On chiral polytopes having a group PSL(3,q) as automorphism group

Abstract: For each prime power $q\geq 5$, we construct a rank four chiral polytope that has a group $PSL(3,q)$ as automorphism group and Schl\"afli type ${q-1,\frac{2(q-1)}{(3,q-1)},q-1}$. We also construct rank five polytopes for some values of $q$ and we show that there is no chiral polytope of rank at least six having a group $PSL(3,q)$ or $PSU(3,q)$ as automorphism group.