Global rigidity of solvable group actions on S¹

In this paper we find all solvable subgroups of Diff ^ω(S¹) and classify their actions. We also investigate the C^r local rigidity of actions of the solvable Baumslag-Solitar groups on the circle. The investigation leads to two novel phenomena in the study of infinite group actions on compact manifolds. We exhibit a finitely generated group Γ and a manifold M such that: • has exactly countably innitely many effective real-analytic actions on M, up to conjugacy in Diff^ω(M); • every effective, real analytic action of Γ on M is C^r locally rigid, for some r ≥ 3, and for every such r, there are infinitely many nonconjugate, effective real-analytic actions of Γ on M that are C^r locally rigid, but not C^r-1 locally rigid.