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.

Abstract. A polygonal complex in Euclidean 3-space E 3 is a discrete polyhedron-like structure with finite or infinite polygons as faces and finite graphs as vertex-figures, such that a fixed number r 2 of faces surround each edge. It is said to be regular if its symmetry group is transitive on the flags. The present paper and its successor describe a complete classification of regular polygonal complexes in E 3 . In particular, the present paper establishes basic structure results for the symmetry groups, discusses geometric and algebraic aspects of operations on their generators, characterizes the complexes with face mirrors as the 2-skeletons of the regular 4-apeirotopes in E 3 , and fully enumerates the simply flag-transitive complexes with mirror vector (1, 2). The second paper will complete the enumeration.

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