Polytopes of large rank for PSL(4,Fq)

This paper examines abstract regular polytopes whose automorphism group is the projective special linear group PSL(4,Fq). For q odd we show that polytopes of rank 4 exist by explicitly constructing PSL(4,Fq) as a string C-group of that rank. On the other hand, we show that no abstract regular polytope exists whose group of automorphisms is PSL(4,F2k).