  We give a way to construct group of pseudo-automorphisms of rational
varieties of any dimension that fix pointwise the image of a cubic hypersurface
of $P^n. These group are free products of involutions, and most of their
elements have dynamical degree >1. Moreover, the Picard group of the varieties
obtained is not big, if the dimension is at least 3.
  We also answer a question of E. Bedford on the existence of birational maps
of the plane that cannot be lifted to automorphisms of dynamical degree >1,
even if we compose them with an automorphism of the plane.
