Regular Polygonal Complexes in Space, I

A polygonal complex in euclidean 3-space E3 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 E3. 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 E3, and fully enumerates the simply flag-transitive complexes with mirror vector (1, 2). The second paper will complete the enumeration.