Structure theorems for subgroups of homeomorphisms groups

Let Homeo(S) represent the full group of homeomorphisms of the unit circle S, and let A represent the set of subgroups of Homeo(S) satisfying the two properties that if G ∈ A then 1) G contains only orientationpreserving homeomorphisms of S and 2) G contains no non-abelian free subgroups. In this article we use classical results about homeomorphisms of the circle and elementary dynamical methods to derive various new and old results about the groups in A; we give a general structure theorem for such groups, a classification of the solvable subgroups of R. Thompson’s group T , and a new proof of Margulis’ Theorem that given G ∈ A the circle S admits a G-invariant probability measure.