Affine extensions of generalized polygons
A partial linear space \((P,L)\) consists of a set \(P\) of points and a set \(L\) of subsets of size at least 2 of \(P\) called lines, such that any two points are contained in at most one line. Two elements \(a,\;b\) of a partial linear space are called opposite if the incidency graph of the space have finite diameter \(d\) and \(\partial (a,b)=d\). In this paper the extensions of generalized polygons by affine planes with the following condition \((*)_{Opp}\) are investigated: \((*)_{Opp}\) or a point \(p\) and any two lines \(m,l\) on \(p\) holds that \(m\) and \(l\) are not opposite in the residue of \(p\) if and only if \(m \subset l^\bot\). Theorem 1.1. Let \(G=(P,L,A)\) be an affine extension of a generalized \(n\)-gon, \(n\geq 5\), in which \((P,L)\) is a thick nonlinear gamma space and \((*)_{Opp}\) holds. Then \(n=6\) and \(G\) is isomorphic to the affine generalized hexagon associated to either the group \(k^7:G_2(k)\) or to the spin representation of a group of type \(B_3(k)\), where \(k\) is a field. Proposition 8.1. Let \(G=(P,L,A)\) be an affine extension of a finite generalized hexagon in which \((P,L)\) is a gamma space and the following holds: \((*)_3\) for any point \(p\), the distance between two different lines \(m,l\) on \(p\) in the hexagon Res\((p)\) equals 2 or 6 if and only if \(m \subset l^\bot\). Then \(G\) is isomorphic to the extended generalized hexagon on 36 points related to the group \(G_2(2)\) or the geometry on 100 points related to the Hall Janko group.
scientific article; zbMATH DE number 1348198
scientific article