1 9 M ar 2 01 5 On semi - finite hexagons of order ( 2 , t ) containing a

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 dualH(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 ∼= H(2) then S must also be a generalized hexagon, and consequently isomorphic to either H(2) or the dual twisted triality hexagon T (2, 8).