On flat flag-transitive \(c.c^*\)-geometries (Q674779)

A \(c.c^*\)-geometry is a rank 3 Buekenhout-geometry with points, lines and planes such that the residue of a point (respectively plane) is a complete graph (respectively dual complete graph) and the residue of a line is a digon. Such a geometry is flat if all points are incident with all planes. The paper under review characterizes all flat flag-transitive \(c.c^*\)-geometries. The result is that there is one sporadic example (related to the symmetric or alternating group of degree 6) and an infinite class obtained by gluing two copies of the point-line geometry of an \(n\)-dimensional affine space over \(GF(2)\). There is some canonical way to do so -- as is explained in the paper -- but also non-canonical ways give rise to flag-transitive examples. There is no explicit classification of these, but some more information (examples, properties) is given in the last section of the paper.