Eisenstein integers and related \(C\)-groups (Q1362562)

An abstract polytope \({\mathcal P}\) is called regular if its automorphism group \(\Gamma({\mathcal P})\) is flag-transitive, or chiral if \(\Gamma({\mathcal P})\) has exactly two orbits on the flags such that adjacent flags are in distinct orbits. The authors employ quotients of linear groups over the Eisenstein integers \(\mathbb{Z}[\omega]\) to construct infinite families of finite regular or chiral 4-polytopes of types \(\{3,3,6\}\), \(\{3,6,3\}\) and \(\{6,3,6\}\). These polytopes are locally toroidal, in the sense that their facets and vertex-figures are spherical or toroidal polyhedra, with at least one of the latter kind.