All polytopes are coset geometries: characterizing automorphism groups of k-orbit abstract polytopes

TL;DR

This paper proves that all abstract polytopes can be represented as coset geometries by characterizing their automorphism groups, extending known results to all k-orbit polytopes and revealing new symmetry properties.

Contribution

It provides a general construction and characterization of automorphism groups for all k-orbit abstract polytopes, broadening the understanding of their symmetry structures.

Findings

Every abstract n-polytope can be constructed as a coset geometry.

Existence of k-orbit n-polytopes with Boolean automorphism groups for all n≥3 and k≠2.

Abstract

Abstract polytopes generalize the classical notion of convex polytopes to more general combinatorial structures. The most studied ones are regular and chiral polytopes, as it is well-known, they can be constructed as coset geometries from their automorphism groups. This is also known to be true for 2- and 3- orbit 3-polytopes. In this paper we show that every abstract $n$-polytope can be constructed as a coset geometry. This construction is done by giving a characterization, in terms of generators, relations and intersection conditions, of the automorphism group of a $k$-orbit polytope with given symmetry type graph. Furthermore, we use these results to show that for all $k\neq 2$, there exist $k$-orbit $n$-polytopes with Boolean automorphism groups, for all $n\geq 3$.