Julien Paupert
email: paupert@asu.edu

Somss

Pierre Will
email: pierre.will@ujf-grenoble.fr

Real reflections, commutators and cross-ratios in complex hyperbolic space

We provide a concrete criterion to determine whether or not two given elements of PU(2,1) can be written as products of real reflections, with one reflection in common. As an application, we show that the Picard modular groups PU(2, 1, O d ) with d = 1, 2, 3, 7, 11 are generated by real reflections up to index 1, 2, 4 or 8.

Introduction

A classical way of studying isometries of a symmetric space S is to decompose them as products of involutions. The most comfortable situation is to have a class C of isometric involutions with the following properties.

1. Any two involutions in C are conjugate in Isom(S).

2. Any element of Isom(S) can be written as a product s 1 s 2 , with s i ∈ C.

For example, if such a family of involutions exists, describing the fixed points of an isometry A = s 1 s 2 of S amounts to studying the relative position of the fixed point sets of s 1 and s 2 which are totally geodesic subspaces and are isometric to one another. If such a family C exists, one usually says that Isom(S) has involution length 2 with respect to C. Of course this requirement is too optimistic in general (for example, it fails in Euclidean space of dimension at least 3).

Assuming that a symmetric space has this property, the next question is to decide when two isometries A and B can be decomposed using a common involution. We will call such a pair decomposable with respect to C (see Definition 2.1; the term linked is also commonly used, see [BM]). This property simplifies the understanding of the group A, B as it reduces to studying the relative position of 3 pairwise isometric totally geodesic subspaces.

One of the most elementary cases is that of the Poincaré disk ∆. There are two classes of involutions in Isom(∆), namely half-turns and reflections. Viewing ∆ as the unit disk in C, half-turns are conjugate in Isom + (∆) to z -→ -z and reflections to z -→ z. It is a classical fact that Isom + (∆) has involution length 2 with respect to reflections, and 3 with respect to half-turns. Moreover, any pair of orientation-preserving isometries of the Poincaré disk can be decomposed in the form A = s 1 s 2 and B = s 2 s 3 ,

(1.1)

with the s i either all half-turns or all reflections. This makes the description of many properties of the group A, B easier. For instance, when the s i are all orientation-reversing, the group A, B has index two in Γ = s 1 , s 2 , s 3 . In particular, A, B is discrete if and only if Γ is. The decomposition property we are interested in shares many features with this example, as our involutions are antiholomorphic.

In [BM], Basmajian and Maskit have studied this question for any space form, that is for Euclidean space, real hyperbolic space and the sphere. In particular, they prove that any orientation-preserving isometry is a product of two involutions. They also show for instance that pairs in Isom + (H n R ) are generically non decomposable when n 4.

In this paper we study the question of decomposability in the complex hyperbolic plane, which can be seen via a projective model as the unit ball in C 2 equipped with a PU(2,1)-invariant metric. In fact, PU(2,1) is the group of holomorphic isometries of H 2 C ; it is the identity component of Isom (H 2 C ), the other connected component consisting of all antiholomorphic isometries. In particular, PU(2,1) has index two in Isom (H 2 C ) and in this context holomorphicity plays the role of preservation of orientation for space forms. The class of involutions we are interested in consists of antiholomorphic involutions, which are all conjugate in PU(2,1) to the map given in affine ball coordinates by σ 0 : (z 1 , z 2 ) -→ (z 1 , z 2 ).

(1.2)

Clearly, σ 0 fixes pointwise the set of real points of H 2 C , and conjugates of σ 0 by elements of PU(2,1) fix pointwise real planes, which are (totally real) totally geodesic embedded copies of the Poincaré disk. We will refer to these antiholomorphic involutions as real reflections.

It has been known since Falbel and Zocca (see [FZ]) that the involution length of PU(2,1) with respect to real reflections is 2, and this fact has been generalized to all dimensions by Choi [Cho]. The main question we address in this paper is the following.

When is a pair of elements of PU(2, 1) decomposable with respect to real reflections?

We will abbreviate this by saying that a pair (A, B) ∈ PU(2, 1) 2 is R-decomposable (see Definition 2.1). There is no hope that generic pairs of elements in PU(2,1) are R-decomposable. A rough argument for this is the following dimension count. The group PU(2,1) has dimension 8, thus the product PU(2,1)×PU(2,1) has dimension 16. On the other hand, the set of real planes in H 2 C has dimension 5 (to see this note that the stabilizer of the set of real points of the ball is PO(2,1) which has dimension 3). As a real reflection is determined by its fixed real plane, the set of triples of real reflections has dimension 15 (and therefore cannot be diffeomorphic to PU(2, 1) × PU(2, 1)). In fact this count leads us to expect that R-decomposability of a pair of isometries is determined by a single (codimension 1) condition, and this will turn out to be the case.

This question has been examined in [W2], where it was proved that, under the assumption that A and B are loxodromic, the pair (A, B) is R-decomposable provided that the trace of the commutator [A, B] is real. However, this result was obtained as a byproduct of a classification of pairs of elements of PU(2,1) by traces; namely the data (TrA, TrB, TrAB, TrA -1 B) determines the pair (A, B) up to PU(2,1)-conjugation (modulo an order two indetermination, corresponding to the sign of the imaginary part of Tr [A, B]). The present approach is more natural, and in particular it allows us to remove the assumption that A and B are loxodromic.

Our main result (Theorem 4.1) is the following.

Theorem. Let A, B ∈ PU(2, 1) be two isometries not fixing a common point in H 2 C . Then: the pair (A, B) is R-decomposable if and only if the commutator [A, B] has a fixed point in H 2 C whose associated eigenvalue is real and positive.

Note that the eigenvalues of [A, B] do not depend on the choice of lifts of A and B to U(2,1). We now sketch our strategy . Any fixed point p 1 of [A, B] in H 2 C gives rise to a cycle of points

p 1 B -1 -→ p 2 A -1 -→ p 3 B -→ p 4 A -→ p 1 . (1.3)
A key fact is that a real reflection σ decomposes A and B if and only if it satisfies σ(p 1 ) = p 3 and σ(p 2 ) = p 4 (Lemma 4.3). Thus we see that the decomposability of (A, B) is equivalent to a specific symmetry of the above 4-cycle. Now, the existence of this symmetry can be detected via cross-ratios. Goldman has proved that an ideal tetrahedron has this symmetry if and only if the cross-ratio X(p 2 , p 4 , p 1 , p 3 ) is real and positive (Lemma 7.2.1 of [G]). We extend this fact to quadruples of points in H n C (Proposition 3.1). The last ingredient is to connect the cross-ratio of the 4-cycle (1.3) to the eigenvalue of [A, B] associated to the fixed point p 1 . Denoting this eigenvalue by λ 1 , we will see that the product λ 1 • X(p 2 , p 4 , p 1 , p 3 ) is real and positive for any pair (A, B) (this is relation 4.1). This means that λ 1 is real if and only if X(p 2 , p 4 , p 1 , p 3 ) is, and both quantities have the same sign, which gives the result. For the sake of completeness we then analyze what happens in the two special cases where λ 1 is negative (section 4.3), and when A and B have a common fixed point (section 4.2).

• When λ 1 is negative, the fixed point of [A, B] is always on the boundary of H 2

C and the group A, B preserves a complex line. This is an example of a maximal representation of the fundamental group of the once punctured torus, in the sense of [BIW].

• To describe the situation when A and B have a common fixed point, the main ingredient is a detailed description of conditions guaranteeing that a real reflection decomposes a given isometry (this is Proposition 2.5). Most cases already appear in the literature, except when A and B are parabolic with a common fixed point. In particular, when A and B are both 3-step unipotent (see the definitions in section 2.1), we use the relative position of their invariant fans, as defined by Goldman and Parker in [GP].

A classical and difficult question in complex hyperbolic geometry is to determine the discreteness or nondiscreteness of a given finitely generated subgroup Γ of PU(2,1), and to obtain a presentation of Γ. Even when discreteness is known from general results (for instance in the case of arithmetic lattices), finding a presentation of the group is a difficult problem. In fact, the most frequent method is to construct a fundamental domain and use the Poincaré Polyhedron theorem. This is a very technical task which requires a detailed understanding of the action of Γ on H 2 C . There are not so many examples of explicit discrete subgroups of PU(2,1), and most of them are obtained from groups with 2 generators. In this case, the existence of a decomposition as in (1.1) connects an algebraic property of the group (being an index 2 subgroup of a group generated by 3 involutions) to a geometric property (the existence of totally geodesic fixed point sets in a certain configuration). In other words, Γ appears as a reflection group. Such decompositions appeared for instance naturally in [START_REF] Falbel | The moduli space of the modular group in complex hyperbolic geometry[END_REF] or [W3] which studied certain representations of Fuchsian groups in PU(2,1). They were also central in the constructions of fundamental domains for Mostow's lattices ( [M]) in [DFP], as well as for the new non-arithmetic lattices obtained in [ParPau], [Pau], [START_REF] Deraux | Census of the complex hyperbolic sporadic groups[END_REF] and [START_REF] Deraux | A family of new non-arithmetic complex hyperbolic lattices[END_REF].

In all these occurrences, the existence of the real reflections decomposing the generators required some work (part of the detailed geometric construction of the fundamental domains), whereas the concrete criterion given by Theorem -d]) are the simplest kind of arithmetic lattice in SU(2, 1). However their explicit algebraic or geometric properties such as generators, fundamental domains and presentations are still unknown in all but very few cases. (Much is known however concerning their volumes, see [H], [St] and more generally Prasad's volume formula [Pra]). More specifically, presentations and fundamental domains have been obtained for PU(2, 1, O d ) when d = 3 (the so-called Eisenstein-Picard modular group) in [FalPar] and when d = 1 (the Gauss-Picard modular group) in [FFP]. More recently, generators for PU(2, 1, O d ) with d = 2, 7, 11 were given in [Z] (these values of d are the ones for which the ring [Bi] that the Bianchi groups are reflective, i.e. generated by reflections up to finite index, for d 19, d = 14, 17. At the end of the 1980's, Shaiheev extended these results in [Sh], using results of Vinberg, proving in particular that only finitely many of the Bianchi groups are reflective. (The finiteness result now follows from Agol's result, [Ag]). The classification of reflective Bianchi groups was recently completed in [BeMc].

O d is Euclidean). Note that, for d = 3, SU(2, 1, O d ) ≃ PU(2, 1, O d )
We prove that the Picard modular groups PU(2, 1, O d ) with d = 1, 2, 3, 7, 11 are reflective; more precisely that they have a subgroup of index 1,2 or 4 which is generated by real reflections (Corollary 5.1).

The paper is organized as follows. We start with some geometric preliminaries in section 2, then study configurations of points and cross-ratios in section 3. Section 4 contains the statement and proofs of our main results, which we then apply to various discrete subgroups of PU(2,1) in section 5.

Geometric preliminaries

Complex hyperbolic space and isometries

The standard reference for complex hyperbolic geometry is [G]. For the reader's convenience we include a brief summary of key definitions and facts. Our main result concerns the case of dimension n = 2, but the general setup is identical for higher dimensions so we state it for all n 1.

Distance function: Consider C n,1 , the vector space C n+1 endowed with a Hermitian form

• , • of signature (n, 1). Let V -= Z ∈ C n,1 | Z, Z < 0 . Let π : C n+1 -{0} -→ CP n denote projectivization. Define H n
C to be π(V -) ⊂ CP n , endowed with the distance d (Bergman metric) given by:

cosh 2 1 2 d(π(X), π(Y )) = | X, Y | 2 X, X Y, Y (2.1)
Different choices of Hermitian forms of signature (n, 1) give rise to different models of H n C . The two most standard choices are the following. First, when the Hermitian form is given by Z Totally geodesic subspaces and related isometries: A complex k-plane is a projective k-dimensional subspace of CP n intersecting π(V -) non-trivially (so, it is an isometrically embedded copy of H k C ⊂ H n C ). Complex 1-planes are usually called complex lines. If L = π( L) is a complex (n -1)-plane, any v ∈ C n+1 -{0} orthogonal to L is called a polar vector of L. Such a vector satisfies v, v > 0, and we will usually normalize v so that v, v = 1.

, Z = |z 1 | 2 +• • •+|z n | 2 -|z n+1 | 2 , the image of V -under projectivization is the unit ball of C n ,
A real k-plane is the projective image of a totally real (k + 1)-subspace W of C n,1 , i. e. a (k + 1)-dimensional real vector subspace such that v, w ∈ R for all v, w ∈ W . We will usually call real 2-planes simply real planes, or R-planes. Every real n-plane in H n C is the fixed-point set of an antiholomorphic isometry of order 2 called a real reflection or R-reflection. The prototype of such an isometry is the map given in affine coordinates by (z 1 , ..., z n ) → (z 1 , ..., z n ) (this is an isometry provided that the Hermitian form has real coefficients).

An elliptic isometry g is called regular if any of its matrix representatives A ∈ U(n, 1) has distinct eigenvalues. The eigenvalues of a matrix A ∈ U (n, 1) representing an elliptic isometry g have modulus one. Exactly one of these eigenvalues has eigenvectors in V -(projecting to a fixed point of g in H n C ), and such an eigenvalue will be called of negative type. Regular elliptic isometries have an isolated fixed point in H n C . A non regular elliptic isometry is called special. Among the special elliptic isometries are the following two types (which exhaust all special elliptic types when n = 2):

1. A complex reflection is an elliptic isometry g ∈ PU(n, 1) whose fixed-point set is a complex (n -1)-plane.

In other words, any lift such an isometry to U(n,1) has n equal eigenvalues, one of which has negative type.

2. A complex reflection in a point is an elliptic isometry having a lift with n equal eigenvalues, the remaining one being of negative type. In other words, such an isometry is conjugate to some λId ∈ U (n), where U (n) is the stabilizer of the origin in the ball model. Complex reflections in a point with order 2 are also called central involutions; these are the symmetries that give H n C the structure of a symmetric space.

A parabolic isometry is called unipotent if it has a unipotent lift in U(n, 1). If not, it is called screw-parabolic, and it can be uniquely decomposed as g = pe = ep with p unipotent and e elliptic (see Theorem 2.2 below). In dimensions n > 1, unipotent isometries are either 2-step or 3-step, according to whether the minimal polynomial of their unipotent lift is (X -1) 2 or (X -1) 3 (see section 3.4 of [ChGr]).

Models in dimension 2

2.2.1 The ball model of H 2

C

The ball model of H 2 C arises from the choice of Hermitian form

H =   1 0 0 0 1 0 0 0 -1   .
It is classical, and we refer the reader to chapter 3 of [G]. We only emphasize the following fact: any elliptic isometry of H 2 C is conjugate to one given in ball coordinates by (z 1 , z 2 ) → (e iα z 1 , e iβ z 2 ) for some α, β ∈ R/Z. A matrix representative in SU(2,1) for the latter is:

E (α,β) =   e i(2α-β)/3 0 0 0 e i(2β-α)/3 0 0 0 e -i(α+β)/3   . (2.2) 2.2.2 The Siegel model of H 2

C

In the presence of parabolic elements, it is very convenient to use the Siegel domain, as the stabilizer of the point at infinity (see below) consists of upper triangle matrices. As mentioned in the previous section, this model corresponds to the Hermitian form given by the matrix:

H =   0 0 1 0 1 0 1 0 0  
In this model, any point m ∈ H 2 C admits a unique lift to C 2,1 of the following form, called its standard lift :

m =   (-|z| 2 -u + it)/2 z 1   with (z, t, u) ∈ C × R×]0, ∞[. (2.3)
The triple (z, t, u) is called the horospherical coordinates of m. The boundary of H 2 C is the level set {u = 0}, together with the distinguished point at infinity, given by

q ∞ ∼   1 0 0   .
Level sets {u = u 0 } with fixed u 0 > 0 are called horospheres based at q ∞ . The boundary ∂H 2 C \ {q ∞ } is a copy of the Heisenberg group N of dimension 3, with group law given in [z, t] coordinates by:

[z 1 , t 1 ] • [z 2 , t 2 ] = [z 1 + z 2 , t 1 + t 2 + 2Im(z 1 z 2 )].
(2.4)

The stabilizer of q ∞ in SU(2, 1) consists of upper triangular matrices, and is generated by the following 3 types of isometries: Heisenberg translations T [z,t] ((z, t) ∈ R × C), Heisenberg rotations R θ (θ ∈ R/2πZ) and Heisenberg dilations D r (r > 0), where:

T [z,t] =   1 -z -(|z| 2 -it)/2 0 1 z 0 0 1   R θ =   e -iθ/3 0 0 0 e 2iθ/3 0 0 0 e -iθ/3   D r =   r 0 0 0 1 0 0 0 1/r   .
(2.5)

In Heisenberg coordinates, they correspond respectively to the following transformations

• T [z,t] is the left multiplication by [z, t],

• R θ is given by [w, s] -→ [e iθ w, s],

• D r is the Heisenberg dilation [w, s] -→ [rw, r 2 s].
Note that Heisenberg translations and rotations preserve each horosphere based at q ∞ whereas Heisenberg dilations permute horospheres based at q ∞ . We will denote by Isom(N) the non-loxodromic stabilizer of q ∞ in SU(2, 1), which is generated by the T [z,t] and R θ . The notation Isom(N) comes from the fact it is the isometry group of the Cygan metric, which we will not use here (see [FalPar]). The group Isom(N) consists exactly of those matrices of the form:

P (z,t,θ) = T [z,t] R θ =   e -iθ/3 -e 2iθ/3 z -e -iθ/3 (|z| 2 -it)/2 0 e 2iθ/3 e -iθ/3 z 0 0 e -iθ/3   (2.6)
It is sometimes more convenient to work with the lift of P (z,t,θ) to U(2,1) given by

  1 -ze iθ -(|z| 2 -it)/2 0 e iθ z 0 0 1   (2.7)
In terms of these parameters the various parabolic conjugacy classes are easily described. Any parabolic isometry is conjugate in PU(2,1) to exactly one of the following:

• P (1,0,0) = T [1,0] if it is 3-step unipotent, • P (0,1,0) if it is 2-step unipotent, • P (0,1,θ) for some non-zero θ ∈ R/2πZ if it is screw parabolic.
Remark 1. Screw and 2-step unipotent parabolics share the property of preserving a (unique) complex line. In the case of P (0,1,θ) , it is the line polar to the vector 0 1 0

T . As we will see in the next section, the situation is slightly more complicated for 3-step unipotent parabolics.

Finally, recall from [FalPar] for future reference that the exact sequence:

1 -→ R -→ N Π -→ C → 1 (2.8)
induces an exact sequence:

1 -→ Isom + (R) -→ Isom(N) Π * -→ Isom + (C) → 1.
(2.9)

Explicitly: Π * (P (z,t,θ) ) = e iθ z 0 1 , acting on C by w → e iθ w + z.

The invariant fan of a 3-step unipotent parabolic

The Siegel model is very well adapted to describing the action of parabolic isometries. To do so, we give a few more details on the structure of the boundary of H 2 C . It is equipped with a (CR) contact structure, which is given in Heisenberg coordinates as the kernel of the 1-form α = dt -2xdy + 2ydx.

We will denote by C m the contact plane at a point m ∈ ∂H 2 C . For any real plane R of H 2 C , the intersection R∩∂H 2 C is a closed curve, which is called an R-circle (see [G]). These curves are Legendrian: they are everywhere tangent to the contact distribution. The following lemma concerns those R-circles containing the point q ∞ , called infinite R-circles. We refer to chapter 4 of [G] for proofs.

Lemma 2.1.

1. Let R be a real plane containing the point q ∞ of the Siegel model. Then in Heisenberg coordinates, the R-circle ∂R is an affine line in the Heisenberg group, which is contained in the contact plane at any of its points, and the restriction to the boundary of the reflection about R is the (Euclidean) half-turn about this affine line.

2. More precisely, an affine line in the Heisenberg group is an infinite R-circle if and only if it is contained in the contact plane at one of its points.

Viewing the boundary of H 2 C as the one point compactification of the Heisenberg group N = C ⋉ R endows the set of complex lines through q ∞ with the stucture of the affine space C. Indeed, for any z ∈ C there exists a unique complex line through q ∞ which contains q ∞ and the point [z, 0] (which is polar to the vector z 1 0

T ). This induces a projection Π : H 2 C \ q ∞ -→ C whose fibers are the complex lines through q ∞ . In restriction to the boundary, this projection is just the vertical projection Π defined in Equation (2.8), which is given in Heisenberg coordinates by: Π

: [z, t] -→ [z, 0].
(2.10)

A fan through q ∞ is the preimage of any affine line in C under the projection Π. In view of (2.10), if L is an affine line in C, the fan Π-1 (L) intersects the boundary of H 2

C along the vertical plane containing L in the Heisenberg group. A general fan is the image of a fan through q ∞ by an element of PU(2,1). These objects were defined by Goldman and Parker in [GoP] (see also chapter 4 of [G]). As stated in [GoP], fans enjoy a double foliation, by real planes and complex lines. We now make this foliation explicit in the case of a fan F through q ∞ in H 2 C , with ∂F projecting to an affine line L ⊂ C ⊂ ∂H 2 C : 1. First, the foliation by complex lines is given by the fibers of Π above the affine line L. In the boundary, this foliation correspond to the foliation of the vertical plane above L by vertical lines.

2. Consider a point m in the vertical plane ∂F . The contact plane at m intersects ∂F along an affine line L ′ = C m ∩ ∂F in the Heisenberg group, which projects vertically onto L. By Lemma 2.1, the line L ′ is the boundary of a real plane. Then ∂F is foliated by the family of lines parallel to L ′ contained in ∂F .

In other words, the foliation of ∂F is obtained by taking all lifts of L to the contact structure. All these lines are boundaries of real planes, and this foliation of ∂F extends inside H 2 C as a foliation of F by real planes.

We can be a bit more explicit.

Lemma 2.2. Let L w,k be the affine line in C parametrized by L w,k = {w(s + ik), s ∈ R}, for some unit modulus w and k ≥ 0. Then the boundary foliation of the fan above L w,k is given by the lines parametrized in Heisenberg coordinates by

L t0 = {[w(s + ik), t 0 + 2sk], s ∈ R}.
Proof. The lines L t0 all project onto L by the vertical projection. A tangent vector to L t0 is given by (Re(w), Im(w), 2k). Evaluating the 1-form α on this vector at the point of parameter s = 1, and using |w| = 1 shows that the line L t0 is in the kernel of α at this point, thus in the contact plane. Therefore L t0 is an R-circle by Lemma 2.1.

Our interest in fans comes from the following fact.

Proposition 2.1. Let P be a 3-step unipotent parabolic isometry of H 2 C . There exists a unique fan F P through the fixed point of P such that 1. The fan F P is stable under P .

2. Every leaf of the foliation of F P by real planes is stable under P . Moreover, when P fixes q ∞ and thus corresponds to left-translation by [z, t] with z = 0 in Heisenberg coordinates, the fan F P is the one above the affine line L w,k , where w = z/|z|, and k = t/(4|z|).

Given a 3-step unipotent parabolic isometry P , we will refer to the fan F P as the invariant fan of P , even though all the fans corresponding to distinct parallel vertical planes are stable under P (these are not stable leaf by leaf).

Proof. First assume that P = T [1,0] , acting on the Heisenberg group as

[z, t] -→ [z + 1, t -2Im(z)].
Then every vertical plane Im(z) = k is globally preserved. The real foliation of the fan corresponding to the vertical plane Im(z) = k is given by the family of lines

L t0 = {[s + ik, t 0 + 2sk], s ∈ R}. Since [1, 0] • [s + ik, t 0 + 2sk] = [s + 1 + ik, t 0 + 2(s -1)k],
we see that the real foliation of the vertical plane Im(z) = k is preserved by P if and only if k = 0, which gives the result for this normalization of P . The first part of the proposition is then obtained by using the fact that any 3-step unipotent parabolic is conjugate in PU(2,1) to T [1,0] . To check the last part, write w = z/|z| and k = t/(4|z|); then by a direct calculation

[z, t] • [w(s + ik, t 0 + 2sk] = [w(|z| + s + ik), t 0 + t + 2sk -2|z|k] = [w(|z| + s + ik), t 0 + 2(|z| + s)k],
which proves that the real leaves of the fan above L w,k are preserved.

In fact the proof of Proposition 2.1 gives us a little more information, which we summarize in the following corollary.

Corollary 2.1. Let P be a 3-step unipotent parabolic.

1. A real plane is stable under P if and only if it is a leaf of the real foliation of its invariant fan.

2. P is characterized by its invariant fan F , and its restriction to F .

For future reference, let us state the following proposition, which is just gathering together what we just exposed.

Proposition 2.2. Let P be a parabolic isometry in PU(2, 1).

1. If P is screw parabolic or 2-step unipotent, it has a unique invariant complex line.

2. If P is 3-step unipotent parabolic, then there exists a unique fan F centered at the fixed point of P which is stable by P and such that every leaf of the foliation of F by real planes is stable by P .

We will also use the following characterization of commuting parabolic isometries:

Lemma 2.3. Let P 1 and P 2 be two parabolic isometries fixing q ∞ . Then P 1 and P 2 commute if and only if one of the following possibilities occurs.

1. Both P 1 and P 2 are either 2-step unipotent or screw parabolics with the same stable complex line.

2. Both P 1 and P 2 are 3-step unipotent with the same fixed point and their invariant fans intersect ∂H 2 C along parallel vertical planes.

Proof. We only prove the second part, the first one being classical. Two 3-step unipotent maps fixing q ∞ are respectively conjugate to the left translations by [z 1 , t 1 ] and [z 2 , t 2 ]. Using (2.4), we see that these two translations commute if and only if z 2 z 1 ∈ R, which is equivalent to saying that their invariant fans are parallel.

Eigenvalues and traces

The following classification of conjugacy classes in U(n, 1) is due to Chen-Greenberg (Theorem 3.4.1 of [ChGr], where the real and quaternionic cases are treated as well):

Theorem 2.2 (Chen-Greenberg). (a) Any elliptic element is semisimple, with eigenvalues of norm 1. Two elliptic elements are conjugate in U(n, 1) if and only if they have the same eigenvalues and the same eigenvalue of negative type.

(b) Any loxodromic element is semisimple, with exactly n-1 eigenvalues of norm 1. Two loxodromic elements are conjugate in U(n, 1) if and only if they have same eigenvalues.

(c) Any parabolic element is not semisimple, and all its eigenvalues have norm 1. It has a unique decomposition g = pe = ep with p strictly parabolic and e elliptic. Two parabolic elements are conjugate in U(n, 1) if and only if their strictly parabolic and elliptic components are conjugate.

(d) There are 2 classes of strictly parabolic elements for n > 1, the vertical Heisenberg translations and the non-vertical Heisenberg translations.

Note that two elliptic elements may be conjugate in GL(n + 1, C) but not in U(n, 1) (if they have the same eigenvalues but different eigenvalues of negative type). It will be useful for future reference to have matrix representatives in SU(2,1) of the different conjugacy classes of isometries when n = 2.

Any elliptic isometry of H 2

C is conjugate to one given in the ball model by the matrix E (α,β) given in (2.2).

Any loxodromic isometry of H 2

C is conjugate to one given in the Siegel model by the matrix D r R θ where D r and R θ are as in (2.5).

Any parabolic isometry of H 2

C is conjugate to one given in the Siegel model by the matrix P (z,t,θ) = T [z,t] R θ . The isometry associated to P (z,t,θ) is unipotent if and only if θ = 0 and 2-step unipotent if and only if θ = 0 and z = 0.

As in the classical case of the Poincaré disc, the isometry type of an isometry is closely related to the trace of a lift to SU(2,1). The characteristic polynomial of a matrix A in SU(2,1) is given by:

χ A (X) = X 3 -z • x 2 + z • x -1, where z = TrA.
(2.11)

Computing its discriminant, we obtain:

f (z) = Res(χ A , χ ′ A ) = |z| 4 -8Re(z 3 ) + 18|z| 2 -27.
(2.12)

This function provides the following classification of holomorphic isometries via the trace of their lifts to SU(2,1) (see ch. 6 of [G]), which is analogous to the classical SL(2,C) case. Denote by C 3 the set of cube roots of unity in C.

Theorem 2.3 (Goldman). Let A ∈ SU(2, 1) and g ∈ PU(2, 1) the corresponding isometry of H 2 C . Then:

• g is regular elliptic ⇐⇒ f (Tr(A)) < 0. • g is loxodromic ⇐⇒ f (Tr(A)) > 0.
• g is special elliptic or screw-parabolic ⇐⇒ f (Tr(A)) = 0 and Tr(A) / ∈ 3C 3 .

• g is unipotent or the identity ⇐⇒ Tr(A) ∈ 3C 3 .

The null-locus of the polynomial f can be seen in Figure 1. We now focus on the special case of elements of SU(2,1) having real trace.

Proposition 2.3. Let A ∈ SU(2, 1) satisfy TrA ∈ R. Then A has an eigenvalue equal to 1. More precisely:

• If A is loxodromic then A has eigenvalues {1, r, 1/r} for some r ∈ R \ [-1, 1].
Figure 1: The null-locus of the polynomial f inscribed in the circle of radius 3 centered at the origin

• If A is elliptic then A has eigenvalues {1, e iθ , e -iθ } for some θ ∈ [0, π]. • If A is parabolic then A has eigenvalues {1, 1, 1} or {1, -1, -1}
Proof. If TrA is real, then χ A has real coefficients, therefore the eigenvalue spectrum of A is stable under complex conjugation. The result follows.

Remark 2. (a) Any loxodromic or parabolic element of SU(2,1) with real trace is conjugate to its inverse; for elliptic isometries this is true under the additional assumption that the eigenvalue 1 has negative type. This follows from Proposition 2.2. An element of a group which is conjugate to its inverse is sometimes called achiral or reversible in the context of isometry groups. In [GonP], Gongopadhyay and Parker have studied and classified these isometries in PU(n, 1) for all n 1.

(b) For later use, let us note that matrix representatives of the conjugacy classes of elements of PU(2,1) having a fixed point in H 2 C with associated positive eigenvalue are given by, in the notation of (2.2) and (2.5): • E (θ,-θ) with θ ∈ R (in the ball model) for elliptic classes,

• D r with r > 1 (in the Siegel model) for loxodromic classes, and

• P (z,t,0) with z ∈ C and t ∈ R (in the Siegel model) for parabolic classes.

Antiholomorphic isometries

The following lemma is useful when computing with antiholomorphic isometries (see [FalPau] in the elliptic case, where the corresponding matrix was called a Souriau matrix for f ). To simplify the statement, we consider a projective model (for example the ball or Siegel model) for which complex conjugation in affine coordinates σ 0 : (z 1 , ..., z n ) → (z 1 , ..., z n ) is an isometry (i.e. the Hermitian form has real coefficients).

Lemma 2.4. Let f be an antiholomorphic isometry of H n C . Then there exists M ∈ U(n, 1) such that for any

m ∈ H n C with lift m ∈ C n+1 : f (m) = M • m.
(2.13)

Any matrix M ∈ U(n, 1) with this property will be called a lift of f .

Proof. f •σ 0 is a holomorphic isometry, i.e. corresponds to an element of PU(n,1). Any of its lifts M ∈ U(n, 1) satisfies the required property.

Definition 2.1.

• Given a real reflection σ and an isometry A ∈ PU(n, 1), we say that σ decomposes A if A = στ for some real reflection τ (equivalently, A = τ ′ σ where τ ′ = στ σ is also a real reflection).

• Given two isometries A, B ∈ PU(n, 1), we say that the pair (A, B) is R-decomposable if there exists a real reflection which decomposes both A and B.

Note that when writing the two isometries A and B as products of real reflections, the order in which the reflections appear is not important. As an application of Lemma 2.4, we obtain a necessary condition on a pair (A, B) to be R-decomposable, namely that the commutator [A, B] must have real trace. Note that the trace of an element of PU(n,1) is not well-defined in general. However, if A and B are in PU(n,1), then the matrix [ Ã, B] does not depend on the choice of lifts à and B made for A and B. This allows us to consider the condition of having real trace for a commutator.

Lemma 2.5. If a pair (A, B) ∈ PU(n, 1) × PU(n, 1) is R-decomposable, then the commutator [A, B] is the square of an anti-holomorphic isometry.

Proof.

If A = σ 1 σ 2 and B = σ 2 σ 3 with each σ i a real reflection, then [A, B] = (σ 1 σ 2 )(σ 2 σ 3 )(σ 2 σ 1 )(σ 3 σ 2 ) = (σ 1 σ 3 σ 2 ) 2 , where σ 1 σ 3 σ 2 is an antiholomorphic isometry. Corollary 2.4. If a pair (A, B) ∈ PU(n, 1) × PU(n, 1) is R-decomposable, then: Tr([A, B]) ∈ R.
Proof. First note that if f 1 and f 2 are antiholomorphic isometries with lifts M 1 and M 2 , then f 1 • f 2 is holomorphic with lift M 1 M 2 ∈ U(n, 1). Using this fact together with Lemma 2.5, we see that if (A, B) is R-decomposable, then the commutator can be lifted to SU(n,1) as M M , with

M = M 1 M 3 M 2 , where M k is a lift of σ k . As a consequence, we see that Tr(M M ) = Tr(M M ) = Tr(M M ) ∈ R
Note that this necessary condition holds in any dimension n. However, to obtain a sufficient condition, we will need Lemma 2.4 below, which is false in dimensions n > 2.

Lemma 2.6. Any antiholomorphic isometry of H 1 C having a fixed point in H 1 C is a real reflection. Proof. Let f be an antiholomorphic isometry of H 1 C with a fixed point p. Then, for any point q = p, the angles (f -1 (q), p, q) and (q, p, f (q)) are opposite (because f is antiholomorphic), therefore f -1 (q) = f (q) (because f is an isometry), and f is an involution and fixes pointwise a geodesic.

Lemma 2.7. If g is an antiholomorphic isometry of H 2 C which exchanges two points of H 2 C , then g is an R-reflection.

Proof. Let p and q be the two points exchanged by g. Then g has a fixed point m on the geodesic (pq) (which is the midpoint of the geodesic segment [pq] when p and q are in H 2 C ). Then g stabilizes the complex line C spanned by p and q, as well as the complex line C ′ orthogonal to C at m. By Lemma 2.6, the restrictions of g to C and C ′ are real reflections, fixing geodesics γ and γ ′ . Now γ and γ ′ are geodesics contained in orthogonal complex lines, therefore they span a Lagrangian plane, which is fixed pointwise by g.

Lemma 2.7 is false as soon as n ≥ 3. Indeed, consider an element M ∈ SU(n,1) given in the Siegel model by

M =   (0) z U 1/z (0)   , (2.14)
where z ∈ C ⋆ and U ∈ U (n -1) is a matrix such that U U = Id, that is a non-symmetric matrix in U (n -1) (such matrices only exist when n > 2). Then

M M =   1 (0) U U (0) . . . 1   = Id
Therefore the antiholomorphic isometry assciated with M is not a real reflection, though it exchanges the two points corresponding to the first and last vectors in the canonical basis of C n+1 , which are both on the boundary of H n C .

The following proposition describes which elements of PU(2,1) can arise as squares of antiholomorphic isometries.

Proposition 2.4. Let A ∈ PU(2, 1) be an isometry admitting a lift à to SU(2, 1) such that for any fixed point A in H 2 C , the corresponding eigenvalue of à is positive. Then A is the square of an antiholomorphic isometry unless A is 2-step unipotent parabolic.

Proof. Being the square of an antiholomorphic isometry is preserved by conjugation, therefore we only need to prove the result for representatives of each conjugacy class. We first provide matrices M such that M M is equal to the lifts listed in Remark 2(b) in the elliptic and loxodromic cases.

1. If A is elliptic the following M satisfies the above condition:

M =   0 e iθ/2 0 e -iθ/2 0 0 0 0 1   2.
If A is loxodromic then we can take:

M =   √ r 0 0 0 1 0 0 0 1/ √ r   .
We now examine the case where A is parabolic, thus unipotent in view of the assumptions. Assume that A is unipotent and that A = φ 2 with φ an antiholomorphic isometry. First, φ has at least one fixed point in the closure of H 2 C and φ 2 has exactly one fixed point there. This implies that φ has only one fixed point, which is the same as that of A. Conjugating by an element of PU(2,1), we can assume that this fixed point is q ∞ . Consider a lift M of φ to SU(2,1) in the sense of Lemma 2.4. The fact that M M is a unipotent map fixing q ∞ , which corresponds to the vector 1 0 0

T implies that M is an upper triangular matrix in SU(2,1) with unit modulus diagonal entries. Therefore M is of the form P (z,t,θ) . Computing M M = P (z,t,θ) P (z,-t,-θ) , we obtain that A has a lift to SU(2,1) of the form:

  1 -z + ze -iθ -z z + ze iθ 0 1 z + ze iθ 0 0 1   (2.15)
This matrix can only be 3-step unipotent or the identity. This proves the result as there is only one PU(2,1)conjugacy class of 3-step unipotents.

Decomposing isometries

The following proposition summarizes results characterizing which real reflections decompose a given holomorphic isometry. Parts (e1-e3) are Proposition 2.4 of [FalPau], part (l) is Proposition 4(2) of [W2] (and follows from Proposition 3.1 of [FZ]), and part (p) is new:

Proposition 2.5. Let A ∈ PU(2, 1) and σ a real reflection with fixed R-plane L.

(e1) If A is a complex reflection with fixed complex line C, then: σ decomposes A if and only if L ∩ C is a geodesic.

(e2) If A is a complex reflection in a point p A , then: σ decomposes A if and only if p A ∈ L.

(e3) If A is regular elliptic with fixed point p A and stable complex lines C 1 , C 2 , then: σ decomposes A if and only if p A ∈ L and L ∩ C i is a geodesic for i = 1, 2.

(l) If A is loxodromic then: σ decomposes A if and only if σ exchanges the 2 fixed points of A.

(p) If A is parabolic the situation is as follows.

1. If A is screw parabolic or 2-step unipotent with fixed point p and stable complex line L then σ decomposes A if and only if σ(p) = p and σ(L) = L.

2. If A is 3-step unipotent with fixed point p and invariant fan F (see Proposition 2.2), then σ decomposes A if and only if σ(p) = p, σ(F ) = F , and the restriction of σ to ∂F is a half-turn.

Proof. Parts (e1-e3) are Proposition 2.4 of [FalPau], part (l) is Proposition 4(2) of [W2] (and follows from Proposition 3.1 of [FZ]). Let us prove part (p).

Denote by p the fixed point of A. Assume that A = σσ ′ where σ ′ is another real reflection. Because both σ and σ ′ are involutions, either σ and σ ′ fix p, or there exists q = p in ∂H 2

C such that σ and σ ′ both swap p and q. In this case, A would fix two distinct points in ∂H 2 C , which is not possible for a parabolic. Thus the fixed real plane of σ contains p. Replacing p by V and using the fact that V is the unique stable complex line or invariant fan of A, we obtain that σ and σ ′ both preserve V .

Conversely, let σ be such a real reflection.

1. If A is 2-step unipotent or screw parabolic, then V is a complex line. The restriction of σ to V is an involution fixing a boundary point of V . As V is a copy of the Poincaré disk, this implies that the restriction σ |V is a symmetry about a geodesic, and therefore L intersects V along a geodesic γ, one of whose endpoints is the fixed point of A. Let us call a the other endpoint of γ, and b = A(a). It is a simple exercise to check that A • σ exchanges a and b. Therefore A • σ, which is an antiholomorphic isometry, exchanges two points of ∂H 2 C . By Lemma 2.7, it is a real reflection and therefore σ decomposes A. 2. If A is 3-step unipotent, we may assume by conjugating that A = T [1,0] ; then the boundary of its invariant fan F A is the vertical plane {y = 0}, on which A acts as a horizontal translation by 1. In view of Lemma 2.8, the fixed R-circle of σ is either a horizontal line {t = t 0 } contained in the plane ∂F A , or an infinite R-circle orthogonal to this vertical plane.

In the first case σ • A acts on ∂F A by (x, t) -→ (x + 1, t 0 -t) which is not an involution, and therefore σ cannot decompose A. In the second case, the restriction of σ to ∂F A is a half-turn, and so is the restriction of σ • A. The latter is also the restriction to ∂F A of a reflection σ ′ about a infinite R-circle; then A = σ • σ ′ by Corollary 2.1, as A and σ • σ ′ agree on their common invariant fan.

The following lemma describes real reflections that preserve a given fan. We state it only for fans through q ∞ , as this is all we will need. Let F be a fan through q ∞ and (L t ) t∈R be the foliation of its boundary by infinite R-circles. Let L be an infinite R-circle not contained in or parallel to ∂F , and L t0 be the unique leaf of ∂F such that L ∩ L t0 is nonempty. We will say that L and F are orthogonal whenever L and L t0 are orthogonal in the contact plane at the point L ∩ L t0 . It is a direct consequence of Lemma 2.1 that the reflection about an infinite R-circle preserves any fan which is orthogonal to it.

Lemma 2.8. Let F be a fan through q ∞ . Let σ be the real reflection about a real plane R. Then σ preserves F if and only R contains the point q ∞ on its boundary and one of the following occurs:

1. The real plane R is one of the leaves of F , in which case σ acts on the vertical plane ∂F as the reflection across the affine line ∂R.

2. The real plane R intersects the Heisenberg group along an affine line orthogonal to the vertical plane ∂F .

In that case, it acts on ∂F as a Euclidean half-turn.

Proof. First, σ fixes the point q ∞ . Indeed, as it preserves F , σ preserves globally the (singular) foliation of F by real planes and therefore fixes the intersection of all leaves, which is q ∞ . This means that R contains q ∞ , and that the corresponding R circle is infinite. Conjugating by an element of Isom(N), we can assume that σ is the real reflection about the real plane H 2 R ∩ H 2 C , which acts on the boundary by σ([z, t]) = [z, -t]. A vertical plane is stable under σ if and only if it is the vertical plane {y = 0}, in which case R is a leaf of the corresponding fan, or a plane orthogonal to the x-axis, which corresponds to the second case.

3 Configurations of points and cross-ratios 3.1 Triples of points Definition 3.1. Given a triple (p 1 , p 2 , p 3 ) of distinct points in H n C ∪ ∂H n C and lifts P i ∈ C n,1 of the p i , the ratio T (p 1 , p 2 , p 3 ) = P 1 , P 2 P 2 , P 3 P 3 , P 1 P 1 , P 3 P 3 , P 2 P 2 , P 1 (3.1)

does not depend on the lifts P i . We will call T (p 1 , p 2 , p 3 ) the triple-ratio of (p 1 , p 2 , p 3 ).

Note that T (p 1 , p 2 , p 3 ) is also well-defined if 2 or more of the points are equal in H n C (but not in ∂H n C ). Observe that holomorphic isometries (elements of PU(n,1)) clearly preserve the triple-ratio, whereas for any antiholomorphic isometry g, we have T (g(p 1 ), g(p 2 ), g(p 3 )) = T (p 1 , p 2 , p 3 ). The triple-ratio is related to the classical Cartan angular invariant A (see [C]) and Brehm's shape invariant σ (see [Br]) for triangles as follows.

• The Cartan angular invariant of three points p 1 , p 2 , p 3 ∈ ∂H n C is defined as: A(p 1 , p 2 , p 3 ) = arg(-P 1 , P 2 P 2 , P 3 P 3 , P 1 ).

It relates to the triple-ratio by

T (p 1 , p 2 , p 3 ) = e 2iA(p1,p2,p3) .

• Brehm's shape invariant σ of three points in H n C is related to the normalized triple product T (p 1 , p 2 , p 3 ) = P 1 , P 2 P 2 , P 3 P 3 , P 1 P 1 , P 1 P 2 , P 2 P 3 , P 3 .

Namely, σ = -Re( T ). Note that:

T (p 1 , p 2 , p 3 ) = T (p 1 , p 2 , p 3 ) T (p 1 , p 3 , p 2 ) .
We refer the reader to chapter 7 of [G] for classical properties of the Cartan invariant. Note in particular that the Cartan invariant satisfies A(p 1 , p 2 , p 3 ) ∈ [-π/2, π/2], and that A(p 1 , p 2 , p 3 ) = ±π/2 (resp. A(p 1 , p 2 , p 3 ) = 0) if and only if the three points are contained in a complex line (resp. a real plane). Also, the Cartan invariant classifies triples of pairwise distinct points in ∂H n C up to holomorphic isometries.

The following classification of triples of points in H n

C is due to Brehm ([Br]). Theorem 3.1. (Brehm) Let (x 1 , x 2 , x 3 ), (y 1 , y 2 , y 3 ) be two triples of points in H n C . There exists g ∈ Isom(H n C ) such that g(x i ) = y i (for i = 1, 2, 3) if and only if:

• d(x 1 , x 2 ) = d(y 1 , y 2 ), d(x 2 , x 3 ) = d(y 2 , y 3 ), d(x 1 , x 3 ) = d(y 1 , y 3 ) and T (x 1 , x 2 , x 3 ) = T (y 1 , y 2 , y 3 ), in which case g is holomorphic, or • d(x 1 , x 2 ) = d(y 1 , y 2 ), d(x 2 , x 3 ) = d(y 2 , y 3 ), d(x 1 , x 3 ) = d(y 1 , y 3 ) and T (x 1 , x 2 , x 3 ) = T (y 1 , y 2 , y 3 ), in which case g is antiholomorphic.
In fact Brehm's formulation is slightly different as he considers -Re( T ) instead of T (so his statement doesn't include our 2 cases). This is equivalent because the norm of T is determined by the 3 side-lengths.

The complex cross-ratio

The following definition is due in this form to Goldman ([G]) (following Koranyi and Reimann ([KR])) in the case of boundary points:

Definition 3.2. Let (p 1 , p 2 , p 3 , p 4 ) be a quadruple of distinct points in H n C ∪ ∂H n C .
The quantity defined by X(p 1 , p 2 , p 3 , p 4 ) = P 3 , P 1 P 4 , P 2 P 4 , P 1 P 3 , P 2 (3.2)

does not depend on the choice of lifts P i of the p i 's, and is called the complex cross-ratio of (p 1 , p 2 , p 3 , p 4 ).

Note that X(p 1 , p 2 , p 3 , p 4 ) is also well-defined when some of the 4 points coincide, as long as at most 2 of them coincide in ∂H n C . The complex cross-ratio of boundary points has been studied in detail in [G] (pp. 224-228), to which we refer the reader for more details. As for the triple-ratio, it is a direct observation that holomorphic isometries preserve X whereas antiholomorphic ones change it to its complex conjugate. One of our main tools will be finding conditions under which such a cross-ratio is real, in the spirit of the following result (Theorem 7.2.1 of [G]): Theorem 3.2 (Goldman). Let (p 1 , p 2 , p 3 , p 4 ) be a quadruple of distinct points in ∂H n C . Then X(p 1 , p 2 , p 3 , p 4 ) is real and positive if and only if there exists a real reflection φ such that φ : p 1 ↔ p 2 and p 3 ↔ p 4 .

Note that, if there exists such a real reflection φ, then:

X(p 1 , p 2 , p 3 , p 4 ) = X(φ(p 1 ), φ(p 2 ), φ(p 3 ), φ(p 4 )) = X(p 2 , p 1 , p 4 , p 3 ) (3.3)
Going back to the definition of X, it is straightforward that X(p 2 , p 1 , p 4 , p 3 ) = X(p 1 , p 2 , p 3 , p 4 ), and we see that the condition that X ∈ R is indeed necessary. In [G] the assumption that X > 0 is omitted, but it must be added for the following reason. X is related to triple products by: X(p 1 , p 2 , p 3 , p 4 ) = P 1 , P 2 P 2 , P 3 P 3 , P 1 P 1 , P 2 P 2 , P 4 P 4 , P 1

• | P 4 , P 2 | 2 | P 3 , P 2 | 2 (3.4)
Because the Cartan invariant belongs to [-π/2, π/2], we see that if X is real and negative the two triple products P 1 , P 2 P 2 , P i P i , P 1 (i = 3, 4) must have arguments either both equal to π/2 or both equal to -π/2. This means that p 3 and p 4 belong to the complex line spanned by p 1 and p 2 , and are on the same side of the geodesic (p 1 p 2 ). See Proposition 2 of [KR] and property # 7 on p. 226 of [G]. However, if p 1 , p 2 , p 3 , p 4 are in such a configuration then there cannot exist a real reflection φ such that φ : p 1 ↔ p 2 and p 3 ↔ p 4 . Indeed, if a real reflection preserves a complex line then it acts on it by reflection in a geodesic.

The following basic observation will allow us to project orthogonally onto the complex sides of the quadrilateral (p 1 , p 2 , p 3 , p 4 ).

Lemma 3.1. Let p 1 , p 2 , p 3 , p 4 ∈ H n C with p 1 = p 2 , and let π 12 denote orthogonal projection onto the complex line L 12 spanned by p 1 and p 2 . Then: X(p 1 , p 2 , p 3 , p 4 ) = X (p 1 , p 2 , π 12 (p 3 ), π 12 (p 4 )).

Proof. Let c 12 be a polar vector for L 12 , normalized so that c 12 , c 12 = 1. Then π 12 is the projectivization of the linear projection in C 2,1 given by: Π 12 (z) = z -z, c 12 c 12 . Then, for any point p in H n C ∪ ∂H n C :

Π 12 (p), p i = p -p, c 12 c 12 , p i = p, p i (i = 1, 2).

The result follows by substituting p 3 and p 4 in this expression. (Here and whenever it is convenient we will slightly abuse notation by using the same letter for points in CP n and their lifts to C n,1 ; we will however insist that lifts of points inside H n C have norm -1 and lifts of points outside H n C have norm 1.)

Cross-ratios and real reflections

Recall that the classical cross-ratio of 4 distinct points in CP 1 = C ∪ {∞} is defined by (see for instance [G]):

[z 1 , z 2 ; z 3 , z 4 ] = (z 4 -z 1 )(z 3 -z 2 ) (z 4 -z 2 )(z 3 -z 1 ) (3.5)
It is invariant under the diagonal action of PGL(2, C), and is real if and only if the 4 points are cocylic or collinear. Moreover, [z 1 , z 2 ; z 3 , z 4 ] is positive if and only if the pairs (z 1 , z 2 ) and (z 3 , z 4 ) do not separate each other on the common circle/line. The KR cross-ratio generalizes the classical cross-ratio in the following sense: Lemma 3.2. If p 1 , p 2 , p 3 , p 4 lie in a common complex line C ⊂ CP n , then

X(p 1 , p 2 , p 3 , p 4 ) = [σ(p 1 ), σ(p 2 ); p 4 , p 3 ]
where σ denotes inversion in the boundary circle of C.

Proof. By applying an element of PU(n,1), we may assume that the complex line C containing the p i is the first coordinate axis of C n (seen as an affine chart of CP n ) in the ball model. Each point of C has a lift to C n,1 of the form [z, 0, ..., 0, 1], and in these coordinates, σ is given by z -→ 1/z. We lift each p i as a vector P i = [z i , 0, ..., 0, 1] T (the standard lift in the ball model), and compute:

X(p 1 , p 2 , p 3 , p 4 ) = (z 3 z 1 -1)(z 4 z 2 -1) (z 4 z 1 -1)(z 3 z 2 -1) = (z 3 -1/z 1 )(z 4 -1/z 2 ) (z 3 -1/z 2 )(z 4 -1/z 1 ) = [σ(z 1 ), σ(z 2 ); z 4 , z 3 ]
From Lemma 3.1 and 3.2 and the properties of the classical cross-ratio we obtain the following reality condition for X:

Proposition 3.1. Let p 1 , p 2 , p 3 , p 4 ∈ H n
C with p 1 = p 2 , and let π 12 denote orthogonal projection onto the complex line L 12 spanned by p 1 and p 2 . Then:

1. The complex cross-ratio X(p 1 , p 2 , p 3 , p 4 ) is real and positive if and only if

• either the points p 1 , p 2 , π 12 (p 3 ), π 12 (p 4 ) are all equidistant from a geodesic γ in L 12 , with p 1 , p 2 on one side of γ and π 12 (p 3 ), π 12 (p 4 ) on the other,

• or the p i are on the boundary of a common complex line and {p 1 , p 2 } does not separate {p 3 , p 4 } on this circle.

2. X(p 1 , p 2 , p 3 , p 4 ) is real and negative if and only if p 1 , p 2 , p 3 , p 4 are on the boundary of a common complex line and {p 1 , p 2 } separates {p 3 , p 4 } on this circle.

Note that the second statement with the p i on the boundary of H n C is one half of Proposition 2 of [KR] and of property 7 on p. 226 of [G]. However the statement of the other half (so, our first statement) is different when some of the p i are in H n C . Proof. Normalize as in Lemma 3.2 so that the complex line L 12 containing p 1 and p 2 is the first coordinate axis in the ball model of H n C . Denote by z 1 , • • • , z 4 the respective coordinates in this unit disk of the points p 1 , p 2 , π 12 (p 3 ), and π 12 (p 4 ). According to Lemma 3.2 and Proposition 3.1,

X(p 1 , p 2 , p 3 , p 4 ) ∈ R ⇐⇒ [1/z 1 , 1/z 2 ; z 4 , z 3 ] ∈ R ⇐⇒ 1/z 1 , 1/z 2 , z 4 , z 3 lie on a common circle C in CP 1 (3.6)
Note that z 1 , z 2 , z 3 , z 4 are in the closed unit disk of C, so that 1/z 1 and 1/z 2 are outside the open unit disk. In particular, either C intersects the unit circle in 2 points p and q, or C is the unit circle.

1. In view of (3.6), if X(p 1 , p 2 , p 3 , p 4 ) < 0 the two pairs (z 1 , z 2 ) and (1/z 3 , 1/z 4 ) separate each other on C.

The latter remark tells us that this is only possible when C is the unit circle. Therefore p 1 , p 2 , p 3 and p 4 all belong to the boundary of L 12 and the pairs (p 1 , p 2 ) and (p 3 , p 4 ) separate each other on C.

Assume that

X(p 1 , p 2 , p 3 , p 4 ) > 0. (a) If C is the unit circle, then z i = 1/z i for i = 1, 2 and thus X(p 1 , p 2 , p 3 , p 4 ) = [z 1 , z 2 , z 4 , z 3 ], which is positive if and only if {p 1 , p 2 } do not separate {p 3 , p 4 } in C.
(b) If C intersects the unit circle in 2 points p and q, let γ denote the geodesic whose endpoints are p and q. Then π 12 (z 3 ) and π 12 (z 4 ) are on a hypercycle with endpoints p and q (the part of C which is inside the unit disk), and z 1 , z 2 are on the image of this hypercycle by reflection in γ (this is the image of the other half of C by inversion in the unit circle), see the left part of Figure 2. Therefore p 1 , p 2 , π 12 (p 3 ), π 12 (p 4 ) are all equidistant from γ, with p 1 , p 2 on one side of γ and π 12 (p 3 ), π 12 (p 4 ) on the other.

The following result is the analogue of Theorem 3.2 in the case where the 4 points are inside H 2 C : 00 11 0 0 1 1 00 11 00 11 0 1 0 0 1 1 00 11 00 11

p 1 p 1 p 2 p 2 π 12 (p 4 ) π 12 (p 3 ) π 12 (p 3 ) π 12 (p 4 ) X(p 1 , p 2 , p 3 , p 4 ) real X(p 1 , p 2 , p 3 , p 4 ) real, d(p 1 , p 3 ) = d(p 2 , p 4 ) and d(p 1 , p 4 ) = d(p 2 , p 3 ) γ γ q p q p Figure 2: Projections of quadruples (p 1 , p 2 , p 3 , p 4 ) to L 12 Theorem 3.3. Let p 1 , p 2 , p 3 , p 4 ∈ H 2 C with p 1 = p 2 or p 3 = p 4 .
There exists a real reflection φ such that φ : p 1 ↔ p 2 and p 3 ↔ p 4 if and only if:

X(p 1 , p 2 , p 3 , p 4 ) > 0, d(p 1 , p 3 ) = d(p 2 , p 4 ) and d(p 1 , p 4 ) = d(p 2 , p 3 ). Proof. Let p 1 , p 2 , p 3 , p 4 ∈ H 2 C satisfy X(p 1 , p 2 , p 3 , p 4 ) > 0, d(p 1 , p 3 ) = d(p 2 , p 4 ) and d(p 1 , p 4 ) = d(p 2 , p 3
). Consider the 2 ordered triples (p 1 , p 2 , p 3 ) and (p 2 , p 1 , p 4 ); these have equal side-lengths by our assumptions. We now compare the corresponding triple-ratios, T 1 = T (p 1 , p 2 , p 3 ) and T 2 = T (p 2 , p 1 , p 4 ). Observe that, taking lifts P 1 , ..., P 4 of p 1 , p 4 in C 2,1 : T 2 = P 2 , P 1 P 1 , P 4 P 4 , P 2 P 1 , P 2 P 4 , P 1 P 2 , P 4 = P 2 , P 1 P 3 , P 2 P 1 , P 3 P 1 , P 2 P 2 , P 3 P 3 , P 1 • P 2 , P 3 P 1 , P 4 P 2 , P 4 P 1 , P 3 • P 3 , P 1 P 4 , P 2 P 3 , P 2 P 4 , P 1

= T 1 • X(p 1 , p 2 , p 3 , p 4 ) X(p 1 , p 2 , p 3 , p 4 ) (3.7)
Now by assumption X(p 1 , p 2 , p 3 , p 4 ) > 0, so that T 2 = T 1 . Then, by Theorem 3.1, there exists an antiholomorphic isometry g sending p 1 to p 2 , p 2 to p 1 and p 3 to p 4 . But such an isometry must be an R-reflection by Lemma 2.7, assuming that p 1 = p 2 . If p 1 = p 2 but p 3 = p 4 , the same argument applies, permuting the indices by the permutation (13)(24).

It may seem that we only used the assumption that X was real in the above proof. Recall however that if X is negative, the four points are on the boundary of a complex line, and the two pairs {p 1 , p 2 } and {p 3 , p 4 } separate each other. In that case, as observed above, there cannot exist a real reflection φ such that φ(p 1 ) = p 2 and φ(p 3 ) = p 4 .

Commutators, decomposable pairs and traces

Main results

Recall that a pair of holomorphic isometries (A, B) ∈ PU(n, 1) 2 is said to be R-decomposable if there exist 3 R-reflections σ 1 , σ 2 and σ 3 such that A = σ 1 σ 2 and B = σ 1 σ 3 . We are now ready to prove our main result. Note that the eigenvalues of elements of PU(2, 1) are not well-defined (up to change of lift in U(2, 1), or even SU(2, 1)), but the eigenvalues of a commutator are well-defined (the commutator itself is independent of lifts). Using Goldman's classification of isometries by trace and Proposition 2.3, this criterion can be reduced to the following:

Theorem 4.2. Let A, B ∈ PU(2, 1) be two isometries not fixing a common point in H 2 C . Then (A, B) is R-decomposable if and only if: • [A, B] is loxodromic and Tr[A, B] > 3, or • [A, B] is unipotent, or • [A, B] is elliptic, Tr[A, B] ∈ R and the eigenvalue 1 of [A, B] is of negative type.
The extra assumption in the elliptic case means that the eigenvalue 1 corresponds to the fixed point of [A, B]. The other eigenvalues of [A, B] are then e ±iθ for some θ, by the assumption that Tr[A, B] ∈ R and Proposition 2.3.

Proof of Theorem 4.1: Let p 1 be a fixed point of [A, B] in H 2 C , P 1 a lift of p 1 in C 2,1 and λ 1 the associated eigenvalue, so that: [A, B]P 1 = λ 1 P 1 . Consider the cycle of four points defined as follows: let P 2 = B -1 P 1 , P 3 = A -1 P 2 and P 4 = B(P 3 ). First assume for simplicity that these 4 points are all distinct. Then opposite sides of the quadrilateral (P 1 P 2 P 3 P 4 ) are identified by A and B as on Figure 3. Note that AP 4 = [A, B]P 1 = λ 1 P 1 . Then: Lemma 4.2. There exists a real reflection φ such that φ(p 1 ) = p 3 and φ(p 2 ) = p 4 if and only if the cross-ratio X(p 2 , p 4 , p 1 , p 3 ) is real and positive.

X(
The following lemma concludes the proof of Theorem 4.1: Lemma 4.3. (A, B) is R-decomposable ⇐⇒ there exists a real reflection φ such that φ : p 1 ↔ p 3 and p 2 ↔ p 4 . Indeed, if A = σ 2 σ 1 and B = σ 3 σ 1 then σ 1 = φ satisfies φ : p 1 ↔ p 3 and p 2 ↔ p 4 . Conversely, if such a φ exists then by Lemma 2.7 above, A • φ and B • φ are real reflections. Indeed they are both antiholomorphic, and A • φ (resp. B • φ) exchanges p 1 and p 2 (resp. p 1 and p 4 ). Therefore (A, B) is R-decomposable.

Finally we examine the case where some of the 4 points p 1 , ..., p 4 are equal. The cross-ratio appearing in Lemma 4.1 is well-defined as long as no three of the points are on ∂H 2 C and equal, in which case all four of them would be equal, contradicting the assumption that A and B do not have a common fixed point. The proofs of Lemmas 4.2 and 4.3 carry through as long as p 1 = p 3 or p 2 = p 4 . Now if p 1 = p 3 = p and p 2 = p 4 = q, then A and B both exchange p and q (if they are distinct). If p and q are in H 2 C then A and B both fix the midpoint of the segment [pq] which is again assumed not to be the case. If p and q are on ∂H 2 C then A and B both have a fixed point on the geodesic line (pq) and act as a half-turn on the complex line spanned by p and q. In that case, on one hand by Proposition 2.5(a1-a3), any real plane containing (pq) decomposes A and B, therefore (A, B) is R-decomposable. On the other hand, the commutator [A, B] is loxodromic (because it acts by translation along the geodeesic (pq)) and has an eigenvalue equal to 1 (because A and B have a common eigenvector as they both preserve the complex line spanned by p, q), therefore [A, B] has 3 real and positive eigenvalues. In particular, the conclusion of Theorem 4.1 holds in this case as well. C then (A, B) is R-decomposable. Proposition 4.1 is the first part of Theorem 2.1 of [FalPau]; it essentially follows from the fact that, given two complex lines in C 2 , there exists a Lagrangian subspace intersecting each of them in a line (see Proposition 2.5 (e1-e3)). Note that the 3 parts of Proposition 4.2 cover all cases where A and B have a common fixed point on ∂H 2 C , because screw-parabolic isometries have 2-step unipotent part (this follows from the fact that their elliptic and unipotent parts commute in the classification theorem of Chen-Greenberg, our Theorem 2.2(c)).

Groups fixing a point

Proof of Proposition 4.2. (a) First note that if A and B are both loxodromic with a common fixed point, then by Proposition 2.5 (l): (A, B) is R-decomposable if and only if A and B have the same fixed points, that is they commute.

-Assume that one of A, B is loxodromic but not the other, say A is loxodromic and B is parabolic or a complex reflection, and that A and B commute. In particular, A has distinct eigenvalues, therefore B must also be diagonalizable by the assumption that A and B commute. Moreover B fixes the fixed points of A, which means that the fixed line of B contains the axis of A. Therefore any real reflection which decomposes A also decomposes B (by Proposition 2.5 (e2) and (l)). In particular, (A, B) is R-decomposable.

-Conversely, assume that A is loxodromic and B is not, and that (A, B) is R-decomposable. By part (p) of Proposition 2.5, B cannot be parabolic. It is thus a complex reflection. Denote by C the complex line fixed by B, and by p A , q A the fixed points of A. Then by Proposition 2.5 (e2) and (l), there exists a real reflection σ with fixed R-plane R such that σ exchanges p A and q A , and such that R ∩ C is a geodesic. Then σ preserves C which contains one of p A and q A , therefore C also contains the other, and A and B commute.

(b) Assume that A is not 3-step unipotent. Conjugating if necessary, we may assume that the common fixed point is q ∞ . Denote by C the stable complex line of A, and pick a point p in the vertical line ∂C. Any affine line through p contained in the contact plane at p is the boundary of a real plane R such that R ∩ C contains p and q ∞ . This implies that for any such R, R ∩ C is the geodesic connecting p to q ∞ , and therefore R decomposes A by part (p) of Proposition 2.5.

-If B preserves a complex line, that is B is a complex reflection or screw parabolic, then its stable complex line C ′ is such that ∂C ′ is a vertical complex line. If C = C ′ then the result is clear. If not, one of the infinite R-circles through p intersects ∂C ′ and therefore the corresponding real reflection decomposes B.

-If B is 3-step unipotent then one of the infinite R-circles through p intersects the invariant fan of B orthogonally, therefore the corresponding real reflection decomposes B.

(c) If both A and B are 3-step unipotent then in view of Proposition 2.5 and Lemma 2.8, the pair (A, B) is R-decomposable if and only if there exists a real plane R such that the infinite R-circle ∂R is orthogonal to both invariant fans of A and B. This is equivalent to saying the these two fans are parallel, which by Lemma 2.3 means that A and B commute.

Maximal representations: C-Fuchsian punctured torus groups

The case where the cross-ratio X(p 2 , p 4 , p 1 , p 3 ) is real and negative corresponds to a rigidity phenomenon, giving the following result which holds in all dimensions:

Proposition 4.3. If [A, B] has a fixed point in H n
C whose associated eigenvalue is real and negative, then the group A, B stabilizes a complex line L in H n C . Moreover in that case the corresponding fixed point of [A, B] is on the boundary ∂H n C , so that A, B is a C-Fuchsian punctured torus group. Proof. Indeed, with the notation of the proof of Theorem 4.1, we have the following variation of lemma 4.1:

X(p 2 , p 4 , p 1 , p 3 ) ∈ R -⇐⇒ λ 1 ∈ R -.
Now by assumption λ 1 < 0, so that by Proposition 3.1 p 1 , p 2 , p 3 , p 4 are on the boundary of a common complex line L and the pairs (p 2 , p 4 ) and (p 1 , p 3 ) separate each other. But A sends (p 3 , p 4 ) to (p 2 , p 1 ) and B sends (p 2 , p 3 ) to (p 1 , p 4 ), so A and B both stabilize L. It is then a simple exercise in the hyperbolic plane to check that this combinatorics implies that A and B must both be loxodromic, with axes meeting inside the ideal quadrilateral Q = (p 1 , p 2 , p 3 , p 4 ). Therefore, Q is disjoint from all its images by elements of the group generated by A and B. This proves that A, B is discrete. Note that in general this quadrilateral is not a fundamental domain for A, B , as its images only tessellate the complex line L if [A, B] is parabolic. See Figure 4.

One can also interpret Proposition 4.3 in terms of the Toledo invariant of the corresponding type-preserving representation of the fundamental group of the once-punctured torus. Given a representation ρ of the fundamental group of a surface Σ into PU(n, 1), the Toledo invariant τ (ρ) is defined as the integral over Σ of the pullback of the Kähler 2-form on H n C by an equivariant map f : Σ -→ H n C :

τ (ρ) = Σ f ⋆ ω. (4.2)
In the case where Σ is non-compact, one should be careful that this integral is well-defined. This is guaranteed for instance by the condition that the map f has finite energy (see [KM]). The existence of a such a finite-energy equivariant map is guaranteed by the assumption that all peripheral loops on Σ are mapped to parabolics by the representation ( [KM]). This assumption being made, it is possible to use an ideal triangulation of Σ = i ∆ i to compute the Toledo invariant, as in [GP]. The result is that the Toledo invariant can be written as where each integral is computed over any 2-simplex with boundary f (∂∆ i ). But the integral of ω over any 2-simplex with boundary ∂∆ is equal to twice the Cartan invariant of ∆ (this is Theorem 7.1.11 of [G]). As a consequence, we see that for a type-preserving representation of the fundamental group of the once-punctured torus, the Toledo invariant is given by:

τ (ρ) = i f (∆i) ω, (4.3)
τ (ρ) = 2 (A(p 1 , p 2 , p 3 ) + A(p 1 , p 3 , p 4 )) , (4.4)
with p 1 , ..., p 4 defined as previously. On the other hand, taking arguments in equation (3.4) gives:

arg (X(p 2 , p 4 , p 1 , p 3 )) = A(p 2 , p 4 , p 1 ) -A(p 2 , p 1 , p 3 ) = A(p 1 , p 2 , p 4 ) + A(p 1 , p 2 , p 3 ) (4.5)
Therefore, if the cross-ratio X(p 2 , p 4 , p 1 , p 3 ) is negative, then A(p 1 , p 2 , p 4 ) and A(p 1 , p 2 , p 3 ) must both be equal to either π/2 or -π/2. In view of (4.4), this means that |τ (ρ)| = 2π. But the Toledo invariant satisfies the Milnor-Wood inequality:

|τ (ρ)| ≤ 2π(2g -2 + p), (4.6)
where equality holds if and only if the representation is discrete and preserves a complex line. In that case, the representation is called maximal (see [T, BIW]). In the case of the once punctured torus g = p = 1, and therefore if |τ (ρ)| = 2π the representation is maximal.

Groups generated by real reflections

We now use the criterion from Theorem 4.2 to show that various subgroups of PU(2, 1) are generated by real reflections. More accurately this means that they are the index 2 holomorphic subgroup of a group of isometries generated by real reflections.

Mostow's lattices and other non-arithmetic lattices in SU(2, 1)

Mostow's lattices from [M] (revisited in [DFP]) as well as the new non-arithmetic lattices in SU(2, 1) studied by Deraux, Parker and the first author (see [ParPau], [Pau], [START_REF] Deraux | Census of the complex hyperbolic sporadic groups[END_REF] and [START_REF] Deraux | A family of new non-arithmetic complex hyperbolic lattices[END_REF]) are all symmetric complex reflection triangle groups. This means that they are generated by 3 complex reflections R 1 , R 2 and R 3 which are in a symmetric configuration in the sense that there exists an isometry J of order 3 such that JR i J -1 = R i+1 (with i mod 3). These groups are in fact contained (with index 1 or 3, depending on the parameters) in the group Γ generated by R 1 and J.

It was shown in [DFP] that Mostow's lattices are generated by real reflections, and in [START_REF] Deraux | A family of new non-arithmetic complex hyperbolic lattices[END_REF] this is extended to all symmetric complex reflection triangle groups. In both cases though, one finds an explicit real reflection which decomposes both holomorphic generators R 1 and J, which requires knowing explicit geometric properties of the group. Now the existence of such a real reflection follows immediately from the following consequence of Theorem 4.1: Proposition 5.1. If R ∈ PU(2, 1) is a complex reflection or a complex reflection in a point, then for any A ∈ PU(2, 1), the pair (R, A) is R-decomposable.

Proof. First assume that R is a complex reflection about a complex line L. Then [R, A] = RR ′ , where R ′ = AR -1 A -1 is a complex reflection conjugate to R -1 . We denote by L ′ its fixed complex line. The extensions of L and L ′ to CP 2 intersect at a unique point p ∈ CP 2 , unless L = L ′ , in which case [R, A] = Id and the result follows from Theorem 4.2. In general, p is fixed by both R and R ′ , thus by [R, A]; we distinguish 2 cases, depending on whether p is in H 2 C or outside of H 2 C .

1. First assume that p ∈ H 2 C . Consider lifts of R and R ′ to SU(2,1) such that the lift of R has eigenvalues e 2iφ , e -iφ , e -iφ (where the rotation angle of R is 3φ), with p corresponding to a e -iφ -eigenvector. Likewise, the lift of R ′ has eigenvalues e -2iφ , e iφ , e iφ , with p corresponding to a e iφ -eigenvector of R

′ . Then p is a fixed point of [R, A] = RR ′ in H 2
C with corresponding eigenvalue 1. The result follows from Theorem 4.1.

2. If p is outside of H 2 C , then L and L ′ are ultraparallel and p is polar to their common perpendicular line, which we denote by L. The isometries R and R ′ act on L by rotation through angles 3φ and -3φ respectively. It is an elementary fact from plane hyperbolic geometry that the product of two elliptic elements with opposite rotation angles and distinct fixed points must be hyperbolic (it follows for instance from Lemma 7.38.2 of [Be]). Therefore RR ′ is loxodromic; moreover its eigenvalue of positive type is 1 (it is the product of those of R and R ′ which are inverse of one another). Therefore its eigenvalue spectrum must be either {r, 1, 1/r} or {-r, 1, -1/r} for some r > 0. Geometrically, this means that the rotation angle of the loxodromic isometry [R, A] is 0 or π. By continuity, the rotation angle of [R, A] must be the same for any value φ and any relative position of L and L ′ (with the restriction that they are ultraparallel). It is 0 when L = L ′ (as R and A are inverse of one another in this case). This means that [R, A] has only positive eigenvalues, and the result follows from Theorem 4.1. Now assume that R is a complex reflection about a point. As above, we write [R, A] = RR ′ with R ′ = AR -1 A -1 . The eigenvalues of R (respectively R ′ ) are {e iφ , e iφ , e -2iφ } (resp. {e -iφ , e -iφ , e 2iφ }), with e 2iφ (resp. e -2iφ ) of negative type. This means that R (resp. R ′ ) acts on any complex line through its fixed point as a rotation of angle 3φ (resp. -3φ). Consider the complex line L spanned by the fixed points of R and R ′ . The action of R and R ′ on L is the same as in the second item above, and this leads to the same conclusion.

Groups with more than two generators

Applying the criterion from Theorem 4.1 to a 2-generator subgroup of PU(2, 1) is completely straightforward. For subgroups generated by more elements (e.g. Picard modular groups) one needs to be a bit more careful, as being generated by real reflections is stronger than having generators which are pairwise R-decomposable. The following observation gives a way to bridge this gap, however its hypotheses are in general too restrictive and in practice we will need some more work to show that a given group is generated by real reflections.

Lemma 5.1. Let Γ be a subgroup of PU(2, 1) generated by A 1 , ..., A k . If there exist real reflections σ 1 , ..., σ k+1 such that (a) A i = σ i σ i+1 for 1 i k, or (b) A i = σ 1 σ i+1 for 1 i k then Γ has index 2 in Γ = σ 1 , ..., σ k+1 . In particular such a Γ is generated by real reflections.

Proof. In each case, pairwise products of the σ i are in Γ, therefore Γ is the index 2 holomorphic subgroup of Γ.

• in case (I), ( T1 , T2 ) -→ ( T1 , T2 2 T1 -1 ),

• in case (II), ( T1 , T2 ) -→ ( T1 2 , T2 ),

• in case (III), ( T1 , T2 ) -→ ( T1 2 , T2

2 ), we obtain finite index subgroups of Γ h ∞ (d) generated by pairs of translations that lift to Γ ∞ (d) and have orthogonal directions. We will denote the lifts of the new T1 and T2 by T 1 and T 2 , and from now on we will only consider the group T 1 , T 2 .

The central involution w -→ -w lifts to Γ ∞ (d) as the transformation given by

R 1 =   -1 0 0 0 1 0 0 0 -1   .
(5.2)

In particular, Γ ∞ (d) is generated by R 1 and its Heisenberg translation subgroup. Note that R 1 is a complex reflection of order 2 about the complex line polar to the vector v 1 = 0 1 0 T .

Lemma 5.3.

1. The real reflection σ 0 about the standard real plane R 2 ∩ H 2 C decomposes R 1 , I 0 and T 2 . 2. The real reflection R 1 σ 0 decomposes T 1 .

Proof.

1. Clearly, σ 0 fixes the polar vector to the mirrors of R 1 and I 0 . Therefore it decomposes these two transformations. Indeed the two transformations I 0 σ 0 and R 1 σ 0 fix the vectors v 0 and v 1 respectively and thus preserve the corresponding complex lines. They are therefore real reflections, by Lemmas 2.6 and 2.7. On the other hand, T 2 is a lift to Γ ∞ (d) of a translation by a multiple of i √ d. This implies that T 2 is a Heisenberg translation by [ik

√ d, t 2 ],
where k is an integer and t 2 is chosen so that T 2 ∈ Γ ∞ (d). In particular, this implies that the invariant fan of T 2 intersects the boundary of H 2C along a vertical plane of the type {x = x 2 }, for some x 2 ∈ R. The fixed R-circle of σ 0 is the x-axis of the Heisenberg group, and is therefore orthogonal to this vertical plane. This implies that σ 0 decomposes T 2 by Proposition 2.5. 2. A direct computation shows that in horospherical coordinates R 1 σ 0 acts as (z, t, u) -→ (-z, -t, u). Its restriction to the boundary is thus the reflection about the infinite R-circle {(iy, 0), y ∈ R}. As T 1 is the Heisenberg translation by some [k, it 1 ], where k ∈ Z, its invariant fan intersects the boundary along a vertical plane {y = y 1 } for some y 1 ∈ R, and is therefore orthogonal to the fixed R-circle of R 1 σ 0 . Again, by Proposition 2.5, this implies that R 1 σ 0 decomposes T 1 .

We will denote the corresponding real reflections as follows:

σ 1 = R 1 σ 0 , σ 2 = T 2 σ 0 , σ 4 = I 0 σ 0 and σ 3 = T 1 σ 1 = T 1 R 1 σ 0 .

(5.3) Proposition 5.2. For all d 1, the subgroup Γ ∞ (d), I 0 of Γ d is generated by real reflections up to finite index.

Proof. Consider the group Γ σ = σ j , j = 0 • • • 4 . By construction Γ σ contains the group T 1 , T 2 , R 1 , I 0 which has finite index in Γ ∞ (d), I 0 . On the other hand, we claim that all pairwise products of the σ i are in Γ ∞ (d), I 0 . Indeed, each of the σ i is of the form A i σ 0 with A i ∈ Γ ∞ (d), I 0 , therefore: σ i σ j = A i σ 0 A j σ 0 = A i A j . The result follows by noting that Γ ∞ (d) and I 0 are preserved under complex conjugation of entries.

Note that from the previous discussion, denoting by T 1 , T 2 the lifts to Γ ∞ (d) of generators of the orthogonal reflection subgroup of Isom + (O d ) (or their square when applicable), the index of T 1 , T 2 , R 1 , I 0 in Γ ∞ (d), I 0 is 2 when d ≡ 3 (mod 4) (and d = 3), 4 when d ≡ 2 (mod 4), and 8 when d ≡ 1 (mod 4) (and d = 1). For d = 3 the index is 6 and for d = 1 it is 16 (as there is an additional rotational part of order 3, 2 respectively). However in the latter cases we can recover the rotational part in the real reflection group by replacing R 1 in the definition of σ 1 by the appropriate complex reflection of order 4 or 6.

For d = 1, 2, 3, 7, 11 it is known from [FalPar], [FFP] and [Z] that Γ ∞ (d), I 0 = Γ d , giving the following:

  4.1 allows us to easily reprove that the Mostow and Deraux-Parker-Paupert lattices are generated by real reflections. More importantly, the careful analysis of decomposability conditions for parabolic isometries in Proposition 2.5 (c) allows us to show that new lattices are generated by real reflections, namely certain Picard modular groups. The Picard modular groups Γ d = SU(2, 1, O d ) (the subgroup of SU(2, 1) consisting of matrices with entries in O d , where d is a positive squarefree integer and O d denotes the ring of integers of Q[ √

  Theorem 4.1. Let A, B ∈ PU(2, 1) be two isometries not fixing a common point in H 2 C . Then: the pair (A, B) is R-decomposable if and only if the commutator [A, B] has a fixed point in H 2 C whose associated eigenvalue is real and positive.

Figure 3 :

 3 Figure 3: The 4-cycle associated to a fixed point of [A, B]

  When A and B have a common fixed point in H 2 C the results are the following: Proposition 4.1. If A, B ∈ PU(2, 1) have a common fixed point in H 2

  Proposition 4.2. Let A, B ∈ PU(2, 1) have a common fixed point on ∂H 2 C . (a) If A or B is loxodromic then (A, B) is R-decomposable if and only if A and B commute. (b) If A and B are both non-loxodromic and one of them is not 3-step unipotent then (A, B) is R-decomposable. (c) If A and B are both 3-step unipotent then (A, B) is R-decomposable if and only if A and B commute.

Figure 4 :

 4 Figure 4: A fundamental fomain for the discrete punctured torus group in the case where λ 1 < 0

  as there are no non-trivial cube roots of unity in O d .

The Picard modular groups PU(2, 1, O d ) are analogous to the Bianchi groups PGL(2, O d ) in PGL(2, C). Bianchi proved in the seminal paper

  seen in the affine chart {z n+1 = 1} of CP n . This model is referred to as the ball model of H n C . Secondly, when Z, Z = 2Re(z1 z n+1 ) + |z 2 | 2 + • • • + |z n | 2, we obtain the so-called Siegel model of H n C , which generalizes the Poincaré upper half-plane. More details on the Siegel model in dimension 2 will be given in the next section.

	Isometries: From (2.1) it is clear that PU(n, 1) acts by isometries on H n C , where U(n, 1) denotes the subgroup
	of GL(n + 1, C) preserving •, • , and PU(n, 1) its image in PGL(n + 1, C). In fact, PU(n,1) is the group of holomorphic isometries of H n C , and the full group of isometries is PU(n, 1)⋉Z/2, where the Z/2 factor corresponds
	to a real reflection (see below). Holomorphic isometries of H n C can be of three types, depending on the number
	and location of their fixed points. Namely, g ∈ PU(n, 1) is :
	• elliptic if it has a fixed point in H n C
	• parabolic if it has (no fixed point in H n C and) exactly one fixed point in ∂H n C
	• loxodromic: if it has (no fixed point in H n C and) exactly two fixed points in ∂H n C

  p 2 , p 4 , p 1 , p 3 ) = P 1 , P 2 P 3 , P 4 P 3 , P 2 P 1 , P 4 The cross ratio X(p 2 , p 4 , p 1 , p 3 ) is real and positive if and only if λ 1 ∈ R + For the next step we use Theorem 3.2 or 3.3, depending on whether p 1 (and hence all otherp i 's) is on ∂H 2 C or in H 2 C .In the latter case, by construction of the 4 points we have d(p 1 , p 2 ) = d(p 3 , p 4 ) and d(p 1 , p 4 ) = d(p 2 , p 3 ).

	This proves the following:	=	λ -1 1 AP 4 , AP 3 P 3 , P 4 P 3 , P 2 BP 2 , BP 3	= λ -1 1	| P 3 , P 4 | 2 | P 3 , P 2 | 2 .	(4.1)
	Lemma 4.1. Therefore Theorem 3.3 tells us:

* Author partially supported by NSF grant DMS 1007340 and SNF grant 200021-131967/1.

Picard modular groups

] when d ≡ 3 (mod 4). The groups Γ d are usually called Picard modular groups. Denote by Γ ∞ (d) the stabilizer of q ∞ in Γ d and I 0 ∈ Γ d given by:

(5.1)

The isometry I 0 is a complex reflection of order 2 about the complex line polar to the vector

Elements of Γ ∞ (d) are upper triangular matrices with units of O d on the diagonal. In particular, Γ ∞ (d) contains no Heisenberg dilations (see (2.5)). In other words Γ ∞ (d) < Isom(N) and therefore it fits into the exact sequence (2.9) as: [FFPn] for d = 1, 3 (see [FFP] for d = 1 and [FalPar]

Proof. We sketch the proof given in [FFPn]. First, note that an element P (z,t,θ) of Isom(N) is in the kernel of Π * if and only if z = θ = 0, see (2.9). This gives part (a). Next, when d = 1, 3, the only units of O d are ±1 and thus the rotational part of an element P (z,t,θ) ∈ Γ ∞ (d) must satisfy θ = 0 or θ = π. In particular the subgroup of Γ ∞ (d) formed by Heisenberg translations has index two. The same observation holds for the translation subgroup Isom + (O d ), which is an extension of its translation subgroup ∆ t d by the central involution w -→ -w. The strategy used in [FFPn] is to try to lift the generators of ∆ t d to Γ ∞ (d). The existence of such a lift can be verified using the explicit form of the map Π * given (2.9). (II) Assume that d ≡ 2 (mod 4). Then ∆ t = T1 , T2 with:

(III) Assume that d ≡ 1 (mod 4) (and d = 1). Then ∆ t = T1 , T2 with:

Remark 3. In cases (II) and (III) above, the translation directions of T 1 and T 2 are orthogonal, and so are the translation directions of the generating pairs of Γ h ∞ (d). In case (I) this is not true. However, the two elements T1 and T2

2 T1 -1 have this property and generate an index two subgroup of Γ h ∞ (d). Therefore in view of the previous discussion if we make the following changes: Corollary 5.1. For d = 3, 7, 11 (respectively 2 and 1), Γ d = PU(2, 1, O d ) has a subgroup of index 2 (respectively 4 and 8) which is generated by real reflections.

Note that for d = 3, the full group Γ 3 is in fact generated by real reflections by Proposition 5.1, as it was shown in [FalPar] (Proposition 5.11) that Γ 3 = J, R 1 with J regular elliptic of order 3 and R 1 a complex reflection of order 6.

Question: Are all Picard modular groups PU(2, 1, O d ) generated by real reflections up to finite index?


Nicolas Cordier

Bjoern Menze

Hervé Delingette

Nicholas Ayache

Patch-based Segmentation of Brain Tissues

We describe our submission to the Brain Tumor Segmentation Challenge (BraTS) at MICCAI 2013. This segmentation approach is based on similarities between multi-channel patches. After patches are extracted from several MR channels for a test case, similar patches are found in training images for which label maps are known. These labels maps are then combined to result in a segmentation map for the test case. The labelling is performed, in a leave-one-out scheme, for each case of a publicly available training set, which consists of 30 real cases (20 highgrade gliomas, 10 low-grade gliomas) and 50 synthetic cases (25 highgrade gliomas, 25 low-grade gliomas). Promising results are shown on the training set, and we believe this algorithm would perform favourably well in comparison to the state of the art on a testing set.

Introduction

We describe our submission to the Brain Tumor Segmentation Challenge (BraTS) at MICCAI 2013. This segmentation approach is based on ideas similar to these developed for human brain labelling in [START_REF] Rousseau | A supervised patch-based approach for human brain labeling[END_REF]. A database of multi-channel patches is first built from a set of training cases, for which label maps are known. Then, given a test case, patches are extracted from several Magnetic Resonance (MR) channels, and similar multi-channel patches are retrieved in the patch database. Since each multi-channel intensity patch from the database is associated with a label patch, a combination of the labels can result in a segmentation map for the test case. First, we detail the pre-processing steps, mainly a global intensity alignment and tumour localization. Second, we recall the currently naive procedure to build the database of patches and retrieve similar patches. Third, we describe the label fusion step. Then, the labelling is performed, in a leave-oneout scheme, for each case of a publicly available training set, which consists of 30 real cases (20 high-grade gliomas, 10 low-grade gliomas) and 50 synthetic cases (25 high-grade gliomas, 25 low-grade gliomas). Finally, results and promising improvements to the current algorithm are discussed.

Method: Patch-based Segmentation

In this section, our patch-based segmentation algorithm is described in its most recent state. Current limitations could be bypassed with several promising improvements, which are still work-in-progress at the time of the submission of the article.

Pre-processing

In order to decrease the computation time, we first sub-sample , with a nearestneighbour interpolation scheme, all the images to 2-mm isotropic resolution, as in [START_REF] Zikic | Decision forests for tissue-specific segmentation of high-grade gliomas in multi-channel mr[END_REF]. In the end, to compare the resulting segmentation to the ground truth, we would perform an up-sampling to 1-mm isotropic resolution.

Moreover, we define a bounding box surrounding the tumour, and we crop the images. There are three main reasons to back this pre-processing step:

computations are faster since there are less patches, -patches are more relevant since we are only interested in segmenting the tumor and we focus on image parts containing a tumor, -we avoid any problem related to image parts missing for some channels, since these happen outside these bounding boxes.

While these bounding boxes are known for the training cases, we do not know their localization for a test case. Given more time, we would intend to roughly localize the bounding box by applying a threshold to T 2 FLAIR images. In the meantime, we currently assume a bounding box around the tumor is known.

Global intensity alignment

Let D be the number of training cases, each of which consists of four MR channels and one label map. Before we build a multi-channel patch database, we would like to normalize each MR image in order to make all the D cases appear more similar to each other. This is a very important step since the similarity criterion, which drives the patch retrieval, is not invariant to affine intensity changes.

Since different Magnetic Resonance Imaging (MRI) scanners could be used for the acquisition of each case, a pre-processing step is mandatory to normalize the image intensities. Similarly to [START_REF] Zikic | Decision forests for tissue-specific segmentation of high-grade gliomas in multi-channel mr[END_REF], we "align the mean intensities within each channel by a global multiplicative factor" K.

Let c be a channel, among T 1 -weighted, T 2 -weighted, FLAIR T 2 -weighted, and contrast enhanced T 1 -weighted images. Let x be a voxel position in the MR image I c . Let < I c > be the mean intensity for channel c.

The global intensity alignment is such that:

∀c, ∀x, I c (x) ← I c (x) × K < I c >
where K is arbitrarily set to 700, which is a value close to the mean intensities. Currently, this step is performed with cropped images.

Building a database of patches and retrieving similar patches

For each case I of the D training cases, for each voxel x of I, we extract a 3x3x3 patch P Ic x per channel c, and we concatenate these patches to get a multi-channel patch P I

x .

Given a new case I test , which consists of the four MR channels previously mentionned, we compute a multi-channel patch P I test x for each voxel x. For each patch in I test , for each of the D reference cases, we find the knearest-neighbours with respect to our similarity measure, which is here the sum of squared differences d(•, •) between multi-channel patches. In the following, k = 5.

For each voxel x of the test case I test , we have retrieved the k most similar patches for each reference case, which makes D ×k similar multi-channel patches in total, for which we know the corresponding 3D label patches.

Label fusion

In order to define the label map S for the test case I, a label fusion method is defined. Aggregation of the votes Given vote weights for each retrieved patch, there are two possibilities to aggregate the votes. First, if we consider a voxel x, the simplest method increments the votes for the label found at the center of each retrieved patch. Second, the method we use in this paper makes use of the labels found in the whole patch, i.e. at the center x and in the 3x3x3 neighbourhood of x of each label patch. Thus, we increment the votes for 27 voxels for each retrieved patch. This results in regularized vote maps.

Vote weight for each retrieved patch

Vote maps For each label, the aggregation of votes results in a vote map, which consists in a map of the aggregated vote weights. These vote maps can be seen as probability maps: they can be scaled by the total of vote increments so that the total of votes for each foreground voxel is 1, in which each vote is a non-negative value between 0 and 1. This rescaling does not affect the segmentation procedure though, as opposed to the rescaling described in the following.

Label selection In the Brain Tumor Segmentation Challenge, we are interested in the segmentation of three regions (complete tumor, tumor core, enhancing tumor) which show an interesting property: these regions are interlocked. Therefore, during label selection, we proceed with three consecutive steps:

1. distinguish between the background and the complete tumor, 2. inside the complete tumor, distinguish between the tumor core and what remains of the complete tumor (the edema), 3. inside the tumor core, distinguish between the enhancing tumor and what remains of the tumor core (the non-enhancing tumor and the necrotic core).

For each of these steps, we want to give more weights to labels which are less represented in the reference dataset, typically tumor classes. Indeed, less represented labels are less likely to be picked at the patch retrieval step, and in the case of tumor classes, the number of false negatives could be increased. Given a pool of two region labels to select from, we proceed as follows in order to achieve a penalization of the biggest region: for each region separately, we divide voxel-wise region votes by the number of voxels for which the region votes are non-zero. Therefore, we slightly favour labels for which only a few voxels have non-zero votes. Finally, for each voxel x, the label with the highest scaled vote is picked.

Evaluation

Our

Discussion

We have described a generic framework for brain tumor segmentation, which relies on similarities between multi-channel patches. The algorithm performs well on the training cases provided for the Brain Tumor Segmentation Challenge. We believe this algorithm would perform favourably well in comparison to the state of the art on a testing set. Promising improvements to the current algorithm are still in progress.

Regarding the pre-processing step, we do not expect the results to differ significantly with slightly different bounding boxes. Indeed, bounding boxes restrict the images in coronal and saggital views, but the axial view is not cropped at all in order to keep more examples of healthy tissues, as is shown in figures 3 to 10. Moreover, let us assume we make an error during the localization of the bounding box:

if we find a bounding box containing the true bounding box, then no false negative would be added, and we would expect the number of additional false positives to be low since the specificity of the classifier is close to 1 in practice, -if we find a bounding box included in the true bounding box, then this may result in less false positives and this would necessarily result in more false negatives since parts of the tumour are necessarily wrongly omitted by the classifier. This is the case for synthetic high-grade cases, due to a human error.

Regarding the global intensity alignment step, we believe results would not be fundamentally different if the full brain mask were used instead of a bounding box. In fact, a global intensity alignment applied to the full images would be the first pre-processing step in order to localize the tumor.

Regarding the retrieval of similar patches in the patch database, a cluster is used since the search is naive and exhaustive. This paper in its current state is mostly a proof-of-concept, and we expect in the foreseeable future to tremendously decrease the computation time by relying on approximate hash-based searches.

  For each voxel x in I test , each retrieved similar patch (P In,m yn,m ) n∈[|1,D|],m∈[|1,k|] contributes to a voting scheme, with a vote weight w(x, •) which depends on the similarity measure d(•, •): w(x, P I y ) = exp(-d(P I y , P I test x ) σ 2 ) where σ 2 is the maximum (over every voxel x ∈ I test and every training case n ∈ [|1, D|]) of the L 2 error made with first-neighbour patches only (m = 1).

Fig. 1 .Fig. 2 .Fig. 4 .Fig. 6 .Fig. 7 .Fig. 8 .Fig. 9 .Fig. 10 .

 124678910 Fig. 1. Per-patient evaluation for the two real BraTS data sets (Real-HG, Real-LG).We show the results for three regions: complete tumor, tumor core, and, for all highgrade cases and a few low-grade cases only, enhancing tumor. We report the following measures: Dice score, Specificity, and Sensitivity.

Table 1 .

 1 Evaluation summary for 20 real high-grade cases.

	Dice score Complete tumor Tumor core Enhancing tumor
	mean	0.79	0.60	0.59
	std-dev	0.17	0.26	0.25
	median	0.87	0.72	0.69
	Dice score Complete tumor Tumor core Enhancing tumor
	mean	0.76	0.64	0.44
	std-dev	0.18	0.21	0.40
	median	0.80	0.71	0.45

segmentation procedure is evaluated on 30 real cases (20 high-grade gliomas, 10 low-grade gliomas) and 50 synthetic cases (25 high-grade gliomas, 25 lowgrade gliomas). The dataset is publicly available through the MICCAI 2013 Brain Tumor Segmentation challenge. We perform the labelling of each image of the training set, in a leave-one-out framework. The Dice score, Jaccard index, specificity and sensitivity are computed by the online evaluation tool provided by the organizers of the Brain Tumor Segmentation Challenge. Average scores are given in tables 1 to 4, per-patient scores are given in figures 1 and 2. The best segmentation maps for each of the four training datasets are shown in figures 3 to 10.

Table 2 .

 2 Evaluation summary for 10 real low-grade cases. Regarding the right-most column, 3 cases show enhancing tumor, and the Dice score is only computed for these.

	Dice score Complete tumor Tumor core
	mean	0.84	0.78
	std-dev	0.08	0.12
	median	0.85	0.83

Table 3 .

 3 Evaluation summary for 25 synthetic high-grade cases.

	Dice score Complete tumor Tumor core
	mean	0.83	0.67
	std-dev	0.04	0.19
	median	0.83	0.74

Table 4 .

 4 Evaluation summary for 25 synthetic low-grade cases.

Acknowledgements

Part of this work was funded by the European Research Council through the ERC Advanced Grant MedYMA 2011-291080 (on Biophysical Modeling and Analysis of Dynamic Medical Images).