Chiral Polyhedra from AGL(1,q)

TL;DR

This paper constructs chiral and regular polyhedra using subgroups of the affine group AGL(1,q), identifying specific automorphism groups and their properties for odd prime powers q.

Contribution

It provides a new method to construct chiral polyhedra from affine groups and classifies which groups can serve as automorphism groups of certain polytopes.

Findings

AGL(1,q) can be the automorphism group of a chiral polyhedron of type {q-1, q-1} for q=1 mod 4

Subgroups of AGL(1,q) cannot be automorphism groups of higher-rank regular or chiral polytopes

The construction captures all polytopes arising from this class of affine groups

Abstract

We present a construction of chiral and regular polyhedra from subgroups of the general affine group AGL(1,q) for odd prime powers q. In particular, we show that the full group AGL(1,q) occurs as the automorphism group of a chiral polyhedron of type {q-1, q-1} when q=1 mod 4, or types {q-1,(q-1)/2} or {(q-1)/2, q-1} when q=3 mod 4, and we compute the genus in each case. We also establish that subgroups of AGL(1,q) cannot serve as full automorphism groups of regular polytopes of rank 3 or higher, nor of chiral polytopes of rank 4 or higher, demonstrating that our construction captures all polytopes that can arise from this class of affine groups.