Four infinite families of chiral $3$-polytopes of type $\{4, 8\}$ with solvable automorphism groups

TL;DR

This paper constructs four infinite families of chiral 3-polytopes of type {4,8} with solvable automorphism groups, providing new examples with automorphism groups of order 2^n for n ≥ 10, partially answering open problems.

Contribution

The authors explicitly construct four infinite families of chiral 3-polytopes of type {4,8} with solvable automorphism groups, expanding known examples and addressing open questions in the field.

Findings

Families have automorphism groups of orders 1024m^4, 2048m^4, 4096m^4, 8192m^4.

Existence of polytopes with automorphism groups of order 2^n for n ≥ 10.

No chiral polytopes of type {4,8} exist for n ≤ 9.

Abstract

We construct four infinite families of chiral $3$-polytopes of type $\{4, 8\}$, with $1024m^4$, $2048m^4$, $4096m^4$ and $8192m^4$ automorphisms for every positive integer $m$, respectively. The automorphism groups of these polytopes are solvable groups, and when $m$ is a power of $2$, they provide examples with automorphism groups of order $2^n$ where $n \geq 10$. (On the other hand, no chiral polytopes of type $\{4, 8\}$ exist for $n \leq 9$.) In particular, our families give a partial answer to a problem proposed by Schulte and Weiss in [Problems on polytopes, their groups, and realizations, {\em Period. Math. Hungar.} 53 (2006), 231-255] and a problem proposed by Pellicer in [Developments and open problems on chiral polytopes, {\em Ars Math. Contemp} 5 (2012), 333-354].