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\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathcal{S}}$$\end{document} 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\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${H \cong H^{D}}$$\end{document}(2) then S\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathcal{S}}$$\end{document} must also be a generalized hexagon, and consequently isomorphic to either HD(2) or the dual twisted triality hexagon T(2, 8).