Rotation numbers of products of circle homeomorphisms

Abstract

Let H = Homeo § (S 1) be the group of orientation preserving homeomorphisms of the circle. Our main question is the following: for given numbers 71 and Y2, what are the possible rotation numbers of a product qlq2 of elements ql, q2 of H with rotation numbers 71, Y2? What if some or all of ql, q2, qlq2 are required to be conjugate to rotations? Our original motivation was the question of which Seifert fibered 3-manifolds admit transverse foliations, which we discuss in Sect. 7. The answers turn out to be much more subtle than we originally expected. We can make the question more precise by working in the universal covering group/7 of H. This is the group of homeomorphisms of F, which lift from an orientation preserving homeomorphism of S 1 = R / Z , that is / 7= ( Q : R ~ R I Q monotonically increasing, Q(r+ 1) = Q(r) + 1 for all r ~ R}. (/7 is simply connected since it is a convex subset of IR•.) For y ~ ~-. define sh(~) ~ H by