TY: THES
T1 - Faithful permutation representations of C-groups
A1 - Piedade, Claudio Alexandre Guerra Silva Gomes da
N2 - 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...
UR - https://ria.ua.pt/handle/10773/33912
Y1 - 2022
PB - No publisher defined