A Class of Automorphism Groups of Polygonal Complexes

In this paper we construct a class of groups together with their actions on certain spaces called polygonal complexes. We give a complete description of the class of all groups which act by automorphisms simply transitively on the oriented edges of non‐positively curved polygonal complexes. This family of group actions is very natural from a geometrical viewpoint, being a kind of higher‐dimensional analogue of the actions of groups on their Cayley graphs (which are simply transitive on vertices). The construction is based on the notion of a non‐positively curved orbihedron and its fundamental group and generalizes the ideas of Ballmann and the author [2].