One-point extensions of generalized hexagons and octagons (Q5901356)

Block designs which are one-point extensions of finite thick generalized polygons are a rather rare species. There is an infinite family of such one-point extensions of Ahrens-Szekeres generalized quadrangles AS\((q)\) of order \((q-1,q+1)\), where \(q\) is an odd prime power. This was found by Thas in 1983 using a construction of \textit{G. Hölz} [Arch. Math. 37, 179--183 (1981; Zbl 0451.05015)]. In addition to this family, there are four more known sporadic examples, one of them being a one-point extension of the split Cayley generalized polygon H(2) of order 2. All of these sporadic examples can be obtained using a construction called affine extension. The authors characterize the affine extension \(S\) of the split Cayley generalized polygon H(2) in terms of the so-called distance property: for any three points \(x\), \(y\) and \(z\) the graph- theoretical distance from \(y\) to \(z\) in the derived generalized hexagon \(S_x\) is the same as from \(x\) to \(z\) in \(S_y\). It is proved that any one-point extension of a generalized hexagon or octagon admitting a flag-transitive group of automorphisms is isomorphic to the affine extension of H(2).