Some combinatorial aspects related to low dimensional manifolds via crystal patterns

Combinatorial features on manifolds are first seen in the description of crystallographic groups. We use this connection to describe discretizations for a class of manifolds in light of possible applications to naturally occurring surfaces. In this paper we consider tessellations on smooth orientable manifolds arising out of certain group actions. The goal is to find appropriate notions in the language of discrete structures to describe the underlying geometry. As a primary motivation a continuous family of lattices on Riemann surfaces is considered. Discrete groups are seen as the central objects in describing various discrete structures on manifolds. Further certain applications have been described.