Title: 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 Teichmüller space of $G$ by
a suitable finite index subgroup of its modular group.