On semi-finite hexagons of order \((2,t)\) containing a subhexagon
In this paper, the authors contribute to the existence problem of semi-finite thick hexagons. Recall the following definitions. A \textit{near \(2d\)-gon} is a partial linear space \(S\) that satisfies the following properties: (NP1) The collinearity graph of \(S\) is connected and has diameter \(d\). (NP2) For every point \(x\) and every line \(L\) there exists a unique point incident with \(L\) that is nearest to \(x\). A \textit{generalised \(2d\)-gon} is a near \(2d\)-gon that has the following additional properties: (GH1) Every point is incident with at least two lines. (GH2) Any two points at distance \(2\) have a unique common neighbour. A partial linear space is called \textit{thick} if it has at least \(3\) points per line and at least \(3\) lines per point. A \textit{semi-finite} thick polygon has a finite number of points on a line, but an infinite number of lines through a point. Semi-finite thick generalised quadrangles with three, four or five points on a line do not exist. In this paper, the authors adress the corresponding question for generalised hexagons. They show that, under the assumption that there does not exists an isometrically embedded subhexagon of order \(2\), such a semi-thick hexagon does not exist. Note that they prove this theorem for the more general class of near hexagons (rather than only for generalised hexagons). There are two different generalised hexagons of order \(2\): the split Cayley hexagon and its point-line dual. The authors are able to exclude both cases by a similar approach using valuations and the valuation geometries arising from the generalised hexagons of order \(2\). There are \(7\) possible valuations for the split Cayley hexagon and \(4\) different valuations for its dual. These valuations have to be excluded, so the proofs possess a computational component. The computer search is described in great detail in the appendix of this paper; the GAP code can also be found online.
scientific article; zbMATH DE number 6644410