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

\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']


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

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