In fact, the employed techniques allow us to prove a stronger result. We show that every near hexagon 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 ∼= HD(2) then S must also be a generalized hexagon, and consequently isomorphic to either HD(2) or the dual twisted triality hexagon T (2, 8).