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

Concerning p-adic Floer Theory

The first few sections of this article are designed to provide precursory evidence and motivation for “arithmetic” analogues of Classical Floer (co)homology theories, with applications to number theory. The work is partially based on Minhyong Kim’s recent paper (mentioned in my last post), and partially inspired by my recent trip to AWS 2018. Later sections in the document are either informal or incomplete.