Abelian and nondiscrete convergence groups on the circle

Abstract:

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 ${{\text {g}}_n}$, such that either (i) ${{\text {g}}_n} \to {\text {g}}$ and ${\text {g}}_n^{ - 1} \to {{\text {g}}^{ - 1}}$ uniformly on the circle where ${\text {g}}$ is a homeomorphism, or (ii) ${{\text {g}}_n} \to {{\text {x}}_0}$ and ${\text {g}}_n^{ - 1} \to {{\text {y}}_0}$ uniformly on compact subsets of the complements of $\{ {{\text {y}}_0}\}$ and $\{ {{\text {x}}_0}\}$, respectively, for some points ${{\text {x}}_0}$ and ${{\text {y}}_0}$ of the circle (possibly ${{\text {x}}_0}{\text { = }}{{\text {y}}_0}$). 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 $f$ such that $f \circ G \circ {f^{ - 1}}$ is a group of Màbius transformations.