Chirality and projective linear groups

Abstract

In recent years the term ‘chiral’ has been used for geometric and combinatorial figures which are symmetrical by rotation but not by reflection. The correspondence of groups and polytopes is used to construct infinite series of chiral and regular polytopes whose facets or vertex-figures are chiral or regular toroidal maps. In particular, the groups PSL 2( Z m) are used to construct chiral polytopes, while PSL 2( Z m[ i]) and PSL 2( Z m[ω]) are used to construct regular polytopes.