## Uniformization of Complex Hyperbolic Curves

I do not intend to prove any results in this post, but for the sake of exposition as I work towards understaning Mochizuki’s ${p}$–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 ${\mathop{\mathbb H}^2}$ (resp. ${\mathop{\mathbb H}^3}$) denote the upper–half plane (resp. hyperbolic 3–space), and denote ${\mathop{\mathbb H}^2\sqcup \partial \mathop{\mathbb H}^2 = \mathop{\mathbb H}^2\sqcup \mathop{\mathbb P}^1{\mathbb R}}$ (resp. ${\mathop{\mathbb H}^3\sqcup \partial \mathop{\mathbb H}^3 = \mathop{\mathbb H}^3\sqcup \mathop{\mathbb P}^1{\mathbb C}}$) by ${\overline{\mathop{\mathbb H}^2}}$ (resp. ${\overline{\mathop{\mathbb H}^3}}$) . Let ${X}$ be a ${r}$–punctured genus ${g}$ curve over ${{\mathbb C}}$ such that ${2g-2+r > 0}$ (i.e has negative Euler characteristic and admits a hyperbolic structure). A hyperbolic structure on ${X}$ is, by definition, a choice of an atlas ${\{U_i\}}$ on ${X}$ with transition maps ${U_i\xrightarrow{\phi_i} \mathop{\mathbb H}^2\xrightarrow{\phi_j^{-1}} U_j\cap U_i}$. We take as fact that such a choice induces an isometry ${\widetilde{X}\cong \mathop{\mathbb H}^2}$, where ${\widetilde{X}}$ is the universal cover of ${X}$.

Fuchsian–Koebe Uniformization: The group ${\mathrm{PSL}_2({\mathbb R})}$ acts transitively on ${\mathop{\mathbb H}^2}$ via fractional linear transformations:
$\displaystyle \left[\begin{smallmatrix} a & b \\ c & d \end{smallmatrix}\right]\cdot z \mapsto \frac{az+b}{cz+d}$

In fact, ${\mathrm{Aut}(\overline{\mathop{\mathbb H}^2}) \cong \mathrm{PSL}_2({\mathbb R})}$. It follows that the identification ${\widetilde{X}\cong \mathop{\mathbb H}^2}$ induces a representation
$\displaystyle \rho_X: \pi_1(X)\rightarrow \mathrm{PSL}_2({\mathbb R})$

which is sometimes called the canonical representation. Set ${\Gamma_F = \rho_X(\pi_1(X))}$. The resulting isometry
$\displaystyle X \cong \mathop{\mathbb H}^2/\Gamma_F$

is called the Fuchsian–Koebe uniformization of ${X}$.

Schottky Uniformization: Observe that ${\rho_X}$ gives a natural ${\mathrm{PSL}_2({\mathbb C})}$–representation after composition with the inclusion ${\mathrm{PSL}_2({\mathbb R})\rightarrow \mathrm{PSL}_2({\mathbb C})}$. Moreover, we may compactify ${X}$ by ordering and filling its set of punctures so that they are viewed as marked points. Let ${X' = \overline{\mathop{\mathbb H}^2}/\Gamma_F}$ denote the compactification of ${X}$. Topologically speaking, ${r}$–pointed and ${r}$–punctured curves carry the same information and define the same point in the relevant moduli stack. The distinction between ${X}$ and ${X'}$ becomes necessary when discussing uniformization. Now recall from the post “Cusps on Bianchi Orbifolds I”, the group ${\mathrm{PSL}_2({\mathbb C})}$ acts on ${\overline{\mathop{\mathbb H}^3}}$ by Poincaré extending the fractional linear transformation on the sphere at ${\infty}$ to all of ${\mathop{\mathbb H}^3}$. Formulaically,
$\displaystyle \left[\begin{smallmatrix} a & b \\ c & d \end{smallmatrix}\right]\cdot (z,t) \mapsto \left(\frac{\overline{(cz + d)}(az+b) + a\overline{c}t^2}{|cz+d|^2 + |c|^2t^2},\frac{t}{|cz+d|^2 + |c|^2t^2}\right)$

Similar to the case in two dimensions, one can show that ${\mathrm{Aut}(\mathop{\mathbb H}^3) \cong \mathrm{Aut}(\mathop{\mathbb P}^1{\mathbb C})\cong \mathrm{PSL}_2({\mathbb C})}$.

Suppose that ${\Gamma_S}$ 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 ${\Gamma_S}$ acts discontinuously on some nonempty connected domain ${D(\Gamma_S)\subset \mathop{\mathbb P}^1{\mathbb C}}$ bounded by ${2g}$ Jordan curves ${A_1,B_1,\dots A_g,B_g}$ whose interiors are pairwise disjoint and such that for each ${i}$ there exists ${\gamma_i\in \Gamma_S}$ with the properties ${\gamma_i(A_i) = B_i}$ and ${\gamma_i(D(\Gamma_S))\cap D(\Gamma_S) = \emptyset}$. Any such ${\Gamma_S}$ is known as a Schottky group and any domain in ${\mathop{\mathbb P}^1{\mathbb C}}$ that is the maximal connected locus of discontinuity for the action of some Schottky group is called a Schottky domain. Set ${\overline{D} = D\cup \bigcup_i \left(A_i\cup B_i\right)}$ and let ${\Omega(\Gamma_S) = \bigcup_{\gamma\in \Gamma_S}\gamma(\overline{D})}$ denote the full locus of discontinuity for the action of ${\Gamma_S}$. The condition that all elements are loxodromic is equivalent to ${\Gamma_S}$ containing no unipotent, i.e parabolic, elements. Passing to the quotient, one obtains homeomorphisms
$\displaystyle (\mathop{\mathbb H}^3\sqcup \Omega)/\Gamma_S \simeq \Theta_g$

and
$\displaystyle \omega:\Omega(\Gamma_S)/\Gamma_S \simeq \partial\Theta_g = \Sigma_g$

where ${\Theta_g}$ denotes the Handlebody obtained by attaching ${g}$ handles to a solid torus, and ${\Sigma_g}$ denotes the boundary surface of genus ${g}$. Intuitively, the ${2g}$ boundary components of ${D(\Gamma_S)}$ are identified in pairs via the action of ${\Gamma_S}$, and the base solid torus of ${\Theta_g}$ manifests in the Poincaré extension of the ${\Gamma_S}$–action to ${\mathop{\mathbb H}^3}$. Note that ${\Sigma_g}$ is compact, ergo closed. The hyperbolic metric on ${\overline{\mathop{\mathbb H}^3}}$ induces a hyperbolic structure on ${\Sigma_g}$, so ${\omega}$ can be promoted to an isometry of Riemann surfaces. The description of ${\Sigma_g}$ as the quotient ${\Omega(\Gamma_S)/\Gamma_S}$ is called the Schottky uniformization of ${\Sigma_g}$.

It turns out that every closed Riemann surface of genus ${g > 1}$ 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 ${\Gamma_S}$ is free on generators ${\gamma_1,\dots, \gamma_g}$. A marked Schottky group is a Schottky group equipped with an ordered choice of free generators. Let ${{\cal S}_g}$ denote the moduli space of marked Schottky groups of genus ${g}$. ${{\cal S}_g}$ consists of classes, under Möbius transformation, of ordered ${g}$–tuples that generate a Schottky group. Let ${{\cal S}_{g,r}}$ denote the holomorphic fibration over ${{\cal S}_g}$ whose fiber over a point ${[\Gamma_S] \in {\cal S}_g}$ is the configuration spaces of ${r}$–points on ${\Omega(\Gamma_S)/\Gamma_S}$. ${{\cal S}_{g,r}}$ can be thought of as the space of equivalence classes of tuples ${(\Gamma_S, \Omega(\Gamma_S)/\Gamma_S, \{s_1,\dots, s_r\})}$, where the ${s_i}$ indicate marked points on ${\Omega(\Gamma_S)/\Gamma_S}$. Note that the ${s_i}$ define a ${\Gamma_S}$–orbit of points, whose representatives are also denoted ${s_i}$, in ${\Omega(\Gamma_S)\subset \mathop{\mathbb P}^1{\mathbb C}}$. Fix ${(\Gamma_S, \Omega(\Gamma_S)/\Gamma_S, \{s_1,\dots, s_r\})\in {\cal S}_{g,r}}$, set ${X' = \Omega(\Gamma_S)/\Gamma_S}$, and let ${X = X'\backslash\{s_1,\dots,s_r\}}$. Then ${X = \mathop{\mathbb H}^2/\Gamma_F}$ for some Fuchsian goup ${\Gamma_F < \mathrm{PSL}_2({\mathbb R})}$. Let ${\overline{\mathop{\mathbb H}^2}_{\Gamma_F}}$ denote the union of ${\mathop{\mathbb H}^2}$ with the cusps of ${\Gamma_F}$ and recall that the ${j}$–invariant defines a bijective Hauptmodul
$\displaystyle j: X \rightarrow \mathop{\mathbb P}^1\backslash\{p_1,\dots, p_r\}$

for some points ${p_i\in \mathop{\mathbb P}^1{\mathbb C}}$. Without loss of generality assume that ${p_i = s_i}$ for each ${i=1,\dots, r}$. Note that ${j}$ lifts to a bijection
$\displaystyle \overline{j}: \overline{\mathop{\mathbb H}^2}_{\Gamma_F}/\Gamma_F \rightarrow \mathop{\mathbb P}^1{\mathbb C}$

that sends the ${\Gamma_F}$–orbit of the ${\{x_i\}}$ to the ${\Gamma_S}$–orbit of ${\{s_i\}}$. In particular, after restricting the codomain, lifting the domain to ${\overline{\mathop{\mathbb H}^2}_{\Gamma_F}}$, and then composing with ${\omega}$, one obtains covering maps
$\displaystyle \overline{\mathop{\mathbb H}^2}_{\Gamma_F}\rightarrow \Omega(\Gamma_S)\rightarrow \Sigma_g$

Suppose ${\Gamma_F = \pi_1(\Sigma_g)}$ is generated by a standard meridian–longitude basis ${\mu_1,\lambda_1,\dots,\mu_g,\lambda_g}$, and let ${N_{\mu}}$ denote the smallest normal subgroup of ${\Gamma_F}$ containing ${\mu_1,\dots,\mu_g}$. One can show that ${\omega}$ is the topological covering space corresponding to the subgroup ${N_{\mu}}$. From ${\omega}$ and the Fuchsian uniformization ${\overline{\mathop{\mathbb H}^2}_{\Gamma_F}/\Gamma_F\rightarrow \Sigma_g}$ it follows that
$\displaystyle \Gamma_S = \Gamma_F/N_{\mu}$

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 ${\Sigma}$ that admits two hyperbolic structures ${\eta_1}$ and ${\eta_2}$. We call a Kleinian group ${\Gamma}$ quasi–Fuchsian of type I if its limit set ${\Lambda \subset \mathop{\mathbb P}^1{\mathbb C}}$ is equal to a ${\Gamma}$–invariant Jordan curve.

Theorem 1: If a compact Riemann surface ${\Sigma}$ admits hyperbolic structures ${\eta_1}$ and ${\eta_2}$, then there exists a quasi–Fuchsian group of type I ${\Gamma(\eta_1,\eta_2)}$ such that
$\displaystyle \left(\mathop{\mathbb P}^1\backslash \Lambda\right)/\Gamma(\eta_1,\eta_2) \cong \Sigma^{\eta_1}\sqcup \Sigma^{\eta_2}$

where ${\Lambda}$ is the invariant Jordan curve of ${\Gamma(\eta_1,\eta_2)}$ and ${\Sigma^{\eta_i}}$ indicates ${\Sigma}$ equipped with the hyperbolic structure ${\eta_i}$.

The main observation towards proving such a result is that ${\Lambda}$ divides ${\mathop{\mathbb P}^1{\mathbb C}}$ into two open discs ${\Omega_1}$ and ${\Omega_2}$, each of which is conformally equivalent to ${\mathop{\mathbb H}^2}$. So it suffices to find Fuchsian groups ${\Gamma_1}$ and ${\Gamma_2}$ such that ${\Omega_i/\Gamma(\eta_1,\eta_2)\cong \mathop{\mathbb H}^2/\Gamma_i}$.

## Cusps on Bianchi Orbifolds II

I need to clarify a minor mistake (now corrected) that was made out of haste in my previous post. Thankfully this leads nicely into a discussion that I was planning on writing about anyways. Recall the following result from part I (Lemma 8):

Lemma 1: If ${M}$ is an orientable hyperbolic 3–orbifold of finite volume, then ${M}$ has finitely many ends (i.e. cusp neighborhoods) and each end is isometric to ${\widetilde{T}\times [0,\infty)}$, where ${\widetilde{T}}$ is some quotient of the 2–torus ${T^2}$. Moreover, the stabilizer subgroup of each cusp ${{\mathbb Z}\oplus {\mathbb Z}}$, generated by a pair of parabolics.

Before the correction, I had written that the cusp neighborhoods are isometric to ${T^2\times [0,\infty)}$. As the proof of the above Lemma implies, this is true if and only if the covering group ${\Gamma}$ does not contain torsion elements that fix the point at ${\infty}$. The existence of such elements introduces interesting delicacies that are relevant to some of the problems I have been working on. In general, if there are elliptic elements that fix a cusp at ${\infty}$, then the cusp is called rigid. This means that it cannot be deformed; and in particular, Dehn surgery cannot be performed (unless the cusp is ${S^2(2,2,2,2)}$). We will explain and investigate these last statements in a later post when we discuss arithmetic knot complements (Cusps on Bianchi Orbifolds III).

Assume from here-on-out that ${\Gamma = \Gamma_d = \mathrm{PSL}_2({\cal O}_d)}$ is a Bianchi group. Then the number of cusps on ${\mathop{\mathbb H}^3/\Gamma_d}$ is equal to the class number of ${K_d}$, so there is always at least one cusp. In what follows we will show that the cusp cross sections of ${\mathop{\mathbb H}^3/\Gamma_d}$ are tori unless ${d \in \{1,3\}}$. Moreover, unless ${d = 3}$, all cusp cross sections are non–rigid.

Lemma 2: If ${\gamma \in \mathrm{PSL}_2({\mathbb C})}$ has finite order, then ${\gamma}$ is an elliptic Möbius transformation.

Proof: Suppose ${\gamma^k = \left[\begin{smallmatrix} 1 & \\ & 1 \end{smallmatrix}\right]}$ for some positive integer ${k}$. Observe that ${\gamma}$ is not conjugate to a matrix of the form ${\left[\begin{smallmatrix} 1 & \alpha \\ & 1 \end{smallmatrix}\right]}$ for some complex number ${\alpha}$ since otherwise ${\gamma}$ would be parabolic, hence have infinite order. So ${\gamma}$ has two distinct fixed points on ${{\mathbb C}}$, therefore is conjugate to a marix of the form ${\left[\begin{smallmatrix} \lambda & \\ & \lambda^{-1} \end{smallmatrix}\right]}$. It is an elementary group–theoretic fact that conjugates of torsion elements are also torsion, which implies that ${\lambda^k = 1}$. It follows that ${\lambda}$ is a root of unity, so ${\lambda + \lambda^{-1}}$ is real and ${|\lambda + \lambda^{-1}|}$ is less than ${2}$. Since the square trace map is invariant under conjugation. and the trace is invariant up to multiplication by ${-1}$, we conclude that ${\gamma}$ is elliptic. $\Box$

The Dirichlet Unit Theorem implies that the group of units in any imaginary quadratic field has rank ${0}$. Hence any unit in ${{\cal O}_d}$ is a root of unity. It turns out that the existence of such elements is rare, as the following Lemma shows.

Lemma 3: ${{\cal O}_d}$ contains a non-real root of unity if and only if ${d \in \{1,3\}}$.

Proof: The “if” direction is clear given the element ${i \in {\cal O}_1}$ and ${\zeta_3 = \frac{1+\sqrt{-3}}{2}}$ in ${{\cal O}_3}$. Conversely, suppose ${{\cal O}_d}$ contains an ${n^{th}}$–root of unity ${\zeta_n\neq \pm 1}$. Then ${K_d}$ contains the subfield ${{\mathbb Q}(\zeta_n)}$, which is Galois of degree ${\phi(n)}$. Recall the fact that ${b\mid a}$ implies ${\phi(b)\mid \phi(a)}$. In particular, one has ${p-1\mid \phi(n)}$ for all primes ${p}$ dividing ${n}$. It follows that ${p-1 \leq 3}$ (otherwise ${K_d}$ contains a subfield of strictly larger degree), and so ${n = 2^{e_1}3^{e_2}}$. Now note if ${n > 6}$, then ${\phi(n) > 2}$. We conclude that ${n \in \{2,3,4,6\}}$. When ${n \in \{2,4\}}$, then ${\zeta_n = \pm 4}$; and when ${n\in \{3,6\}}$, ${\zeta_n \in \{\frac{\pm 1 \pm \sqrt{-3}}{2}\}}$. Any imaginary quadratic field containing ${\pm i}$ (resp. ${\pm \zeta_3}$ or ${\pm \overline{\zeta_3}}$) must also contain ${K_1}$ (resp. ${K_3}$), hence be equal to ${K_1}$ (resp. ${K_3}$) by comparing degrees. $\Box$

Lemma 4: ${\Gamma_d}$ contains a non–identity element fixing ${\infty}$ if and only if ${d \in \{1,3\}}$.

Proof: Let ${\gamma = \left[\begin{smallmatrix} a & b \\ c & d \end{smallmatrix}\right] \in \mathrm{PSL}_2({\cal O}_d)}$ and suppose ${\gamma}$ is an element fixing ${\infty}$. Observe that we can identify the point ${\infty}$ with ${\frac{z}{0}}$ in ${\mathop{\mathbb P} {\mathbb C}}$ for any ${z\in {\mathbb C}}$. Consider ${\gamma \cdot \infty = \frac{a \frac{z}{0} + b}{c \frac{z}{0} + d}}$. It is easy to see that ${\gamma \cdot \frac{z}{0} = \frac{z'}{0}}$ if and only if ${b = c = 0}$, so ${\gamma = \left[\begin{smallmatrix} a & 0 \\ 0 & d \end{smallmatrix}\right]}$. Then ${d = a^{-1}}$ since ${\mathrm{det }\gamma = 1}$, hence ${a \in {\cal O}_d}$ is a unit. By Dirichlet’s unit Theorem, ${a}$ is necessarily a root of unity; and since ${\gamma \neq \left[\begin{smallmatrix} 1 & 0 \\ 0 & 1 \end{smallmatrix}\right]}$ by hypothesis, Lemma~3 implies ${d \in \{1,3\}}$. For future reference we note that these elements are ${\left[\begin{smallmatrix} i & 0 \\ 0 & -i \end{smallmatrix}\right]}$ and ${\left[\begin{smallmatrix} \omega & 0 \\ 0 & \omega^2 \end{smallmatrix}\right]}$, respectively, where ${\omega = \frac{-1 + \sqrt{-3}}{2}}$. $\Box$

Theorem 5: The cusp cross sections of the Bianchi orbifold ${\mathop{\mathbb H}^3/\Gamma_d}$ are tori unless ${d\in \{1,3\}}$. When ${d = 1}$ (resp. ${d = 3}$), the cusp cross section is a pillowcase (resp. ${S^3(3,3,3)}$).

Proof: When ${d \not \in \{1,3\}}$, Lemma~4 shows that there are no torsion elements in ${\Gamma_d}$ that fix the cusp cross sections, hence each such cross section is a torus by Lemma~1. When ${d = 1}$, the unique tosion element fixing the cusp at ${\infty}$ is ${\gamma = \left[\begin{smallmatrix} i & 0 \\ 0 & -i \end{smallmatrix}\right]}$, which acts as ${-1}$ on ${H_1(T^2,{\mathbb Z})}$. In particular the action of ${\gamma}$ on ${T^2}$ induces a degree two quotient map ${g: T^2 \rightarrow P = T^2/\langle \gamma \rangle}$. So ${P}$ is a pillowcase, which has orbifold structure ${S^2(2,2,2,2)}$. This last statement can be seen directly by noting that ${\gamma}$ stabilizes the group of ${4^{th}}$–roots of unity in ${{\mathbb C}}$, and the successive pairwise dihedral angles of the corresponding lines in ${\mathop{\mathbb P} {\mathbb C}}$ have cone angle ${\frac{\pi}{2}}$. If ${d = 3}$, then ${\gamma = \left[\begin{smallmatrix} \omega & 0 \\ 0 & \omega^2 \end{smallmatrix}\right]}$ fixes ${\infty}$, where ${\omega = \frac{-1 + \sqrt{-3}}{2}}$. Note that ${\gamma}$ has order ${3}$ and that it permutes the third roots of unity in ${{\mathbb C}}$. Passing to ${\mathop{\mathbb P} {\mathbb C}}$ we see that ${\gamma}$ permutes the lines ${\{{\mathbb R} e^{\frac{i\pi}{3}}, {\mathbb R} e^{\frac{i\pi}{3}}, {\mathbb R} e^{\frac{i\pi}{3}}}$, which clearly have successive pairwise cone–angle ${\frac{\pi}{3}}$. It follows that the cusp cross section in ${\mathop{\mathbb H}^3/\Gamma_d}$ is topologically ${S^2}$ with three singular points of order ${3}$, i.e ${S^2(3,3,3)}$. $\Box$