On nilpotent groups of real analytic diffeomorphisms of the torus

Let Diff ω 0 ( T 2 ) denote the connected component of the identity of the group of real analytic diffeomorphisms of the torus of dimension 2 . We consider nilpotent subgroups Diff ω 0 ( T 2 ) and we show that they are metabelian. This implies in particular that any homomorphism from a subgroup of finite index of SL (n, Z ) (n≥4) to Diff ω 0 ( T 2 ) has a finite image.