A Tits alternative for surface group diffeomorphisms and Abelian-by-Cyclic actions on surfaces containing an Anosov diffeomorphism

We obtain three main results about smooth group actions on surfaces. Our first theorem states that if a group of diffeomorphisms of a surface contains an Anosov diffeomorphism then the group contains a free subgroup or preserves one of the stable or unstable foliations up to finite index. We consider this result as a version of Tits alternative for diffeomorphism group. This theorem combined with various techniques including properties of Misiurewicz-Ziemian rotation sets, Herman-Yoccoz Theory of circle diffeomorphisms and the Ledrappier-Young entropy formula, etc, give us our second theorem, which is a global rigidity result about Abelian-by-Cyclic group actions on surfaces in the presence of an Anosov diffeomorphism. This gives a complete classification of Abelian-by-Cyclic group actions on the two-torus up to topological conjugacy and up to finite covers. Furthermore, the group structure combined with the theory of SRB measures and measures of maximal entropy yields the third main result: for a full measure set of rotation vectors, we get a complete classification up to smooth conjugacy.