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

Isabel Hubard, Elías Mochán

Published 2022-08-01Version 1

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-know 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 groups (elementary abelian 2-groups) as automorphism group, for all $n\geq 3$.