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 E q : ( q − 1)/(2, q − 1), where E q denotes an elementary abelian group of order q, or D 2 n ( q ) , the dihedral group of order 2 n ( q ) where n ( q ) := ( q ± 1)/gcd(2, q − 1) for some prime-power q .