Regular polygonal complexes in space, I
Daniel Pellicer, Egon Schulte · Transactions of the American Mathematical Society · 2010
A polygonal complex in Euclidean $3$-space $\mathbb {E}^3$ is a discrete poly- hedron-like structure with finite or infinite polygons as faces and finite graphs as vertex-figures, such that a fixed number $r\geqslant 2$ 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 $\mathbb {E}^3$. 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 $\mathbb {E}^3$, and fully enumerates the simply flag-transitive complexes with mirror vector $(1,2)$. The second paper will complete the enumeration.