Noncompact surfaces, triangulations and rigidity

It is shown that every noncompact surface has a (3,6)-tight triangulation, and a characterisation of such triangulations is given in terms of length constraints on the boundary walks of superfaces of a given genus. Applications are given to the generic rigidity of associated countable bar-joint frameworks in R^3. In particular, every noncompact surface has a (3,6)-tight triangulation that is minimally 3-rigid. A proof of Ker\'ekj\'art\'o's classification of noncompact surfaces is also given.