Abstract
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).
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