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 to construct a visual extension of any given rational map.

25 pages, 1 figure