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 , which is called the Poincaré extension. In this paper, we construct extensions of rational maps on the Riemann sphere over endomorphisms of 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 that allows to construct a visual extension of any given rational map.

On hyperbolic cobordisms and Hurwitz classes of holomorphic coverings

“…The null cobordism where S and S ′ are connected is related to the extension of a single holomorphic covering to the respective 3-hyperbolic spaces. This situation has been studied in [3] with applications to holomorphic dynamical systems. In particular, in [3] the authors gave the construction of a geometric extension for generic rational maps.…”

“…This situation has been studied in [3] with applications to holomorphic dynamical systems. In particular, in [3] the authors gave the construction of a geometric extension for generic rational maps. The present article develops the geometrical part of [3] in the case of a collection of holomorphic coverings.…”

“…Every finite collection of non-univalent rational (polynomial) maps forms a cobordant family. This is not clear even in the case of a single rational map R (for more details on this problem see [3]).…”

“…The results in this section develop ideas in [3], and the Schreier representation of semigroups as treated in [2]. We start with a brief introduction to Schreier representations.…”

“…Finally, similar ideas allow us to consider the Hurwitz space as a representation space of a special class of holomorphic correspondences. This follows using results from [2] with [3]. So we can construct a Teichmüller space of correspondences of the form R −1 •R, called the deck correspondence associated to R. From [3] it follows that the Speisser class of R fibers over the Moduli space of the deck with fiber equivalent to the conformal Hurwitz class of R. Let G R = R −1 •R, C be the semigroup of holomorphic correspondences generated by the deck correspondence associated to R and constant maps.…”