A finite flag-transitive geometry of extended \(G_ 2\)-type
The author constructs a finite flag-transitive GAB (a geometry that is almost a building [terminology of \textit{W. M. Kantor}, see Eur. J. Comb. 2, 239-247 (1981; Zbl 0514.51006)] or geometry of type M [terminology of \textit{J. Tits}, see The geometric vein, The Coxeter Festschr., 519-547 (1982; Zbl 0496.51001)] belonging to an (affine) extended \(G_ 2\) diagram. The group of its automorphism is the classical group \(G_ 2(3)\). All residues have order 2. The construction is explicit and uses the split octave algebra over the field GF(3) and its automorphism group \(G_ 2(3)\). The generalized hexagons in the residues are recognized as the geometry of the non-isotropic points of a unitary space over GF(9) where the lines are the maximal cliques. The author also shows that the GAB has no apartments in the sense of Kantor (see reference above), but it satisfies the intersection property.
scientific article; zbMATH DE number 4105730