Let Homeo(S^1) represent the full group of homeomorphisms of the unit circle
S^1, and let A represent the set of subgroups of Homeo(S^1) satisfying the two
properties that if G is in A then (1) G contains only orientation-preserving
homeomorphisms of S^1 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 in A the circle S^1 admits a
G-invariant probability measure.