Abelian and nondiscrete convergence groups on the circle

A group G of homeomorphisms of the unit circle onto itself is a convergence group if every sequence of elements of G contains a subsequence, say g_n, such that either (i) g_n → g and g;;1 → g-I uniformly on the circle where g is a homeomorphism, or (ii) g_n → Xo and g;; 1 → Y₀ uniformly on compact infsets of the complements of (Y₀) and (x_o), respectively, for some points X_o and Y₀ of the circle (possibly X_o = Y₀). For example, a group of K-quasisymmetric maps, for a fixed K, is a convergence group. We show that if G is an abelian or nondiscrete convergence group, then there is a homeomorphism I such that loGo I-I is a group of Mobius transformation.