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 $\mathbb {H}^3$ which is called the Poincaré extension. In this paper, we construct extensions of rational maps on the Riemann sphere over endomorphisms of $\mathbb {H}^3$ 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 $\mathbb {H}^3$ that allows us to construct an extension of any given rational map which is right equivariant with respect to the action of $PSL(2,\mathbb {C})$.