Global rigidity of solvable group actions on S^1

Abstract

In this paper we find all solvable subgroups of Diff^omega(S^1) 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 Gamma and a manifold M such that: * Gamma has exactly countably infinitely many effective real-analytic actions on M, up to conjugacy in Diff^omega(M); * every effective, real analytic action of Gamma 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 Gamma on M that are C^r locally rigid, but not C^(r-1) locally rigid.