On Poincar\'e extensions of rational maps

TL;DR

This paper extends rational maps on the Riemann sphere into hyperbolic space, creating conformally natural homomorphisms and visual extensions by leveraging M"obius automorphisms and complex multiplication in hyperbolic geometry.

Contribution

It introduces a novel method to extend rational maps into hyperbolic space using M"obius automorphisms and complex multiplication, expanding the classical Poincaré extension.

Findings

Constructed extensions of rational maps into hyperbolic space.

Defined conformally natural homomorphisms on subsemigroups of Blaschke maps.

Extended complex multiplication to hyperbolic space for visual extensions.

Abstract

There is a classical extension, of M\"obius automorphisms of the Riemann sphere into isometries of the hyperbolic space $\mathbb{H}^3$, which is called the Poincar\'e 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 M\"obius 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.