A cellular parametrization for closed surfaces with a distinguished point.

The convex hull construction in Minkowski space is used here to parametrize Dirichlet fundamental polygons by L -lengths of the edges of the corresponding convex hull. The basic conditions have simple geometric interpretations by projection to the Poincare model, and the presentation is done also in terms of concepts in the Poincare model. The parametrization leads to a cell-decomposition of conformal structures of closed surfaces of genus g with a distinguished point. A connection between the entries of the matrix of a Mobius transformation and the corresponding L -length, with distinguished point at the origin, is obtained. A necessary and sufficient condition for discreteness is obtained in terms of the matrices of the generators of the group.