I do not intend to prove any results in this post, but for the sake of exposition as I work towards understaning Mochizuki’s –adic uniformization theory, I would like to review the uniformization of complex–analytic hyperbolic curves. There are three main ways to realize such curves as the quotient of a topological space by the action of some group that acts discontinuously. We will focus primarily on the first two uniformizations described below.

Let (resp. ) denote the upper–half plane (resp. hyperbolic 3–space), and denote (resp. ) by (resp. ) . Let be a –punctured genus curve over such that (i.e has negative Euler characteristic and admits a hyperbolic structure). A hyperbolic structure on is, by definition, a choice of an atlas on with transition maps . We take as fact that such a choice induces an isometry , where is the universal cover of .

**Fuchsian–Koebe Uniformization:** The group acts transitively on via fractional linear transformations:

In fact, . It follows that the identification induces a representation

which is sometimes called *the canonical representation*. Set . The resulting isometry

is called the *Fuchsian–Koebe uniformization* of .

**Schottky Uniformization:** Observe that gives a natural –representation after composition with the inclusion . Moreover, we may compactify by ordering and filling its set of punctures so that they are viewed as marked points. Let denote the compactification of . Topologically speaking, –pointed and –punctured curves carry the same information and define the same point in the relevant moduli stack. The distinction between and becomes necessary when discussing uniformization. Now recall from the post “Cusps on Bianchi Orbifolds I”, the group acts on by Poincaré extending the fractional linear transformation on the sphere at to all of . Formulaically,

Similar to the case in two dimensions, one can show that .

Suppose that is a free, finitely generated Kleinian group that consists only of loxodromic elements: see Cusps on Bianchi Orbifolds I for the classification of Möbius transformations. A Theorem of Maskit in *A Characterization of Schottky Groups* implies that acts discontinuously on some nonempty connected domain bounded by Jordan curves whose interiors are pairwise disjoint and such that for each there exists with the properties and . Any such is known as a *Schottky group* and any domain in that is the maximal connected locus of discontinuity for the action of some Schottky group is called a *Schottky domain*. Set and let denote the full locus of discontinuity for the action of . The condition that all elements are loxodromic is equivalent to containing no unipotent, i.e parabolic, elements. Passing to the quotient, one obtains homeomorphisms

and

where denotes the Handlebody obtained by attaching handles to a solid torus, and denotes the boundary surface of genus . Intuitively, the boundary components of are identified in pairs via the action of , and the base solid torus of manifests in the Poincaré extension of the –action to . Note that is compact, ergo closed. The hyperbolic metric on induces a hyperbolic structure on , so can be promoted to an isometry of Riemann surfaces. The description of as the quotient is called the *Schottky uniformization* of .

It turns out that every closed Riemann surface of genus admits a Schottky uniformization; however, the closedness condition excludes punctured surfaces. There is a way to handle surfaces with an even number of punctures, using so–called extended Schottky groups, but we will not discuss these here. Instead, we consider marked surfaces. This will allow us to relate the Fuchsian–Koebe and Schottky uniformizations on the level of covering maps and fundamental groups.

Suppose is free on generators . A marked Schottky group is a Schottky group equipped with an ordered choice of free generators. Let denote the moduli space of marked Schottky groups of genus . consists of classes, under Möbius transformation, of ordered –tuples that generate a Schottky group. Let denote the holomorphic fibration over whose fiber over a point is the configuration spaces of –points on . can be thought of as the space of equivalence classes of tuples , where the indicate marked points on . Note that the define a –orbit of points, whose representatives are also denoted , in . Fix , set , and let . Then for some Fuchsian goup . Let denote the union of with the cusps of and recall that the –invariant defines a bijective Hauptmodul

for some points . Without loss of generality assume that for each . Note that lifts to a bijection

that sends the –orbit of the to the –orbit of . In particular, after restricting the codomain, lifting the domain to , and then composing with , one obtains covering maps

Suppose is generated by a standard meridian–longitude basis , and let denote the smallest normal subgroup of containing . One can show that is the topological covering space corresponding to the subgroup . From and the Fuchsian uniformization it follows that

**Bers (Simultaneous) Uniformization:** We conclude this discussion by mentioning the Bers simultaneous uniformization Theorem, which looks similar to the Schottky uniformization, but applies to a compact Riemann surface that admits two hyperbolic structures and . We call a Kleinian group quasi–Fuchsian of type I if its limit set is equal to a –invariant Jordan curve.

**Theorem 1:** If a compact Riemann surface admits hyperbolic structures and , then there exists a quasi–Fuchsian group of type I such that

where is the invariant Jordan curve of and indicates equipped with the hyperbolic structure .

The main observation towards proving such a result is that divides into two open discs and , each of which is conformally equivalent to . So it suffices to find Fuchsian groups and such that .