Triangle geometries
The aim of this paper is to study triangle geometries. A triangle geometry is a rank 3 incidence geometry or, using the recent concept of a chamber system, a rank 3 chamber system whose rank 2 residues are projective planes. (By a theorem of Tits, every triangle geometry arises as quotient of a suitable affine building). The author investigates two special classes of triangle geometries: the trivalent ones (whose rank 1 residues have three chambers) and the tight ones (containing exactly three rank 2 residues). For the first class he shows that a regular automorphism group of a trivalent geometry can only be of four types and gives an example for three of these types. For the second class the author gives a method of construction using projective planes; in particular he proves that any finite desarguesian plane gives rise to a tight triangle geometry which admits the full Singer group of the plane, together with its multipliers. Special types of tight triangle geometries are studied, too.
scientific article; zbMATH DE number 3907238