A remark on Z-orientability of Kleinian groups

Juan Carlos García
Universidad de La Frontera (Chile)
Abstract:
The notion of
\[Z\]
-orientability, for
\[2\]
-cell decomposition of a closed Riemann surface, was considered by Zapponi to decide if a given Strebel quadratic form has square roots. He also used this notion in the setting of dessins d'enfants to obtain certain unicellular dessins d'enfants in genus zero (a generalization of Leila's flowers) with the property that such a family is Galois-invariant and it contains at least two Galois orbits. Recently, it has been proved that
\[Z\]
-orientability provides a new Galois invariant for dessins d'enfants. In this paper, we extend this notion for Kleinian groups in any dimension and, as an application, this notion is used to provide a necessary and sufficient geometrical condition for a non-constant surjective meromorphic map
\[\varphi:S \to \widehat{\mathbb C}\]
, where
\[S\]
is a connected Riemann surface, to admit an square root, that is, a meromorphic map
\[\psi:S \to \widehat{\mathbb C}\]
such that
\[\varphi=\psi^{2}\]
.
This is part of the results obtained for my Ph. D. Thesis under the supervision of R. A. Hidalgo and S. Quispe.