Constructions of k-orbit abstract polytopes

Abstract

The desire to derive new polytopes from old polytopes dates back to the classical study of polytopes, as many of the Archimedian solids can be obtained from Platonic solids through the act of truncation. In this dissertation, we apply these ideas to the setting of abstract polytopes. We present a number of constructions of abstract polytopes, which will generally share some properties with the polytopes from which they were derived. Most notably, we are interested in the circumstances under which the automorphism group of the derived polytope is isomorphic to the automorphism group of the original polytope. We construct polytopes called k-bubbles which generalize truncated polytopes, and which will generally have k flag-orbits and retain the automorphism group of the polytopes from which they are derived. Additionally, among the polytopes that we will construct will be examples of two-orbit polytopes, as well as semiregular polytopes, which we can construct given a preassigned automorphism group. We will also construct polytopes with k flag-orbits, for arbitrary k, with a preassigned automorphism group. Finally, we focus on k-orbit polytopes whose automorphism groups are certain quotients of Coxeter groups called string C-groups. We will prove that there almost always exists a k-orbit polytope whose automorphism group is a given string C-group. In particular, we will prove that in every odd rank there is exactly one counterexample.