On Sobolev infinitesimal rigidity of linear hyperbolic actions on the 2-torus*

Abstract

Let A be a symmetric hyperbolic matrix in SL(2, ℤ) and Γ the subgroup of SL(2, ℤ) generated by A. We aim to study the infinitesimal rigidity of the standard action of Γ on the torus $${\Bbb T^2}$$ . More precisely, we will consider the Sobolev W s –infinitesimal rigidity of this action (that is to determine if the cohomology space H1(Γ,W s (T M)) is trivial or not), and show that it is W s –infinitesimally rigid only if 0 ≤ s < 1. A consequence will be that this action is not C∞–infinitesimally rigid.