Setup: Let be an imaginary quadratic field with discriminant and class number , and denote its ring of integers by . Set and consider the Bianchi orbifold , where denotes hyperbolic –space. The goal of this note is to prove the following Theorem:
Theorem 1: The cusp set of is in bijection with , viewed as a subset of . Moreover, the number of ends of is equal to .
First recall that acts on via the fractional linear transformation . Geometrically, each such is the product of an even number of inversions within circles and lines in . Suppose acts by inversions in circles and lines . The action of can be extended to
as follows. First note that we can identify with . Then, given any circle (resp. line ) in , observe that there exists a unique hemisphere (resp. plane ) in that is simultaneously orthogonal to and intersects at (resp. ). The Poincaré extension of to is obtained by applying the corresponding inversions in and . We now recall the classification of elements in :
is elliptic if and .
is parabolic if .
is loxodromic otherwise.
It is an easy exercise to show that is parabolic if and only if it has a unique fixed point on , in which case is conjugate to the standard translation . acts transitively on points in , so the stabilizer of any point is conjugate to that of , which can easily be worked out to be . Hence we recover as the symmetric space . It is well-known that , which in turn is diffeomorphic to the 3–sphere . Similarly, acts transitively on , the sphere at infinity, so all point stabilizers (in ) are conjugate to the subgroup of upper–triangular matrices
We are particularly interested in point stabilizers inside discrete subgroups of .
Definition 3: A Kleinian group (resp. Bianchi group) is a discrete subgroup of (resp. for some ). A hyperbolic orbifold is called a Kleinian orbifold (resp. Bianchi orbifold) if is commensurable with a Kleinian group (resp. Bianchi group).
Definition 4: Let be any number field with exactly one complex place and ring of integers . Suppose is a quaternion algebra over that is ramified at all real places and let be a –embedding. Then a subgroup of is an arithmetic Kleinian group if it is commensurable with some , where is an –order in and 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 act discontinuously on . In particular, the –stabilizer of any point in is finite and the stabilizer of a point on the sphere at infinity is conjugate to a discrete subgroup . By inspection one observes that can take on of the three forms:
Finite cyclic, a finite extension of generated by a parabolic or loxodromic element, or a finite extension of generated by a pair of parabolics.
The only delicate part about the above classification involves noting that any loxodromic element in is conjugate to a matrix of the form with . So if there were two loxodromic elements, they would be –translates of eachother, meaning that item (3) can only occur if the two summands are generated by parabolic elements.
Definition 6: A point is a cusp if its stabilizer subgroup contains a free abelian group of rank .
Note that we can always take to be one of the parabolic generators.
In order to gain traction on the cusp set of hyperbolic orbifolds , we need a “smallness" condition on . The theory of Tamagawa numbers shows that Kleinian groups of the form have covolume . Hence,
Lemma 7: Arithmetic Kleinian groups have finite covolume.
Lemma 8: If is an orientable hyperbolic 3–orbifold of finite volume, then has finitely many ends (i.e. cusp neighborhoods) and each end is isometric to , where is some quotient of the 2–torus . Moreover, the stabilizer subgroup of each cusp is of type above.
Proof: If there were infinitely many ends, then would not have finite volume. By the classification of discrete subgroups of and the definition of a cusp, the stabilizer of each end is generated by a pair of parabolics and . Without loss of generality we may assume that and , where . In particular, and represent independent translations in . Let denote the torsion subgroup of , and set . A consequence of the fact that is the free product of its torsion-free and torsion parts is we can factor the orbit space as
Since is torsion–free, is a manifold. In particular, the translations and , viewed as elements of , generate independent copies of in that intersect at a single point. In otherwords, they generate a copy of . By construction of the Poincaré extension of to , one sees that the stabilization locus of the in is isometric to . Yet again appealing to the Poincaré extension, it follows that the cusp neighborhoods in are isometric to .
Corollary 9: All Bianchi orbifolds have at least one cusp.
Proof: Let be a Bianchi group and consider the standard –basis for , where
Then and are independent parabolic elements in . By Lemma~8 their stabilization locus is an end of .
In order to prove Theorem~1, it suffices now to prove the following Lemma. Recall that each can be described as a fraction , written in lowest terms, with . Recall also that elements can be described as equivalence classes of points in modulo the equivalence relation if and only if there exists such that . Since is a rank –module, every fractional ideal can be generated by a pair of elements. Let denote the equivalence class of ideals (in the class group of ) with reprepresentative the ideal .
Lemma 10: Let and be points in . Then there exists such that if and only if .
Proof: First assume there exists such that . Let for some satisfying . Then . By definition there exists some nonzero such that and . Hence as ideals, which implies . Set and . It suffices now to show that . Indeed the containment is clear. Note that
So for all . It follows that , ergo . We conclude that in the class group of . Conversely, suppose . By definition there exists nonzero and in such that . Note this is equivalent to . It is a well-known fact that ideals in an imaginary quadratic number field embed as lattices in . We fix such embeddings of and , and denote their images by and , respectively. Since by hypothesis, there exists some linear transformation such that ; in particular, . Without loss of generality we can assume since and have coordinates in . As is a constant, one has . So on the level of one has
Proof of Theorem~1: Any Bianchi group has finite covolume by Lemmas~5 and~7. Lemma~8 then implies that has finitely many cusps. Let denote the cusp set of . From the previous exposition it is clear that every cusp in (parabolic element) gives rise to an element . Conversely, given , the parabolic element
fixes . This proves that is bijective with , which is the first claim of the Theorem. Next define the map by , where is the ideal class group of . As previously noted, every –ideal in can be generated by two elements; so if is any ideal class, there exists such that is a representative for . Hence , proving that is surjective. Using the previous paragraph and Lemma~10, one sees that descends to a bijection
We conclude that , completing the proof.