Non-commutative groups as prescribed polytopal symmetries

TL;DR

This paper explores how non-commutative groups can be realized as symmetries of convex polytopes, establishing existence results, limitations, and bounds on the dimensions needed for such realizations.

Contribution

It demonstrates that non-abelian groups with a central involution can be realized as automorphism groups of centrally symmetric polytopes, and provides bounds on the dimensions required for such realizations.

Findings

Non-abelian groups with a central involution can be realized as automorphism groups of centrally symmetric polytopes.

Certain groups cannot be realized as automorphism groups of convex polytopes in low dimensions.

An optimal lower bound for the dimension of group realizations as isometry groups of polytopes.

Abstract

We study properties of the realizations of groups as the combinatorial automorphism group of a convex polytope. We show that for any non-abelian group $G$ with a central involution there is a centrally symmetric polytope with $G$ as its combinatorial automorphisms. We show that for each integer $n$, there are groups that cannot be realized as the combinatorial automorphisms of convex polytopes of dimension at most $n$. We also give an optimal lower bound for the dimension of the realization of a group as the group of isometries that preserves a convex polytope.