On chiral polytopes having a group PSL(3,q)$\mathrm{PSL}(3,q)$ as automorphism group

Abstract

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