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Chiral Polyhedra from AGL(1,q)

Evan Angelone, Egon Schulte

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.

Chiral Polyhedra from AGL(1,q)

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.
Paper Structure (4 sections, 10 theorems, 48 equations)

This paper contains 4 sections, 10 theorems, 48 equations.

Table of Contents

  1. Introduction
  2. Abstract polyhedra
  3. The group AGL(1, q)
  4. Construction of chiral polyhedra

Key Result

Lemma 3.1

Let $a\in\mathbb{F}_q^*$, $b\in\mathbb{F}_q$. Then, (a) $\alpha(a,b)^{-1} = \alpha(a^{-1},-a^{-1}b)$. (b) $\alpha(1,b)^k = \alpha(1,kb)$ for $k\geq 0$. (c) $\alpha(a,b)^k = \alpha(a^k, \frac{a^k-1}{a-1}b)$ for $a\neq 1$, $k\geq 0$. (d) The order of $\alpha(a,b)$, $a\neq 1$, in $\mathrm{AGL}(1,q)$ co

Theorems & Definitions (22)

  • Lemma 3.1
  • proof
  • Lemma 3.2
  • proof
  • Lemma 3.3
  • proof
  • Lemma 3.4
  • proof
  • Lemma 3.5
  • proof
  • ...and 12 more