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.

This paper attempts to relate some ideas of Grothendieck in his Esquisse d'un programme and some of the recent results on 2-dimensional topology and geometry. Especially, we shall discuss the Teichmller theory, the mapping class groups, $ SL (2,\bold C) $ representation variety of surface groups, and Thurston's theory of measured laminations.

We discuss some applications of an intrinsic multipication in the space of simple loops in a surface.