On Poincaré extensions of rational maps

Abstract

There is a classical extension of Möbius automorphisms of the Riemann sphere into isometries of the hyperbolic space H 3 \mathbb {H}^3 which is called the Poincaré extension. In this paper, we construct extensions of rational maps on the Riemann sphere over endomorphisms of H 3 \mathbb {H}^3 exploiting the fact that any holomorphic covering between Riemann surfaces is Möbius for a suitable choice of coordinates. We show that these extensions define conformally natural homomorphisms on suitable subsemigroups of the semigroup of Blaschke maps. We extend the complex multiplication to a product in H 3 \mathbb {H}^3 that allows us to construct an extension of any given rational map which is right equivariant with respect to the action of P S L ( 2 , C ) PSL(2,\mathbb {C}) .