Two infinite families of chiral polytopes of type \{4,4,4\} with solvable automorphism groups

TL;DR

This paper constructs two infinite families of chiral polytopes of type {4,4,4} with solvable automorphism groups, providing new examples and partial answers to an open problem in polytope theory.

Contribution

The authors introduce two infinite families of chiral polytopes with solvable automorphism groups of specific orders, advancing understanding of polytope automorphism group structures.

Findings

Automorphism groups are solvable and have orders 1024m^2 and 2048m^2.

Existence of polytopes with automorphism groups of order 2^n for n > 9 when m is a power of 2.

No chiral polytopes of type [4,4,4] exist for n ≤ 9.

Abstract

We construct two infinite families of locally toroidal chiral polytopes of type $\{4,4,4\}$, with $1024m^2$ and $2048m^2$ 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 $n$ can be any integer greater than $9$. (On the other hand, no chiral polytopes of type $[4,4,4]$ exist for $n \leq 9$.) In particular, our two families give a partial answer to a problem proposed by Schulte and Weiss in [Problems on polytopes, their groups, and realizations, {\em Periodica Math.\ Hungarica\} 53 (2006), 231-255].