$2$-groups behaving as automorphism groups of regular $3$-polytopes

TL;DR

This paper classifies certain regular polytopes with automorphism groups of order a power of two, focusing on specific Schl"afli types for large n, thereby addressing a problem posed by Schulte and Weiss.

Contribution

It provides a classification of regular polytopes with automorphism groups of order 2^n for specific Schl"afli types, advancing understanding in polytope symmetry groups.

Findings

Classified regular polytopes with automorphism groups of order 2^n

Focused on Schl"afli types {4, 2^{n-3}}, {4, 2^{n-4}}, {4, 2^{n-5}}

Addressed a problem posed by Schulte and Weiss

Abstract

In this paper, we classify regular polytopes with automorphism groups of order $2^n$ and Schl\"afli types $\{4, 2^{n-3}\}, \{4, 2^{n-4}\}$ and $\{4, 2^{n-5}\}$ for $n \geq 10$, therefore giving a partial answer to a problem proposed by Schulte and Weiss in [Problems on polytopes, their groups, and realizations, Periodica Math. Hungarica 53(2006) 231-255].