Yes you are right, I'll correct that.

And by the way, does anyone know a free set of generators of $D\pi_1(S)$ ?

Andy answered the right question, I've just edited my post this morning :)

Thanks for the answer. Actually I am realizing I'm interested in the action of $G$ on $G^1/G^2$ !

The trivial action of $SL(2,\mathbb{Z})$ on $\mathbb{R}$ is the action by linear fractional transformations ?

By linear transformation. Namely a matrix $ \left ( \begin{array}{cc} a & b \\ c & d \end{array} \right ) $ acts by $ \left ( \begin{array}{cc} a & b \\ c & d \end{array} \right ) \cdot \left ( \begin{array}{c} u \\ v \end{array} \right ) = \left ( \begin{array}{c} au + bv \\ cu + dv \end{array} \right ) $. I thought it was obvious :)

Igor : I am mainly interested in propetries of closed manifolds

Misha : You are right, my question is not specific enough. In the jungle of differentiable manifolds, let's try to locate those which can carry a riemanniann metric of negative sectionnal curvature. It seems that this condition is very restrictive. I would like to understand how much it is. So I think the relevant question is : what are the topological properties which a manifold of variable curvature must have, and in a second time what are the topological properties manifolds of constant curvature have and variable curvature might not have.