Chiral polyhedra from AGL(1, q)

An abstract polyhedron is a partially ordered set characterized by four properties which altogether generalize the face lattice of a Euclidean polyhedron. Abstract polyhedra (with finite faces) are maps on closed surfaces, and most maps on surfaces are abstract polyhedra. The \emph{regular} abstract polyhedra (that is, those with a flag-transitive automorphism group action, hence maximal "rotational" and "reflective" symmetry) in particular have received the most attention. Chiral polyhedra, on the other hand, possessing maximum rotational symmetry without reflective symmetry, are trickier to construct, and their groups are evidently rarer among groups of a given order. Unlike regular polyhedra (which are isomorphic to their mirror images), chiral polyhedra always arise in mirror–image pairs called "enantiomorphs." In this talk, we present Angelone & Schulte's construction of chiral polyhedra whose automorphism group is a subgroup of the general affine group over a field. We also show that our construction is the only way that chiral polyhedra can be derived from the affine group, and that no chiral ddd-polytopes can be constructed from the affine group for ranks d≥4d \geq 4d≥4. The polyhedra arising from this construction end up being minimal under the covering relationship between chiral polyhedra, and we leverage this fact to show that all groups of chiral polyhedra of a certain type are extensions of this affine-group construction.