Internal and External Invariance of Abstract Polytopes A dissertation presented by Gabe Cunningham to The Department of Mathematics In partial fulfillment of the requirements for the degree of Doctor of Philosophy in the field of Mathematics Northeastern University Boston, Massachusetts May, 2012 1 Internal and External Invariance of Abstract Polytopes by Gabe Cunningham ABSTRACT OF DISSERTATION Submitted in partial fulfillment of the requirements for the degree of Doctor of Philosophy in Mathematics in the Graduate School of Science of Northeastern University, May 2012 2 Abstract In addition to the usual symmetries by reflections and rotations, abstract polytopes can have external symmetries such as self-duality and self-Petriality. In this dissertation, we describe and study a way of measuring the invariance of abstract polytopes under such external operations. We then present methods for constructing abstract polytopes with specified external symmetries. In particular, we describe how to construct polyhedra that are self-dual and self-Petrie, and how to construct polytopes that are self-dual and chiral. 3 Acknowledgements I would like to thank the members of my committee for their time and attention. I would like to thank Professor Schulte for all of the help he has given me over the years. He has helped proofread my papers, given me many interesting articles to read, and invited me to rewarding seminars and conferences. He has been instrumental in getting my career started. Finally, I would like to thank my wife, Nomeda. She inspires me to work hard, not only by her words, but by her example. Furthermore, if it hadn’t been for her urging, there is a good chance I would never have even applied to graduate school and found out how rewarding it could be. 4 Contents Abstract 2 Acknowledgements 4 Contents 5 1 Background 7 1.1 Introduction . 7 1.2 Abstract polytopes . 9 1.3 Regularity . 17 1.4 Direct regularity . 28 1.5 Chirality . 30 1.6 Duality . 34 1.7 Petrie duality . 36 2 The mixing operation 39 2.1 Mixing finitely generated groups . 39 5 2.2 Variance groups and the structure of the mix . 45 2.3 Mixing polytopes . 51 2.4 Polytopality of the mix . 55 3 Measuring invariance 66 3.1 External and internal invariance . 66 3.2 Variance of the mix . 71 4 Constructing self-dual and self-Petrie polytopes 80 4.1 Self-dual polytopes . 81 4.2 Self-dual, self-Petrie polyhedra . 83 4.3 Self-dual, self-Petrie covers of universal polyhedra . 87 5 Constructing chiral polytopes 92 5.1 Building chiral polytopes of high rank . 92 5.2 Self-dual chiral polytopes . 96 6 Appendix 103 Bibliography 105 6 Chapter 1 Background 1.1 Introduction The study of abstract polytopes is a growing field, uniting combinatorics with geometry and group theory. Abstract polytopes are combinatorial structures with a distinctly geometric flavor, generalizing the face-lattices of convex polytopes. The result is a rich theory that encompasses not only polytopes, but also face-to-face tilings of the plane, tessellations of space-forms, and other fascinating structures. As in the classical theory, the abstract polytopes that have been most extensively stud- ies are the regular polytopes: those whose automorphism group acts transitively on the set of their flags. These polytopes possess maximal symmetry by abstract reflections. Closely related are the chiral polytopes, which possess two flag-orbits under the action of the auto- morphism group. Intuitively, a chiral polytope has full rotational symmetry, but it occurs in two mirror-image forms. Compared to regular polytopes, relatively little is known about chiral polytopes. In particular, we know very few examples of chiral polytopes of high rank (dimension). 7 In addition to the study of polytopes with a high degree of internal symmetry – that is, polytopes with a large number of polytope automorphisms – the study of external symmetries of polytopes and maps on surfaces has been gaining attention; for example, see [15, 32]. The idea is to consider operations that transform one polytope into another, and to then look for (or construct) polytopes that are invariant under this transformation. The classic example is self-duality, but there are many others. In this dissertation, we focus on external symmetries arising from certain outer auto- morphisms of groups. In particular, we consider outer automorphisms of the automorphism group of the universal polytope of rank n, a polytope that covers all others of the same rank. There are three principal external symmetries this gives rise to: duality, Petrie duality, and mirror symmetry. Our goal is then to measure to what extent a polytope is invariant under these symmetries. Our main tool for constructing new polytopes is the mixing construction, which creates the minimal common cover of two polytopes [24]. The mix of a polytope with its dual is self-dual, and in general, the mix of a polytope with all of its images under a group of transformations is invariant under that group of transformations. The result, however, is not always a polytope; we must check that separately. We start by giving background on (abstract) polytopes, including regularity, chirality, duality, and Petrie duality, in Chapter 1. In Chapter 2, we describe the mixing construction for groups and for polytopes, and we examine in detail the structure of the mix of two polytopes. In particular, we describe several circumstances that guarantee that the mix of two polytopes will again be a polytope. Next, we describe how to use mixing to measure the invariance of a polytope under external symmetries in Chapter 3. In Chapter 4, we focus on duality and Petrie duality, and we explicitly describe many self-dual, self-Petrie polyhedra. Finally, in Chapter 5, we construct chiral polytopes that are self-dual – in other 8 words, self-dual polytopes that lack mirror symmetry. 1.2 Abstract polytopes Abstract polytopes are essentially generalizations of the face-lattice of a convex polytope, and they retain much of the geometric flavor of convex polytopes. As we build up the definition, given in [23], we will give some geometric justification for the properties we require. Some of our later constructions produce objects that are not quite polytopes, and so we start with these more general definitions. Definition 1.1. A flagged poset of rank n is a partially ordered set P such that there is a unique maximal element, a unique minimal element, and such that each maximal chain of P has n + 2 elements. We use Fn to denote the maximal element, and F−1 to denote the minimal element. We call the elements of a flagged poset faces, and we call the maximal chains flags. Since each flag of a flagged poset contains the same number of faces, every flagged poset has a natural rank function. In particular, we define the rank of F−1 to be −1 and the rank of Fn to be n, with the rank of any other face obtained in the natural way from its position in any flag. A face of rank j is called a j-face. The faces F−1 and Fn are called the improper faces of P; the remaining faces are the proper faces. If F and G are faces of a flagged poset P such that F ≤ G, then we define the section G/F to be the poset consisting of those faces H such that F ≤ H ≤ G. In fact, if F is a j-face and G is a k-face, then the section G/F is a flagged poset of rank k − j − 1, where G plays the role of the maximal face, and F plays the role of the minimal face. We think of the rank of a face as its dimension, informally speaking. Thus, in analogy with convex polytopes, we refer to 0-faces as vertices and 1-faces as edges. Faces of rank 9 n−1 are called facets. The requirement that a flagged poset of rank n must contain a unique n-face and a unique (−1)-face is analogous to the fact that a convex n-polytope is viewed to have a single face of dimension n (namely, the whole polytope) and a single face of dimension −1 (namely, the empty set). There is a natural identification between each j-face F and its section F/F−1, and we occasionally blur the distinction between these two concepts. If F is a k-face, then the co- k-face at F (or sometimes simply the co-face at F ) is the section Fn/F . In particular, if F is a vertex, then we call the co-face at F a vertex-figure. If G is a facet and F is a vertex, then the section G/F is called a medial section. Note that a medial section is a facet of the vertex-figure Fn/F and also a vertex-figure of the facet G/F−1. The requirement that all flags of a flagged poset have the same length ensures that we get no “holes” and that all maximal proper faces have the same rank. Essentially, we are forcing the set of proper faces to form a polytopal complex [13] where the maximal polytopes have the same dimension (except, of course, our polytopes will be abstract instead of convex). We note that there is one flagged poset of rank −1 (where the minimal and maximal element coincide), and one flagged poset of rank 0 (where the only two elements are the minimal and maximal elements). In rank 1, there are already infinitely many flagged posets. Flagged posets are really too general to have much geometric flavor.