Faithful permutation representations of C-groups - Piedade, Claudio Alexandre Guerra Silva Gomes da (2022)
The study of regular objects, such as polytopes, and their symmetries is a subject that attracts researchers from different areas of mathematics, such as geometers and algebraists, but also researchers from other areas of knowledge such as chemistry, thanks to the high symmetry of the molecules. An abstract polytope is a structure that combinatoricaly describes a classical polytope (a generalization of polygons and polyhedra to higher dimensions). Abstract regular polytopes can be described as a poset, as an incidence geometry or as C-group with linear diagram. A hypertope was introduced as a polytope-like structure where its group of symmetries is a C-group however it does not need to have a linear diagram. Grünbaum’s problem, one of the classical problems of the theory of abstract polytopes, not yet completely solved, consists in the classification of locally toroidal polytopes. The problem is extensible to hypertopes of rank 4 with toroidal rank 3 residues. Locally toroidal hypertopes are constructed from toroidal regular hypermaps {4, 4}, {6, 3}, {3, 6} or (3, 3, 3). The groups of these toroidal regular hypermaps can be represented as faithful transitive permutation representation graphs, which can be then used either to classify locally toroidal polytopes or to construct new polytopes/hypertopes with toroidal residues. In this thesis, a classification of all the possible degrees of faithful transitive permutation representations of the toroidal regular hypermaps and of the locally toroidal regular polytopes of type {4, 4, 4} is given...