The Tits alternative problem for real analytic diffeomorphism groups

I discuss the Tits alternative problem for the group of real analytic diffeomorphisms.
I will provide a proof for a weak version of this problem in dimension one (the circle, the real line
intervals and germs). More precisely, I prove a finitely generated subgroup of the group of real analytic diffeomorphisms of the mentioned unidimensional manifolds either is virtually solvable or contains a non abelian free Semigroup. As an immediately consequence of it, one can obtain such a group has not intermediate growth. It is a joint work with Meysam Nassiri.