From geometry to groups and back: The study of highly symmetric polytopes

This dissertation deals with highly symmetric abstract polytopes. Abstract polytopes are combinatorial structures that generalize the classical notion of convex polytopes. Particular attention is given to the polytopes whose automorphism groups (or groups of symmetries) have exactly two orbits on the set of flags. We classify the two-orbit polytopes of rank n and find a set of generators for the automorphism groups of these polytopes. We give a characterization of two-orbit and fully-transitive polyhedra in terms of its automorphism group; that is, we find necessary and sufficient conditions for a group to be the automorphism group of a two-orbit or fully-transitive polyhedron. We also consider a different kind of "symmetry" of certain abstract polytopes. We generalize the concept of a self-dual n-polytope to that of a self-invariant polytope with respect to d, where d is an automorphism of the Coxeter group C = [infinity, ..., infinity] of rank n. As an application, we study self-dual two-orbit and fully-transitive polyhedra. Using a twisting operation on the extended group of a self-dual chiral 4-polytope, we obtain chiral quotients of the Petrie-Coxeter polyhedra. We reinterpret the classical definition of the medial of a map in order to define the medial of a polyhedron and classify regular and two-orbit medial polyhedra. We give examples of finite chiral polytopes of rank 5, by using the software MAGMA to find normal subgroups of the rotational subgroup of the universal Coxeter group C = [infinity, ..., infinity] of rank 5, that are not normal in C.