On the rank two geometries of the groups PSL(2, q): part I

Abstract

We determine all firm and residually connected rank 2 geometries on which PSL(2,q) acts flag-transitively, residually weakly primitively and locally two-transitively, where one of the maximal parabolic subgroups is isomorphic to Eq : (q − 1)/(2, q − 1), where E_q denotes an elementary abelian group of order q, or D_2n(q), the dihedral group of order 2n(q) where n(q) := (q ± 1)/gcd(2, q − 1) for some prime-power q.