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

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 $n\geq 3$.