We study incidence geometries that are thin and residually connected. These geometries generalise abstract polytopes. In this generalised setting, guided by the ideas from the polytopes theory, we introduce the concept of chirality, a property of orderly asymmetry occurring frequently in nature as a natural phenomenon. The main result in this paper is that automorphism groups of regular and chiral thin residually connected geometries need to be C-groups in the regular case and C + -groups in the chiral case.
We derive some general results on the symmetries of equivelar toroids and provide detailed analysis of the subgroup lattice structure of the dihedral group D 4 and of the octahedral group to complete classification by symmetry type of those in ranks 3 and 4. Keywords Symmetries of toroids • Map • Polytope • Cubical tessellation 1 Introduction Over the last few decades numerous papers dealt with polytopes and maps that have large automorphism groups but are not necessarily regular (see for example [4, 6, 8, 14, 15]). In particular, a lot of research has been done on chiral polytopes which are
A k-orbit map is a map with k flag-orbits under the action of its automorphism group. We give a basic theory of k-orbit maps and classify them up to k 4. "Hurwitz-like" upper bounds for the cardinality of the automorphism groups of 2-orbit and 3-orbit maps on surfaces are given. Furthermore, we consider effects of operations like medial and truncation on k-orbit maps and use them in classifying 2-orbit and 3-orbit maps on surfaces of small genus.
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