Extensions of holomorphic motions and holomorphic families of Möbius groups

Abstract

A normalized holomorphic motion of a closed set in the Riemann sphere, defined over a simply connected complex Banach manifold, can be extended to a normalized quasiconformal motion of the sphere, in the sense of Sullivan and Thurston. In this paper, we show that if the given holomorphic motion, defined over a simply connected complex Banach manifold, has a group equivariance property, then the extended (normalized) quasiconformal motion will have the same property. We then deduce a generalization of a theorem of Bers on holomorphic families of isomorphisms of Möbius groups. We also obtain some new results on extensions of holomorphic motions. The intimate relationship between holomorphic motions and Teichmüller spaces is exploited throughout the paper.