Structure theorems for subgroups of homeomorphism groups

Let Homeo(S1) represent the full group of homeomorphisms of the unit circle S1, and let A represent the set of subgroups of Homeo(S1) satisfying the two properties that if G ? A then 1) G contains only orientation preserving homeomorphisms of S1 and 2) G contains no non-abelian free subgroups. This expository article uses 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 within a family of such results by Beklaryan, Malyutin, and Solodov, a new proof of Margulis’ Theorem that given G ? A the circle S1 admits a G-invariant probability measure, and we classify the solvable subgroups of R. Thompson’s group T