Triple factorisations of the general linear group and their associated geometries

Triple factorisations of finite groups G   of the form G=PQPG=PQP are essential in the study of Lie theory as well as in geometry. Geometrically, each triple factorisation G=PQPG=PQP corresponds to a G  -flag transitive point/line geometry such that ‘each pair of points is incident with at least one line’. We call such a geometry collinearly complete, and duality (interchanging the roles of points and lines) gives rise to the notion of concurrently complete geometries. In this paper, we study triple factorisations of the general linear group GL(V)GL(V) as PQP where the subgroups P and Q either fix a subspace or fix a decomposition of V   as V1⊕V2V1⊕V2 with dim⁡(V1)=dim⁡(V2)dim⁡(V1)=dim⁡(V2).