Uniformizations of stable $(γ,n)$-gonal Riemann surfaces

Abstract: A $(\gamma,n)$-gonal pair is a pair $(S,f)$, where $S$ is a closed Riemann surface and $f:S \to R$ is a degree $n$ holomorphic map onto a closed Riemann surface $R$ of genus $\gamma$. If the signature of $(S,f)$ is of hyperbolic type, then there is pair $(\Gamma,G)$, called an uniformization of $(S,f)$, where $G$ is a Fuchsian group acting on the unit disc ${\mathbb D}$ containing $\Gamma$ as an index $n$ subgroup, so that $f$ is induced by the inclusion of $\Gamma <G$. The uniformization is uniquely determined by $(S,f)$, up to conjugation by holomorphic automorphisms of ${\mathbb D}$, and it permits to provide natural complex orbifold structures on the Hurwitz spaces parametrizing (twisted) isomorphic classes of pairs topologically equivalent to $(S,f)$. In order to produce certain compactifications of these Hurwitz spaces, one needs to consider the so called stable $(\gamma,n)$-gonal pairs, which are natural geometrical deformations of $(\gamma,n)$-gonal pairs. Due to the above, it seems interesting to search for uniformizations of stable $(\gamma,n)$-gonal pairs, in terms of certain class of Kleinian groups. In this paper we review such uniformizations by using noded Fuchsian groups, which are (geometric) limits of quasiconformal deformations of Fuchsian groups, and which provide uniformizations of stable Riemann orbifolds. These uniformizations permit to obtain a compactification of the Hurwitz spaces with a complex orbifold structure, these being quotients of the augmented Teichm\"uller space of $G$ by a suitable finite index subgroup of its modular group.