Title: Structure Theorems for Subgroups of Homeomorphisms Groups, in preparation, with Collin Bleak and Martin Kassabov, (slides) Abstract:  We produce structure Theorems for a class of subgroups of the group Homeo_+(S^1) of all orientation-preserving homeomorphisms of the circle: the class of subgroups G with no free subgroups, for instance, solvable groups. The key tool is proving that the rotation number map is a group homomorphism and it is done by relating the dynamics of G and its group structure. Applications include new proofs of known results, such as the Margulis' Theorem on the existence of a G-invariant probability measure on S^1 and Ghys' description of solvable analytic subgroups of Homeo_+(S^1). Back to top.