A classification of $\mathbb C$-Fuchsian subgroups of Picard modular groups
Given an imaginary quadratic extension of , we give a classification of the maximal nonelementary subgroups of the Picard modular group preserving a complex geodesic in the complex hyperbolic plane . Complementing work of Holzapfel, Chinburg-Stover and Möller-Toledo, we show that these maximal -Fuchsian subgroups are arithmetic, arising from a quaternion algebra for some explicit and the discriminant of . We thus prove the existence of infinitely many orbits of -arithmetic chains in the hypersphere of .
Let be a Hermitian form with signature on . The projective special unitary subgroup of contains two conjugacy classes of Lie subgroups isomorphic to . The subgroups in one class preserve a complex projective line for the projective action of on the projective plane , and those of the other class preserve a totally real subspace. The groups and act as the groups of orientation preserving isometries, respectively, on the upper halfplane model of the real hyperbolic space and on the projective model of the complex hyperbolic plane defined using the form . If is a discrete subgroup of , the intersections of with the Lie subgroups isomorphic to are its Fuchsian subgroups and the Fuchsian subgroups preserving a complex projective line are called -Fuchsian subgroups. We refer to Section 2 for more precise definitions and comments on the terminology.
Let be an imaginary quadratic number field, with discriminant and ring of integers . We consider the Hermitian form defined by
The Picard modular group is a nonuniform arithmetic lattice of , see for instance [Hol2, Chap. 5] and subsequent works of Falbel, Parker, Francsics, Lax, Xie-Wang-Jiang, and many others, for information on these groups. In this paper, we classify the maximal -Fuchsian subgroups of , and we explicit their arithmetic structures.
When , there is exactly one conjugacy class of Lie subgroups of isomorphic to . When is the Bianchi group , the analogous classification is due to Maclachlan and Reid (see [Mac, MR1] and [MR2, Chap. 9]). They proved that the maximal nonelementary Fuchsian subgroups of are commensurable up to conjugacy with the stabilisers of the circles for , when acts projectively (by homographies) on the projective line , and that all these subgroups arise from explicit quaternion algebras over . For information on Bianchi groups, see for instance [Fin] and the references of [MR1].
More generally, given a semisimple connected real Lie group with finite center and without compact factor, there is a nonempty finite set of infinite conjugacy classes of Lie subgroups of locally isomorphic to , unless itself is locally isomorphic to . The structure of the set of these subgroups plays an important role for the classification of the linear representations of , and for the classification of the groups themselves, see for instance [Kna, Ser] among others. Given a discrete subgroup of , it is again interesting to study the Fuchsian subgroups of , that is, the intersections of with these Lie subgroups, to classify the maximal ones and to see, when is arithmetic, if its maximal Fuchsian subgroups are also arithmetic (see Proposition 3.1 for a positive answer) with an explicit arithmetic structure. From now on, .
We first prove (see Proposition 3.2 and just after) that a nonelementary -Fuchsian subgroup of preserves a unique projective point with relatively prime in . We define the discriminant of as . For any positive natural number , let
Let . The set of -conjugacy classes of maximal nonelementary -Fuchsian subgroups of with discriminant is finite and nonzero. Every maximal nonelementary -Fuchsian subgroup of with discriminant is commensurable up to conjugacy in with .
In the course of the proof of this result, we prove a criterion for when two groups for are commensurable up to conjugacy in . A further application of this condition shows that every maximal nonelementary -Fuchsian subgroup of is commensurable up to conjugacy in with for a squarefree natural number .
Recall (see for instance [Gol]) that a chain
with a complex projective line (if nonempty and not a singleton). It is -arithmetic if its stabiliser in has a dense orbit in it.
There are infinitely many -orbits of -arithmetic chains in the hypersphere .
The figure below shows part of the image under vertical projection in the Heisenberg group of the orbit under of a -arithmetic chain whose stabiliser has discriminant , when .
We say that a subgroup of arises from a quaternion algebra defined over if it is commensurable with for some -algebra morphism . In Section 4, we prove the following result (see [MR2, Thm. 9.6.3] in the Bianchi group case).
Every nonelementary -Fuchsian subgroup of of discriminant is conjugate in to a subgroup of arising from the quaternion algebra .
The classification of the quaternion algebras over then allow to classify up to commensurability and conjugacy the maximal nonelementary -Fuchsian subgroups of : two such groups, with discriminant and are commensurable up to conjugacy if and only if the quaternion algebras and are isomorphic. This holds for instance if and only if the quadratic forms and are equivalent over .
As was mentioned to us by M. Stover after we posted a first version of this paper on ArXiv, the existence of a bijection between wide commensurability classes of -Fuchsian subgroups of and isomorphism classes of quaternion algebras over unramified at infinity and ramified at all finite places which do not split in is a particular case of the 2011 unpublished preprint [CS] (see Theorem 2.2 in its version 3), which proves such a result for all arithmetic lattices of simple type in . In particular, the existence of this bijection (and our Corollary 4.3) should be attributed to Chinburg-Stover (although they say it was known by experts). Möller-Toledo in [MT] (a reference we were also not aware of for the first draft of this paper) also give a description of the quotients by the maximal -Fuchsian subgroups of the real hyperbolic planes they preserve, and more generally of all Shimura curves in Shimura surfaces of the first type. We believe that our precise correspondence brings interesting effective and geometric informations.
Acknowledgements: The first author thanks the Väisälä foundation and the FIM of ETH Zürich for their support during the preparation of this paper. The second author thanks the Väisälä foundation and its financial support for a fruitful visit to the University of Jyväskylä and the nordic snows. This work is supported by the NSF Grant no 093207800, while the second author was in residence at the MSRI, Berkeley CA, during the Spring 2015 semester. We thank Y. Benoist and M. Burger for interesting discussions on this paper. We warmly thank M. Stover for informing us about the paper [CS] and many other references, including [MT].
2 The complex hyperbolic plane
Let be the nondegenerate Hermitian form
of signature on with coordinates , and let be the associated Hermitian product. The point and the corresponding element (using homogeneous coordinates) is negative, null or positive according to whether , or . The negative/null/positive cone of is the subset of negative/null/positive elements of .
The negative cone of endowed with the distance defined by
is the complex hyperbolic plane . The distance is the distance of a Riemannian metric with pinched negative sectional curvature . The linear action of the special unitary group of
on induces a projective action on with kernel , where is the group of third roots of unity. This action preserves the negative, null and positive cones of , and is transitive on each of them. The restriction to of the quotient group of is the orientation-preserving isometry group of .
The null cone of is the Poincaré hypersphere , which is naturally identified with the boundary at infinity of . The Heisenberg group
acts isometrically on and simply transitively on by the action induced by the matrix representation
of in . The projective transformations induced by these matrices are called Heisenberg translations.
If a complex projective line meets , its intersection with is a totally geodesic submanifold of , called a complex geodesic. The intersection of a complex projective line in with the Poincaré hypersphere is called a chain, if nonempty and not reduced to a point. Each complex projective line in meeting (or its associated complex geodesic , or its associated chain ) is polar to a unique positive point , that is, for all (or equivalently or ). This element is the polar point of the projective line , of the complex geodesic and of the chain . Conversely, for each positive point , there is a unique complex projective line polar to , the polar line of . The intersection of with is a complex geodesic.
An easy computation (using for instance Equation (42) in [PP1]) shows that
In particular, is isomorphic to , and is also isomorphic to . More generally, if is a positive point in , then by Equation (1), its stabiliser in is the direct product of a Lie group embedding of in preserving the complex geodesic polar to , with the group of complex reflections with fixed point set the projective line polar to .
The polar chain of is
that is is the set of satisfying the equation
When , in the coordinates of , this is the equation of an ellipse, whose image under the vertical projection is the circle with center and radius in given by the equation
If , then is the vertical affine line over .
We refer to Goldman [Gol, p. 67] and Parker [Par] for the basic properties of . These references use different Hermitian forms of signature to define the complex hyperbolic plane, and the curvature is often normalised differently from our definitions. Our choices are consistent with [PP1] and [PP2].
3 Classification of -Fuchsian subgroups of
Before starting to study Fuchsian subgroups of discrete subgroups of , let us mention that it is a very general fact that the maximal nonelementary (that is, not virtually cyclic) Fuchsian subgroups of arithmetic subgroups of are automatically (arithmetic) lattices of the copy of containing them.
Let be a semisimple connected real Lie group with finite center and without compact factor, and let be a maximal nonelementary Fuchsian subgroup of an arithmetic subgroup of . Then is an arithmetic lattice in the copy of the group locally isomorphic to containing it.
One of the main points of what follows will be to determine explicitly the arithmetic structure of , that is the -structure thus constructed on the group locally isomorphic to containing it, relating it to the arithmetic structure of , that is the given -structure on .
Proof. Let be a semisimple connected algebraic group defined over , let be an algebraic subgroup of defined over locally isomorphic to , and assume that is nonelementary in . As a nonelementary subgroup of a group locally isomorphic to is Zariski-dense in it, and as the Zariski closure of a subgroup of is defined over , we hence have that is defined over . Therefore by the Borel-Harish-Chandra theorem [BHC, Thm. 7.8], is an arithmetic lattice in . Since the copies of subgroups of locally isomorphic to are algebraic, the result follows.
Let be an imaginary quadratic number field, with its discriminant, its ring of integers, its trace and its norm. The Picard modular group of , that we denote by , consists of the images in of matrices of with coefficients in . It is a nonuniform arithmetic lattice by the result of Borel and Harish-Chandra cited above.
A discrete subgroup of is an extended -Fuchsian subgroup if it satisfies one of the following equivalent conditions
preserves a complex projective line of meeting ,
fixes a positive point in ,
preserves a chain.
Many references, see for example [FaP1], do not use the word “extended”. But as defined in the introduction, in this paper, a -Fuchsian subgroup is a discrete subgroup of preserving a complex geodesic in and inducing the parallel transport on its unit normal bundle. It is the image of a Fuchsian group (that is, a discrete subgroup of ) by a Lie group embedding of in . The extended -Fuchsian subgroups are then finite extensions of -Fuchsian subgroups by finite groups of complex reflections fixing the projective line or positive point or chain in the definition above. In particular, up to commensurability, the notions of extended -Fuchsian subgroups and of -Fuchsian subgroups coincide. The -Fuchsian lattices have been studied under a different viewpoint than our differential geometric one, as fundamental groups of arithmetic curves on ball quotient surfaces or Shimura curves in Shimura surfaces, by many authors, see for instance [Kud, Hol1, Hol2, MT] and their references.
An element of is -irreducible if it does not preserve a point or a line defined over in . An element of is rational if it lies in . Note that the polar line of a positive rational point of is defined over . The group , image of in , preserves , but in general acts transitively on neither the positive, the null nor the negative points of .
The Galois group acts on by , and fixes pointwise. A positive point is Hermitian cubic over if it is cubic over (that is, if its orbit under has exactly three points), and if its other Galois conjugates over are null elements in the polar line of .
A nonelementary extended -Fuchsian subgroup of fixes a unique rational point in . This point is positive and it is the polar point of the unique complex geodesic preserved by .
Proof. If is loxodromic, let be its repelling and attracting fixed points, and let be its positive fixed point. Since the two projective lines tangent to the hypersphere at and are invariant under , their unique intersection point is fixed by , therefore is equal to . In particular, is polar to the complex projective line through (see also [Par, Lemma 6.6] for a more analytic proof).
Let be the complex projective line preserved by , which meets . As is not elementary, there are loxodromic elements such that their sets of fixed points in are disjoint. Since passes through as well as through , and by the uniqueness of the polar point to , we hence have .
As and have infinite order, one of them cannot be -irreducible. Otherwise, if both were -irreducible, then by [PP2, Prop. 18], the point would be Hermitian cubic and its orbit under would be , a contradiction. Assume then for instance that preserves a line or a point defined over . As any projective subspace preserved by is a combination of , and , and as and are not defined over , it follows that is rational.
Let be a nonelementary extended -Fuchsian subgroup of . By the previous proposition, fixes a unique rational point in , which may be written with relatively prime. Such a writing is unique up to the simultaneous multiplication of by a unit in . Since the units in have norm , and since the trace and norm of take integral values on the integers of , the number
is well defined, we call it the discriminant of . As is positive, we have . The radius of the vertical projection of the polar chain of is hence . The discriminant of depends only on the conjugacy class of in : for every , since by uniqueness we have , we have
A chain is (-)arithmetic if its stabiliser in has a dense orbit in . The following result along with Proposition 3.2 justifies this terminology. This result is well known, and it is the other direction of [MT, Lem. 1.2], see also [Hol1, Prop. 1.5,§III.1] and [Kud, §3]. We give a proof, which is a bit different, for the sake of completeness.
The stabiliser of any positive rational point is a maximal nonelementary extended -Fuchsian subgroup of , whose invariant chain is arithmetic.
Proof. Let be the linear algebraic group defined over , such that and . We endow with the -structure whose -points are so that the action of on is defined over .
As seen in Section 2, the set of real points of is isomorphic to as a real Lie group. The group is reductive and it has a (semisimple) Levi subgroup defined over , such that is isomorphic to . By a theorem of Borel-Harish-Chandra [BHC], the group is an arithmetic lattice in , which (preserves the projective line polar to and) is contained in . As is a lattice in , the group is nonelementary and has a dense orbit in the chain .
Recall that in the coordinates of , the chains are ellipses whose images under the vertical projection are Euclidean circles (see also [Gol, §4.3]). The figure in the introduction is the vertical projection of part of the orbit under of the chain when , so that is the Gauss-Picard modular group, whose generators have been given by [FFP]. The figure shows the square in with projections of chains whose diameter is at least . In the figures below, , where is a primitive third root of unity, so that is the Eisenstein-Picard modular group, whose generators have been given by [FaP2]. The first figure shows part of the orbit of and the second figure shows part of the orbit of . They both show the square in with projections of chains whose diameter is at least in the first figure and at least 0.75 in the second.
The first part of Theorem 1.1 in the introduction concerns the classification up to conjugacy of the maximal nonelementary -Fuchsian subgroups of . Consider the set of maximal nonelementary -Fuchsian subgroups of , on which the group acts by conjugation. We will prove that the discriminant map on induces a finite-to-one map from onto .
The second part of Theorem 1.1 concerns the classification up to commensurability and conjugacy. Given a group and a subgroup of , recall that two subgroups of are commensurable if has finite index in and in , and are commensurable up to conjugacy in (or commensurable in the wide sense) if there exists such that and are commensurable. For any positive natural number , let
The group is, by Proposition 3.3, a maximal nonelementary extended -Fuchsian subgroup, which preserves the projective line . Its discriminant is . We will prove that every element of with discriminant is commensurable up to conjugacy in with .
Proof of Theorem 1.1. (1) Let and let
By Proposition 3.3, the stabiliser in of the positive rational point is a maximal nonelementary extended -Fuchsian subgroup of , with discriminant . Hence is nonempty.
Let be the semisimple connected linear algebraic group defined over such that and . Let be the rational representation such that , and is the linear action of on . Let be the closed algebraic submanifold of with equation . In particular, is defined over , and is homogeneous under , by Witt’s theorem. The map
which to associates the stabiliser of in (which is the image of by the canonical map ), is well defined by Proposition 3.3 and -equivariant, and its image contains . Hence the finiteness of follows from the finiteness of the number of orbits of on , see [BHC, Thm. 6.9].
(2) Let , and let be its discriminant. By Propositions 3.2 and 3.3, and by maximality, there is a unique positive rational point with relatively prime in such that is contained with finite index in , and .
Claim. There exists such that .
Assuming this claim for the moment, we conclude the proof of the second part of Theorem 1.1: The groups and are commensurable, since
and since is contained in the commensurator of in by a standard argument of reduction to a common denominator.
The following result, useful for the proof of the above claim, also gives a natural condition for when two such groups for are commensurable up to conjugacy in . A necessary and sufficient condition will be given as a consequence of Section 4.
If satisfy , then and are commensurable up to conjugacy in .
Proof. Let and . As seen above, we only have to prove that there exists such that .
Assume first that , so that . Since , there exists such that . It is easy to check using Equation (1) and since that the matrix
belongs to . Let be its image in . It is easy to check that as wanted .
If , so that , the same argument works when in the above proof is replaced by the matrix
and the equation with .
Proof of the claim. As the lattice does not preserve the complex geodesic with equation , we may assume that is nonzero, up to replacing by an element in its orbit under , which does not change the discriminant of . Let be the Heisenberg translation by the element
which belongs to . An easy computation shows that
Let be the image in of the diagonal element in . Then maps to . By the previous lemma, there exists such that . Hence the claim follows with .
4 Quaternion algebras
Let with and . The quaternion algebra is the -dimensional central simple algebra over with standard generators satisfying the relations , and . The (reduced) norm of an element of is
The map defined by
is a morphism of -algebras and is a discrete subgroup of the stabiliser of in .
is a morphism of -algebras and that the image of is a discrete subgroup of . The map
is a morphism of -algebras, sending into the stabiliser of in (see Equation (1)). The claim follows by noting that .
Proof of Theorem 1.3. By Theorem 1.1, we only have to prove that the maximal -Fuchsian subgroup of stabilising (which has finite index in ) arises from the quaternion algebra . It is easy to check that the element
belongs to and maps to . Hence, using Equation (1), a matrix has its image (by the canonical projection ) in if and only if there exists and with such that . A straightforward computation gives
This matrix has coefficients in if and only if