# Cusps on Bianchi Orbifolds

Setup: Let ${K_d = {\mathbb Q}(\sqrt{d})}$ be an imaginary quadratic field with discriminant ${D}$ and class number ${h_d}$, and denote its ring of integers by ${{\cal{O}}_d}$. Set ${\Gamma_d = \mathrm{PSL}_2({\cal{O}}_d)}$ and consider the Bianchi orbifold ${\mathop{\mathbb H}^3/\Gamma_d}$, where ${\mathop{\mathbb H}^3}$ denotes hyperbolic ${3}$–space. The goal of this note is to prove the following Theorem:

Theorem 1: The cusp set of ${\Gamma_d}$ is in bijection with ${\mathbb{P}K_d}$, viewed as a subset of ${\mathbb{P}{\mathbb C}}$. Moreover, the number of ends of ${\mathop{\mathbb H}^3/\Gamma_d}$ is equal to ${|\mathop{\mathbb P} K_d/\Gamma_d| = h_d}$.

First recall that ${\gamma = \left[\begin{smallmatrix} a & b \\ c & d \end{smallmatrix}\right]\in \mathrm{PSL}_2({\mathbb C})}$ acts on ${\mathbb{P}{\mathbb C} = {\mathbb C} \cup \{\infty\}}$ via the fractional linear transformation ${\gamma \cdot z \mapsto \frac{az + b}{cz + d}}$. Geometrically, each such ${\gamma}$ is the product of an even number of inversions within circles and lines in ${{\mathbb C}}$. Suppose ${\gamma}$ acts by inversions in circles ${C_1,\dots. C_{i}}$ and lines ${l_1,\dots, l_{j}}$. The action of ${\gamma}$ can be extended to

$\displaystyle \mathop{\mathbb H}^3 = \{(x,y,t) \in {\mathbb C} \times {\mathbb R} : t > 0\}$

as follows. First note that we can identify ${\partial \mathop{\mathbb H}^3 = \{(x,y,t)\in \mathop{\mathbb H}^3: t = 0\}}$ with ${\mathbb{P}{\mathbb C}}$. Then, given any circle ${C}$ (resp. line ${l}$) in ${{\mathbb C} \subset \mathbb{P}{\mathbb C}}$, observe that there exists a unique hemisphere ${\widetilde{C}}$ (resp. plane ${\widetilde{l}}$) in ${\mathop{\mathbb H}^3}$ that is simultaneously orthogonal to ${{\mathbb C}}$ and intersects ${{\mathbb C}}$ at ${C}$ (resp. ${l}$). The Poincaré extension of ${\gamma}$ to ${\mathop{\mathbb H}^3}$ is obtained by applying the corresponding inversions in ${\widetilde{C_1},\dots,\widetilde{C_i}}$ and ${\widetilde{l_1},\dots,\widetilde{l_j}}$. We now recall the classification of elements ${\gamma \neq I_2}$ in ${\mathrm{PSL}_2({\mathbb C})}$:

Definition 2:
${\gamma}$ is elliptic if ${\mathrm{tr }\gamma \in {\mathbb R}}$ and ${|\mathrm{tr }\gamma|< 2}$.
${\gamma}$ is parabolic if ${\mathrm{tr }\gamma=\pm 2}$.
${\gamma}$ is loxodromic otherwise.

It is an easy exercise to show that ${\gamma}$ is parabolic if and only if it has a unique fixed point on ${\mathop{\mathbb P}{\mathbb C}}$, in which case ${\gamma}$ is conjugate to the standard translation ${z \mapsto z + 1}$. ${\mathrm{PSL}_2({\mathbb C})}$ acts transitively on points in ${\mathop{\mathbb H}^3}$, so the stabilizer of any point is conjugate to that of ${(0,0,1)}$, which can easily be worked out to be ${\mathrm{PSU}_2({\mathbb C})}$. Hence we recover ${\mathop{\mathbb H}^3}$ as the symmetric space ${\mathrm{PSL}_2({\mathbb C})/\mathrm{PSU}_2({\mathbb C})}$. It is well-known that ${\mathrm{PSU}_2({\mathbb C}) \cong \mathrm{SO}_3({\mathbb R})}$, which in turn is diffeomorphic to the 3–sphere ${S^3}$. Similarly, ${\mathrm{PSL}_2({\mathbb C})}$ acts transitively on ${\mathop{\mathbb P}{\mathbb C}}$, the sphere at infinity, so all point stabilizers (in ${\mathrm{PSL}_2({\mathbb C})}$) are conjugate to the subgroup of upper–triangular matrices

$\displaystyle B_{\infty} = \left\{\left[\begin{smallmatrix} a & b \\ & a^{-1} \end{smallmatrix}\right]: a\in {\mathbb C}^{\times}, b \in {\mathbb C}\right\}$

We are particularly interested in point stabilizers inside discrete subgroups of ${\mathrm{PSL}_2({\mathbb C})}$.

Definition 3: A Kleinian group (resp. Bianchi group) is a discrete subgroup of ${\mathrm{PSL}_2({\mathbb C})}$ (resp. ${\mathrm{PSL}_2({\cal{O}}_d)}$ for some ${d < 0}$). A hyperbolic orbifold ${\mathop{\mathbb H}^3/\Gamma}$ is called a Kleinian orbifold (resp. Bianchi orbifold) if ${\Gamma}$ is commensurable with a Kleinian group (resp. Bianchi group).

Definition 4: Let ${k}$ be any number field with exactly one complex place and ring of integers ${{\cal{O}}}$. Suppose ${A}$ is a quaternion algebra over ${k}$ that is ramified at all real places and let ${\rho: A \rightarrow \mathrm{M}_2({\mathbb C})}$ be a ${k}$–embedding. Then a subgroup ${\Gamma}$ of ${\mathrm{PSL}_2({\mathbb C})}$ is an arithmetic Kleinian group if it is commensurable with some ${\rho({\cal{A}}^1)/\{\pm I_2\}}$, where ${{\cal{A}}}$ is an ${{\cal{O}}_d}$–order in ${A}$ and ${{\cal{A}}^1}$ is its elements of norm 1.

The following Lemma can be found in Shimura or Maclachlan–Reid's book, and for brevity we state it without proof.

Lemma 5: Bianchi groups are arithmetic Kleinian groups.
Definition~4 is the same as that in the theory of Shimura varieties. The discreteness condition implies that such ${\Gamma}$ act discontinuously on ${\mathop{\mathbb H}^3}$. In particular, the ${\Gamma}$–stabilizer of any point in ${\mathop{\mathbb H}^3}$ is finite and the stabilizer of a point on the sphere at infinity is conjugate to a discrete subgroup ${\Gamma_{\infty} < B_{\infty}}$. By inspection one observes that ${\Gamma_{\infty}}$ can take on of the three forms:

Finite cyclic, a finite extension of ${{\mathbb Z}}$ generated by a parabolic or loxodromic element, or a finite extension of ${{\mathbb Z}\oplus {\mathbb Z}}$ generated by a pair of parabolics.

The only delicate part about the above classification involves noting that any loxodromic element in ${\Gamma_{\infty}}$ is conjugate to a matrix of the form ${\left[\begin{smallmatrix} \lambda & \\ & \lambda^{-1} \end{smallmatrix}\right]}$ with ${|\lambda + \lambda^{-1}|^2 \not\in [0,4]}$. So if there were two loxodromic elements, they would be ${\Gamma_{\infty}}$–translates of eachother, meaning that item (3) can only occur if the two summands are generated by parabolic elements.

Definition 6: A point ${\zeta \in \mathop{\mathbb P}{\mathbb C}}$ is a cusp if its stabilizer subgroup contains a free abelian group of rank ${2}$.
Note that we can always take ${\left[\begin{smallmatrix} 1 & 1 \\ & 1 \end{smallmatrix}\right]}$ to be one of the parabolic generators.

In order to gain traction on the cusp set of hyperbolic orbifolds ${\mathop{\mathbb H}^3/\Gamma}$, we need a “smallness" condition on ${\Gamma}$. The theory of Tamagawa numbers shows that Kleinian groups of the form ${\rho({\cal{A}}^1)/\{\pm I_2\}}$ have covolume ${1}$. Hence,

Lemma 7: Arithmetic Kleinian groups have finite covolume.

Lemma 8: 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 is of type ${(3)}$ above.

Proof: If there were infinitely many ends, then ${M}$ would not have finite volume. By the classification of discrete subgroups of ${B_{\infty}}$ and the definition of a cusp, the stabilizer of each end is generated by a pair of parabolics ${\gamma_1}$ and ${\gamma_2}$. Without loss of generality we may assume that ${\gamma_1 = \left[\begin{smallmatrix} 1 & 1 \\ & 1 \end{smallmatrix}\right]}$ and ${\gamma_2 = \left[\begin{smallmatrix} 1 & \omega \\ & 1 \end{smallmatrix}\right]}$, where ${\omega \in {\mathbb C}\backslash {\mathbb R}}$. In particular, ${\gamma_1}$ and ${\gamma_2}$ represent independent translations in ${\mathop{\mathbb P}{\mathbb C}}$. Let ${\mathrm{Tors}(\Gamma)}$ denote the torsion subgroup of ${\Gamma}$, and set ${\overline{\Gamma} = \Gamma/\mathrm{Tors}(\Gamma)}$. A consequence of the fact that $\Gamma$ is the free product of its torsion-free and torsion parts is we can factor the orbit space ${\mathop{\mathbb H}^3/\Gamma}$ as

$\displaystyle (\mathop{\mathbb H}^3/\overline{\Gamma})/\mathrm{Tors}(\Gamma)$

Since ${\overline{\Gamma}}$ is torsion–free, ${\mathop{\mathbb H}^3/\overline{\Gamma}}$ is a manifold. In particular, the translations ${\gamma_1}$ and ${\gamma_2}$, viewed as elements of ${\overline{\Gamma}}$, generate independent copies of ${S^1}$ in ${\mathop{\mathbb P}{\mathbb C}}$ that intersect at a single point. In otherwords, they generate a copy of ${T^2}$. By construction of the Poincaré extension of ${\gamma_i}$ to ${\mathop{\mathbb H}^3}$, one sees that the stabilization locus of the ${\gamma_i}$ in ${\mathop{\mathbb H}^3/\overline{\Gamma}}$ is isometric to ${T^2 \times [0,\infty)}$. Yet again appealing to the Poincaré extension, it follows that the cusp neighborhoods in ${\mathop{\mathbb H}^3/\Gamma}$ are isometric to ${(T^2 \times [0,\infty))/\mathrm{Tors}(\Gamma) = T^2/\mathrm{Tors}(\Gamma) \times [0,\infty)}$. $\Box$

Corollary 9: All Bianchi orbifolds have at least one cusp.
Proof: Let ${\Gamma_d}$ be a Bianchi group and consider the standard ${{\mathbb Z}}$–basis ${\{1,\omega\}}$ for ${{\cal{O}}_d}$, where

$\displaystyle \omega = \begin{cases} \sqrt{d} & d\not \cong 1\pmod 4 \\ \frac{1+\sqrt{d}}{2} & d \cong 1\pmod 4 \end{cases}$

Then ${\gamma_1 = \left[\begin{smallmatrix} 1 & 1 \\ & 1 \end{smallmatrix}\right]}$ and ${\gamma_2 = \left[\begin{smallmatrix} 1 & \omega \\ & 1 \end{smallmatrix}\right]}$ are independent parabolic elements in ${\Gamma_d}$. By Lemma~8 their stabilization locus is an end of ${\mathop{\mathbb H}^3/\Gamma_d}$. $\Box$

In order to prove Theorem~1, it suffices now to prove the following Lemma. Recall that each ${p \in K_d}$ can be described as a fraction ${\frac{x}{y}}$, written in lowest terms, with ${x,y \in {\cal{O}}_d}$. Recall also that elements ${[x,y]\in \mathop{\mathbb P} K_d}$ can be described as equivalence classes of points in ${K_d^2}$ modulo the equivalence relation ${[x,y]\sim [x',y']}$ if and only if there exists ${\lambda \in K_d}$ such that ${\lambda[x,y] = [x',y']}$. Since ${{\cal{O}}_d}$ is a rank ${2}$ ${{\mathbb Z}}$–module, every fractional ideal can be generated by a pair of elements. Let ${\left[(x,y)\right]}$ denote the equivalence class of ideals (in the class group of ${{\cal{O}}_d}$) with reprepresentative the ideal ${(x,y)}$.

Lemma 10: Let ${[x,y]}$ and ${[x',y']}$ be points in ${\mathop{\mathbb P} K_d}$. Then there exists ${\gamma \in \Gamma_d}$ such that ${\gamma [x,y] = [x',y']}$ if and only if ${\left[(x,y)\right]= \left[(x',y')\right]}$.

Proof: First assume there exists ${\gamma \in \Gamma_d}$ such that ${\gamma[x,y] = [x',y']}$. Let ${\gamma = \left[\begin{smallmatrix} a & b\\ c & d \end{smallmatrix}\right]}$ for some ${a,b,c,d \in {\cal{O}}_d}$ satisfying ${ad - bc = 1}$. Then ${\gamma[x,y] = [ax+by,cx + dy] = [x',y']}$. By definition there exists some nonzero ${\lambda \in K_d}$ such that ${ax+by = \lambda x'}$ and ${cx + dy = \lambda y'}$. Hence ${(ax+by,cx+dy) = \lambda(x,y)}$ as ideals, which implies ${[(ax+by,cx+dy)] = [(x,y)]}$. Set ${x_0 = ax+by}$ and ${y_0 = cx+dy}$. It suffices now to show that ${(x,y) = (x_0,y_0)}$. Indeed the containment ${(ax+by, cx+dy)\subset (x,y)}$ is clear. Note that

$\displaystyle dx_0 - by_0 = dax + dby - bcx - bdy = x$

and
$\displaystyle -cx_0 + ax_0 = -cax - cby + acx + ady = y$

So ${mx + ny = m(dx_0 - by_0) + n(-cx_0 + ax_0) \in (x,y)\cap (x_0,y_0)}$ for all ${m,n\in {\cal{O}}_d}$. It follows that ${(x,y)\subset (x_0,y_0)}$, ergo ${(x,y) = (x_0,y_0)}$. We conclude that ${[(x,y)] = [(x',y')]}$ in the class group of ${K_d}$. Conversely, suppose ${\left[(x,y)\right]= \left[(x',y')\right]}$. By definition there exists nonzero ${\alpha}$ and ${\beta}$ in ${{\cal{O}}_d}$ such that ${\alpha(x,y) = \beta(x',y')\subset {\cal{O}}_d}$. Note this is equivalent to ${(x,y) = \frac{\beta}{\alpha}(x',y')}$. It is a well-known fact that ideals in an imaginary quadratic number field embed as lattices in ${{\mathbb C}}$. We fix such embeddings of ${\alpha(x,y)}$ and ${\beta(x',y')}$, and denote their images by ${L}$ and ${L'}$, respectively. Since ${L = L'}$ by hypothesis, there exists some linear transformation ${\gamma \in PSL_2({\mathbb C})}$ such that ${\gamma(L) = L'}$; in particular, ${\gamma\left((\alpha x,\alpha y)\right) = (\pm \beta x,\pm \beta y)}$. Without loss of generality we can assume ${\gamma \in PSL_2({\cal{O}}_d)}$ since ${L}$ and ${L'}$ have coordinates in ${{\cal{O}}_d^2}$. As ${\alpha}$ is a constant, one has ${\gamma(x,y) = (\pm \frac{\beta}{\alpha}x',\pm \frac{\beta}{\alpha}y')}$. So on the level of ${\mathop{\mathbb P} K_d}$ one has $\displaystyle \gamma[x,y] = [\pm \frac{\beta}{\alpha}x',\pm \frac{\beta}{\alpha}y'] = [x',y']$

$\Box$

Proof of Theorem~1: Any Bianchi group ${\Gamma_d}$ has finite covolume by Lemmas~5 and~7. Lemma~8 then implies that ${\mathop{\mathbb H}^3/\Gamma_d}$ has finitely many cusps. Let ${{\cal{C}}_d}$ denote the cusp set of ${\Gamma_d}$. From the previous exposition it is clear that every cusp in ${{\cal{C}}_d}$ (parabolic element) gives rise to an element ${[x,y]\in \mathop{\mathbb P} K_d}$. Conversely, given ${[x,y]\in \mathop{\mathbb P} K_d}$, the parabolic element
$\displaystyle \gamma = \left[\begin{smallmatrix} 1+xy & -x^2 \\ y^2 & 1 - xy \end{smallmatrix}\right]$

fixes ${[x,y]}$. This proves that ${{\cal{C}}_d}$ is bijective with ${\mathop{\mathbb P} K_d}$, which is the first claim of the Theorem. Next define the map ${\widetilde{\phi}: {\cal{C}}_d \rightarrow C_d}$ by ${[x,y] \mapsto [(x,y)]}$, where ${C_d}$ is the ideal class group of ${K_d}$. As previously noted, every ${{\cal{O}}_d}$–ideal in ${K_d}$ can be generated by two elements; so if ${I \in C_d}$ is any ideal class, there exists ${x,y\in K_d}$ such that ${[(x,y)]}$ is a representative for ${I}$. Hence ${\widetilde{\phi}\left([x,y]\right) = I}$, proving that ${\phi}$ is surjective. Using the previous paragraph and Lemma~10, one sees that ${\widetilde{\phi}}$ descends to a bijection
$\displaystyle \mathop{\mathbb P} K_d/\Gamma_d = {\cal{C}}_d/\Gamma_d \rightarrow C_d$

We conclude that ${|\mathop{\mathbb P} K_d/\Gamma_d| = |{\cal{C}}_d/\Gamma_d| = |C_d| = h_d}$, completing the proof. $\Box$