This paper examines abstract regular polytopes whose automorphism group is the projective special linear group PSL(4,q). For q odd we show that polytopes of rank 4 exist by explicitly constructing PSL(4,q) 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(4,2k).
Journal of Algebra
Brooksbank, Peter A. and Leemans, Dimitri. "Polytopes of Large Rank for PSL(4,q)." Journal of Algebra (2016) : 390-400.