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 generalise work of Young-Eun Choi to the setting of ideal triangulations with vertex links of arbitrary genus, showing that the set of all (possibly incomplete) hyperbolic cone-manifold structures realised by positively oriented hyperbolic ideal tetrahedra on a given topological ideal triangulation and with prescribed cone angles at all edges is (if non-empty) a smooth complex manifold of dimension the sum of the genera of the vertex links. Moreover, we show that the complex lengths of a collection of peripheral elements give a local holomorphic parameterisation of this manifold.

Several identities similar to the Schlaefli formula are established for tetrahedra in a space of constant curvature.

We give a brief introduction to some of the recent works on finding geometric structures on triangulated surfaces using variational principles.

We show that solutions of Thurston's equation on triangulated 3-manifolds in a commutative ring carry topological information. We also introduce a homogeneous Thurston's equation and a commutative ring associated to triangulated 3-manifolds. It is shown that solutions to homogeneous equations over the real numbers are critical points of an entropy function.