Symmetric polytopes whose automorphism groups are 2-groups

TL;DR

This paper explores polytopes with automorphism groups that are 2-groups, revealing their structure, variety, and hereditary properties under universal extensions across different ranks.

Contribution

It characterizes the structure and variety of polytopes with 2-group automorphism groups and proves the hereditary nature of these groups under universal extensions.

Findings

Existence of small regular and semiregular polytopes with 2-group automorphism groups

Prescribing sections of rank 3 in toroidal polytopes with 2-group automorphism groups

Hereditary property of 2-groups under universal extensions to higher ranks

Abstract

The present work investigates regular, semiregular, and chiral polytopes of any rank $d\geq 3$, whose automorphism groups are 2-groups. There is a large variety of rather small finite regular or alternating semiregular polytopes with automorphism groups of 2-power order: for such polytopes with toroidal sections of rank 3, the various sections of rank 3 can be entirely prescribed (possibly with one exception in the semiregular case). It is also shown that having a 2-group as automorphism group is hereditary under taking universal extensions: the universal extension of a given regular, chiral, or alternating semiregular polytope with a finite or infinite 2-group as automorphism group, is a polytope of one rank higher with an infinite 2-group an automorphism group.