Locally affine geometries of order 2 where shrinkings are affine expansions

Abstract

Given a locally a affine geometry Γ of order 2 and a flag-transitive subgroup $G ≤ Aut(Γ)$, suppose that the shrinkings of Γ are isomorphic to the a affine expansion of the upper residue of a line of Γ by a homogeneous representation in a 2-group. We shall prove that, under certain hypotheses on the stabilizers G_p and G_l of a point p and a line l, we have G=R{G_p} for a representation group R of $Res(p)$. We also show how to apply this result in the classification of flag-transitive c-extended P- and T-geometries.