Uniformly quasiregular semigroups in two dimensions

Let G be a semigroup of K-quasiregular or K-quasimeromorphic functions mapping a given open set U in the Riemann sphere into itself, for a fixed K, the semigroup operation being the composition of functions. We prove that if G satisfies an algebraic condition, which is true for all abelian semigroups, then there exists a K -quasiconformal homeomorphism of U onto an open set V such that all the functions in f oGo f^-1 are meromorphic functions of V into itself. In particular, if U is the whole sphere then the elements of f oGo f^-1 are rational functions. We give an example of a semigroup generated by two functions on the sphere, each quasiconformally conjugate to a quadratic polynomial, that cannot be quasiconformally conjugated to a semigroup of rational functions. We give another such example of a semigroup of K-quasiconformal homeomorphisms. These results extend and complement a similar positive conjugacy result of Tukia and of Sullivan for groups of K-quasiconformal homeomorphisms.