Almost simple groups with socle PSL(2,q) acting on abstract regular polytopes

Our paper deals with the classification of abstract regular polytopes for almost simple groups with socle PSL(2,q). We consider all almost simple groups PSL(2,q)≤G≤PΓL(2,q) and determine the maximal rank of string C-group representations for G, i.e. the maximal rank of an abstract regular polytope with automorphism group G, as well as the existence of string C-group representations of lower ranks. Similar results have already been obtained by various authors in the cases G≅PSL(2,q) and G≅PGL(2,q) and they are summarized in this paper.