On \(Af.Af^*\) geometries (Q1904251)

This paper initiates the determination of the flag-transitive rank 3 geometries of type \(L.L^*\) (Buekenhout geometries with a linear diagram and the point-residues are dual linear spaces, the plane-residues are linear spaces). If the (dual) linear spaces are (dual) affine planes, then we have a geometry of type \(Af.Af^*\). It is this special case that has been considered in the paper under review. The authors classify for instance the flag-transitive geometries of type \(Af.Af^*\) of diameter 1. Other partial results are obtained. It is worth mentioning that a lot of properties in this paper do not depend on the flag-transitivity. In that respect, the paper may also be of geometrical and combinatorial interest.