The residually weakly primitive and locally two-transitive rank two geometries for the groups PSL(2, q)

of rank two on which some group PSL(2,q), q a prime power, acts flag-transitively.

Actually we require that the action be RWPRI (residually weakly primitive) and (2T)1

(doubly transitive on every residue of rank one). In fact our definition of RWPRI requires

the geometry to be firm (each residue of rank one has at least two elements) and RC

(residually connected).

The main goal is achieved in this thesis.

It is stated in our "Main Theorem". The proof of this theorem requires more than 60pages.

Quite surprisingly, our proof in the direction of the main goal uses essentially the classification

of all subgroups of PSL(2,q), a famous result provided in Dickson’s book "Linear groups: With an exposition of the Galois field theory", section 260, in which the group is called Linear Fractional Group LF(n, pn).

Our proof requires to work with all ordered pairs of subgroups up to conjugacy.

The restrictions such as RWPRI and (2T)1 allow for a complete analysis.

The geometries obtained in our "Main Theorem" are bipartite graphs; and also locally 2-arc-transitive

graphs in the sense of Giudici, Li and Cheryl Praeger. These graphs are interesting in their own right because of

the numerous connections they have with other fields of mathematics.