Noncompact surfaces, triangulations and rigidity

Abstract

Every noncompact surface is shown to have a (3,6)-tight triangulation, and applications are given to the generic rigidity of countable bar-joint frameworks in $R^3$. In particular, every noncompact surface has a (3,6)-tight triangulation that is minimally 3-rigid. A simplification of Richards' proof of Kerékjártó's classification of noncompact surfaces is also given.