We give a characterization of the action of the mapping class group on Thurston's space of measured laminations.

The goal of this chapter is an attempt to relate some ideas of Grothendieck in his Esquisse dun programme [10] and some of the recent results on 2-dimensional topology and geometry. Especially, we shall discuss Teichmller theory, the mapping class groups, the SL (2, C) representation variety of surface groups, and Thurstons theory of measured laminations.

We consider a volume maximization program to construct hyperbolic structures on triangulated 3-manifolds, for which previous progress has lead to consider angle assignments which do not correspond to a hyperbolic metric on each simplex. We show that critical points of the generalized volume are associated to geometric structures modeled on the extended hyperbolic space-the natural extension of hyperbolic space by the de Sitter space-except for the degenerate case where all simplices are Euclidean in a generalized sense.