This is the accepted version of the journal article: Porti, Joan; Tillmann, Stephan. «Projective structures on a hyperbolic 3- orbifold». Acta Mathematica Vietnamica, Vol. 46, Issue 2 (June 2021), p. 347-355. DOI 10.1007/s40306-021-00419-0 This version is available at https://ddd.uab.cat/record/256250 under the terms of the license Projective structures on a hyperbolic 3–orbifold Joan Porti and Stephan Tillmann Abstract We compute and analyse the moduli space of those real projective structures on a hyperbolic 3–orbifold that are modelled on a single ideal tetrahedron in projective space. Pa- rameterisations are given in terms of classical invariants, traces, and geometric invariants, cross ratios. AMS Classification 57M25, 57N10 Keywords 3–orbifold, projective geometry 1 Introduction This note studies certain real projective structures on a 3-dimensional orbifold, O, which is obtained by taking the one tetrahedron triangulation with two vertices of S 3 , deleting the vertices and modelling the edge neighbourhoods on IR3 /hri, where r is a rotation by 120◦ (see Figure 1). This orbifold supports a unique complete hyperbolic structure; this has two Euclidean (3, 3, 3)–turnover cusps and is of finite volume. In [6], the philosophy was put forward that strictly convex projective manifolds behave like hyperbolic manifolds sans Mostow rigidity. This paper computes a moduli space of projective structures on O that are modelled on an ideal tetrahedron. This moduli space, denoted Mod(O), turns out to be a smooth, open 2–dimensional disc. We obtain two parameterisations of Mod(O): one in terms of algebraic invariants, traces, and one in terms of geometric invariants, cross-ratios. The complete hyperbolic structure on O is singled out as the only structure on Mod(O) having standard cusps, whilst the remaining structures on Mod(O) all have generalised cusps. It is also characterised as the unique fixed point of a natural involution on Mod(O). The problem to decide which of the structures on Mod(O) are properly convex appears to be difficult by elementary means, but is completely solved by the theoretical results of Cooper, Long and Tillmann [6, 5], and of Ballas, Danciger and Lee [2]; the answer is all of them. This problem has also motivated some of Choi’s work [4]. Acknowledgements. We thank the anonymous referee for suggestions that improved the paper. This research was supported through the programme Research in Pairs (No. 1811) by the Centre International de Rencontres Mathmatiques Luminy in 2017. The authors would like to thank the staff at CIRM for an excellent collaboration environment. Research of the first author is supported by FEDER-MEC (grant number PGC2018-095998-B-I00). Research of the second author is supported by an Australian Research Council Future Fellowship (project number FT170100316). 1 2 Projective structures modelled on triangulations In n–dimensional real projective space, n–simplices are overly congruent. Given any two n–simplices, there is a projective transformation taking one to the other. Given an n– simplex, there is a n–dimensional family of projective transformations taking it to itself whilst fixing each of its vertices. The following notions can be defined in all dimensions, but we restrict to the case n = 3. The space IRP3 will be viewed as the set of 1–dimensional vector subspaces of IR4 with the induced topology. The set of projective transformations, PGL(4, IR), then corresponds to the quotient of GL(4, IR) by its centre, the group of all non-zero multiples of the identity matrix. If ∆ is the 3–simplex with P vertices corresponding to the standard unit vectors e1 , . . . , e4 in IR4 and containing ei in its interior, then the family of projective transformations stabilising ∆ and fixing its vertices corresponds to the set of diagonal matrices in GL(4, IR) having all entries positive or all entries negative. This gives a 3– dimensional family of projective transformations. Let M be an arbitrary, ideally triangulated 3–orbifold of finite topological type. We recall that an ideal 3–simplex is a 3–simplex with its four vertices removed, and an ideal triangulation of M is an expression of M as a finite collection of ideal 3–simplices with their faces glued in pairs, so that the branching locus of the orbifold is contained in the 2–skeleton. We assume that the interior of each ideal i–simplex is embedded and that M has finitely many ends, each homeomorphic to the product of a 2-orbifold with a half line [0 + ∞). The end-compactification M of M is the result of adding the vertices of the ideal 3–simplices, topologically it is the one-point compactification of each end of M . A real projective structure on M is (an equivalence class of) a pair (dev, ρ), where f → IRP3 is a locally injective map and ρ : π1 (M ) → PGL(4, IR) is a representa- dev : M tion of the orbifold fundamental group which makes dev ρ-equivariant. Since the orbifold universal cover Mf is non-compact, some of its ends may be homeomorphic to IR2 × (0, 1). We therefore make some additional assumptions. 1. Structure is modelled on projective simplices: Let Mf be the orbifold universal cover of M . Assume that M is a good orbifold, so that M f is a manifold. The ideal simplices of M lift to M f. Let M c be the result of adding the vertices to the lifted simplices, with the corresponding identifications induced by side parings. For instance, for an ideally triangulated hyperbolic 3-orbifold of finite volume, f is the hyperbolic 3-space IH3 and M M c consists in adding the set of points in the ideal 3 ∼ 2 boundary ∂∞ IH = S fixed by parabolic isometries, which is a dense countable subset of ∂∞ IH3 . In general, the set Mc\M f is a countable set of points in which the triangulation is not locally finite. Furthermore, the end-compactification M of M is the quotient of M c by the action of the orbifold fundamental group of M . f → IRP3 extends to a continuous, equivariant map We assume that dev : M dev d: M c → IRP3 . In this case, dev d is ρ0 –equivariantly homotopic to a map dev d 0 with the property that dev d 0 (∆) is a projective simplex (of dimension 0, 1, 2 or 3) for every simplex ∆ in M c. In particular, the map is possibly not locally injective. 2. Structure is non-collapsed: Let ∆ a 3–simplex in M c. 2 We assume that the images of the vertices of ∆ under dev d are in general position. In this case, the above homotopy between dev d 0 and dev d can be assumed to preserve M f, and dev0 maps any simplex to a simplex of the same dimension. Moreover, it may be d assumed to do so by a linear map and in particular, it is locally injective at interior points of 3–simplices. Definition 1 Let M be an ideally triangulated 3–orbifold with the property that the ideal triangulation restricts to an ideal triangulation of the singular locus. A real projective structure modelled on the triangulation is a real projective structure (dev, ρ) on M which is modelled on projective simplices and non-collapsed. If the triangulation consists of a single 3–simplex, we will also say that the structure is modelled on a 3–simplex. 3 The moduli space Figure 1: To obtain O, first glue the faces meeting along one of the edges with cone angle 2π/3 to obtain a spindle, and then identify the boundary discs of the spindle. The result is S 3 minus two points, with the labelled graph (minus its vertices) as the singular locus. The hyperbolic structure can be obtained√ by identifying the ideal 3–simplex with an ideal hyperbolic 3–simplex with shape parameter 21 + 6−3 . The fundamental group of O admits, up to conjugation, exactly two irreducible representations into SL(2, C). They are complex conjugates and correspond to holonomies for the hyperbolic structure. Theorem 2 The set of real projective structures on O modelled on a 3–simplex is pa- rameterised by a connected component, X , of the set of all (w, x, y, z) ∈ IR4 subject to the following two equations: w + x + y + z = 3 + wy, (3.1) wy = zx. (3.2) The structures corresponding to any two distinct points of X are neither isotopic nor projectively equivalent. Moreover, X is diffeomorphic to an open disc. The involution (w, x, y, z) → (y, z, w, x) on X has exactly one fixed point, (3, 3, 3, 3) ∈ X , that corresponds to the complete hyperbolic structure on O . Before proving the theorem, we describe the zero set of (3.1) and (3.2). 3 Lemma 3 The zero set of (3.1) and (3.2) in IR4 is a smooth surface with two compo- nents, each diffeomorphic to a disc. One of the components, X , contains the point with coordinates w = x = y = z = 3, and it is distinguished from the other component by the inequalities x > 1 and z > 1 (equivalently y > 1 and w > 1). Proof of the lemma From (3.1) we replace z = w y − w − x − y + 3 in (3.2), so the set we want to describe is diffeomorphic to {(x, y, w) ∈ IR3 | −wxy + wx + wy + x2 + xy − 3x = 0}. We change variables to simplify computations: set w = a + 1, x = b + 1, y = c + 1. Then the subvariety becomes {(a, b, c) ∈ IR3 | b2 − abc + a + c = 0}. We view the defining equation as a quadratic equation on b, whose discriminant is Disc = a2 c2 − 4a − 4c. It can be checked that the vanishing locus of Disc consist of precisely two smooth curves, each homeomorphic to IR (for instance, by computing c for each value of a, as Disc is quadratic on any of the variables). Also by elementary methods, one can show that the set {(a, c) ∈ IR2 | Disc ≥ 0} is homeomorphic to two half planes. Then the set we want to describe is the 2–to–1 branched covering of these half planes, branched along their bound- aries, and therefore the subvariety is diffeomorphic to two smooth discs. Furthermore the open quadrant {(a, b) ∈ IR2 | a > 0, c > 0} contains one of the components of Disc ≥ 0 and is disjoint from the other one. Finally, the points (a, c) = (0, 0) and (a, c) = (2, 2) lie in Disc = 0, but in different components. Proof of Theorem 2 Denote the ideal 3–simplex by [v1 , v2 , v3 , v4 ]. The face pairings are α[v1 , v2 , v3 ] = [v1 , v2 , v4 ] and β[v2 , v3 , v4 ] = [v1 , v3 , v4 ]. The orbifold fundamental group is generated by the face pairings, and we have the following presentation: π1orb (O) = hα, β : α3 = β 3 = (αβα−1 β −1 )3 = 1i. (3.3) To determine all projective structures of O modelled on a 3–simplex up to projective equivalence, it suffices to fix a projective 3–simplex, ∆, and to determine all representa- tions ρ of π1 (O) with ρ(α) and ρ(β) as the corresponding face pairings. Here we have used that all 3–simplices are projectively equivalent. Choosing ∆ = [e1 , e2 , e3 , e4 ], the most general form of lifts of the face pairings is:     s1 0 0 a1 b1 t1 0 0  0 s2 0 a2  b2 0 0 0  A=  0 0 0 a3  and B = b3 0 t3 0  ,    (3.4) 0 0 s4 a4 b4 0 0 t4 subject to s1 s2 s4 a3 6= 0 and t1 t3 t4 b2 6= 0. Since the above does not take division by the centre into account, the equation A3 = cI4 for c 6= 0 is projectively equivalent to A3 = I4 , since the equation c3 = 1 always has a non-zero real root. Similarly for B. This gives: −1 −b−1     1 0 0 a1 2 0 0 0 1 0 a2   and B =  b2 0 0 0  A= 0 0 , (3.5) 0 a3   b3 0 1 0 0 0 −a−1 3 −1 b4 0 0 1 4 subject to a3 b2 6= 0. Note that both matrices are elements of SL(4, IR). Since we are interested in representations up to conjugacy, one may conjugate the above to give:     1 0 0 a1 −1 1 0 0  and B = −1 0 0 0 . 0 1 0 a2    A= 0 0 0 −1  b3 0 1 0 (3.6) 0 0 1 −1 b4 0 0 1 It now remains to analyse (ABA−1 B −1 )3 = cI4 . With (3.6), one first notes that c = 1. In particular, any representation of π1 (O) into PGL(4, IR) lifts to a representation into SL(4, IR). By a case-by-case analysis, one obtains the following cases: Case 1: a1 = a2 = b3 = b4 = 0. In this case there is a single representation which in fact satisfies A3 = B 3 = ABA−1 B −1 = I4 . The corresponding developing map is not locally injective, and hence that there is no corresponding real projective structure. Case 2: One obtains a single equation: (a1 + a2 )(b3 + b4 ) = 3 + a1 a2 b3 b4 . (3.7) Analysis of which representations are conjugate yields that pairs A, B and A0 , B 0 give conjugate representations in PGL(4, IR) if and only if they are conjugate by M = diag(m, m, m−1 , m−1 ) for some m 6= 0. The effect on the quadruples is: (a01 , a02 , b03 , b04 ) = (ma1 , ma2 , m−1 b3 , m−1 b4 ). (3.8) Note that M corresponds to a projective transformation stabilising any subsimplex of ∆. Since no face pairing is a reflection, we need to check local injectivity at the edges of the triangulation. It follows from inspection that local injectivity at the axis of A is equivalent to not both a1 and a2 to be contained in (−∞, 0], and local injectivity at the axis of B is equivalent to not both b1 and b2 to be contained in (−∞, 0]. Thus, the corresponding pair of equivariant map and representation, (dev, ρ), can be replaced by a projective structure given by (M ◦ dev, M ◦ ρ ◦ M −1 ) with M = diag(m, m, m−1 , m−1 ) that is locally injective at both the axes of A and B unless (a1 , a2 ≤ 0 and b1 , b2 ≥ 0) or (a1 , a2 ≥ 0 and b1 , b2 ≤ 0). But equation (3.7) has no solutions of this form. It remains to check local injectivity around the axis of the commutator C = ABA−1 B −1 . The description of dev around the axis of C is obtained by gluing translates of ∆ following a cyclic order. Knowing that dev is locally injective on the edges of A and B , we have that this process of gluing simplices around the axis of C turns either once or several times: namely either dev is locally injective or it is locally a cyclic branched covering, branched on the axis of C . Moreover, since we require that C 3 = I4 , by continuity the degree of this branched covering is locally constant along the parameter space. This degree is the criterion for choosing one of the components of the parameter space and discarding the other. Before that, we describe the parameter space, then we will check whether it is locally injective or a branched covering in a given point on each component. The coordinates given in the statement of the theorem can be expressed in terms of classical 5 invariants of the chosen lift of ρ: w = a1 b4 = 2 + tr AB, x = a1 b3 = 2 + tr A−1 B, y = a2 b3 = 2 + tr A−1 B −1 , z = a2 b4 = 2 + tr AB −1 , and (3.7) can be expressed in terms of these, giving the first equation given in (3.1). The second arises from wy = a1 a2 b3 b4 = xz. It can now be verified that the zero set of (3.1) and (3.2) corresponds to the quotient of the action of IR \ {0} given in (3.8) on the set of all (a1 , a2 , b3 , b4 ) subject to (3.7). We apply Lemma 3 to conclude that the parameter space has two components. Next we check wheter local injectivity of dev holds or not on a point on each component of this parameter space. On the component X we consider (w, x, y, z) = (3, 3, 3, 3): it is a straightforward compu- tation that it corresponds to the holonomy of the hyperbolic structure. In particular the devoloping map is locally injective along the axis of C = ABA−1 B −1 . On the other component of the parameter space we consider (w, x, y, z) = (1, 1, 1, 1): we take a1 = a2 = b3 = b4 = 1. We look at the cycle of consecutive translates of ∆ than gives a full turn around e1 e4 , the axis of C = ABA−1 B −1 : (∆, A(∆), AB(∆), ABA−1 (∆), C(∆), CA(∆), CAB(∆), CABA−1 (∆), C 2 (∆), BAB −1 (∆), BA(∆), B(∆)). (3.9) This cycle has 12 elements, because C has order 3 and the edge has four representatives in ∆. The matrix C acts on this cycle by a cyclic permutation. By direct computation, when a1 = a2 = b3 = b4 = 1:   0 1 0 0 1 0 0 0 CAB = 0 0 0 1  0 0 1 0 Hence CAB(∆) = ∆ and the cycle (3.9) turns at least twice around the edge, i.e. dev is locally a branched covering of degree ≥ 2. (We prove in the Remark 5 that the degree is precisely 2.) Hence we discard the whole component. Let X be the component of the parameter space containing (w, x, y, z) = (3, 3, 3, 3). We have shown that it parameterizes projective structures, we aim to show that these structures are properly convex. For that purpose, we look at the peripheral subgroups of π1orb (O). There are two conjugacy classes of peripheral subgroups, corresponding to the ends of O and represented respectively by the stabilizer of v1 and the stabilizer of v4 . The stabilizer of v1 is the group generated by α and γ = [α, β], it is the fundamental group of the link of the end of O corresponding to v1 (which is a turnover S 2 (3, 3, 3)): π1orb (S 2 (3, 3, 3)) ∼ = hα, γ | α3 = γ 3 = (α2 γ)3 = 1i. Similarly, for v4 the peripheral group is generated by β and γ . The maximal torsion-free subgroup of the peripheral group of v1 has index 3, it is generated by αγ and γα and it is isomorphic to Z2 . 6 Lemma 4 For (w, x, y, z) ∈ X \ {(3, 3, 3, 3)}, up to conjugation we have     0 0 1 0 λ1 0 0 0 1 0 0 0  0 λ2 0 0 A = ρ(α) =  0 1 0 0  and AC = ρ(αγ) =   0 0 λ3  0 0 0 0 1 0 0 0 1 with λi ∈ IR, λi > 0 and λ1 6= ±1. Assuming the lemma, we prove that the set of properly convex projective structures on X is open and closed. For openess, the holonomy of hαγ, γαi ∼ = Z2 preserves a flag in R4 , for any structure on X . We can argue similarly for v4 and the peripheral subgroup generated by β and γ . Hence, we may apply Theorem 0.2 of [5] to say that set of properly convex structures is open in X . For closedness, we apply Theorem 1.5 of [2]. Namely, the compact orbifolds obtained by truncating the ends have totally geodesic boundary (two turnovers). Ballas, Danciger and Lee show in [2] that these structures yield a convex structure on the double of the truncated orbifolds, and apply a theorem of Benoist for convex structures on closed orbifolds [3]. Alternatively, one can directly refer to Theorem 0.1 of the forthcoming work [7] to obtain closedness. Thus every structure on X is properly convex. Proof of Lemma 4 First we claim that all eigenvalues of AC = ρ(αγ) are real, for each value of the parameters (w, x, y, z) ∈ X . For that purpose, we compute the characteristic polynomial of ρ(αγ) = AC , by a long but explicit computation: (λ − 1)(λ3 + (−yx − w + 2 x + 2 y − z) λ2 + (zw − 2 w + x + y − 2 z) λ − 1) and we check that the discriminant of its degree three factor is nonnegative. To check that, we write w and z in terms of x and y from (3.1) and (3.2); this only parameterizes an open dense subset of X , but it is sufficient to determine the non-negativity of the discriminant. With this parameterization of a subset X , the discriminant is 2 2 2 2 y2 − 3 y + 3 x − 3 x + 3 (−y + x)2 x2 y 2 − 3 x2 y − 3 xy 2 + 3 x2 + 3 xy + 3 y 2 , (xy − x − y)6 which is always nonnegative. From the formula of the characteristic polynomial of AC = ρ(αγ), an elementary compu- tation shows that for (w, x, y, z) ∈ X \ {(3, 3, 3, 3)}, AC has a (real) eigenvalue λ1 6= 1. Up to replacing it by 1/λ21 , we assume that λ1 has multiplicity 1. Let p1 ∈ IRP3 be the projective point fixed by AC corresponding to Eλ1 (AC), the eigenspace of eigenvalue λ1 . Set p2 = A(p1 ) and p3 = A(p2 ), we have p1 = A(p3 ). By construction, p1 = Eλ1 (AC), p2 = Eλ1 (A2 CA2 ), p3 = Eλ1 (CA), (3.10) where Eλ denotes the eigenspace with eigenvalue λ. Next we aim to show that p1 , p2 and p3 span a 3-dimensional linear subspace. We first show that p1 6= p2 . Seeking a contradiction, assume p1 = p2 , then also p3 = p2 = p1 . As the product of the matrices CA, A2 CA2 , AC is 1, (3.10) would imply that λ31 = 1, which contradicts λ1 6= 1 and λ1 ∈ IR. Let V ⊂ IR4 be the span of the three affine lines p1 , p2 and p3 . Commutativity between AC and CA implies that AC(p2 ) = p2 and AC(p3 ) = p3 . Thus V contains three different lines that are invariant by AC . If V had dimension 2, then this would imply that AC acts as a multiple of the identity on V , contradicting that λ1 has multiplicity 1. 7 Hence dim V = 3, as claimed. As dim(IR4 /V ) = 1 and A and C have order three, both A and C act trivially on IR4 /V . Thus every matrix in hA, Ci has an eigenvector equal to 1. By looking at the action of hA, Ci on the vector subspace V , one can find a matrix in hAC, CAi ∼= Z2 whose restriction to V has all eigenvalues different from 1. This gives the fourth vector for a basis of IR4 so that the matrices with respect to these basis are as claimed. Finally, the assertion on the sign of the eigenvalues follows from continuity by looking at the representation at (w, x, y, z) = (3, 3, 3, 3). Remark 5 When (w, x, y, z) = (1, 1, 1, 1), hA, Bi acts as Alt(5) by permuting five points in IRP3 . More precisely, we take a1 = a2 = b3 = b4 = 1, and both A and B preserve the set of five projective points {he1 i, he2 i, he3 i, he4 i, he1 + e2 − e3 − e4 i} in IRP3 . The action on this set determines the group action, as a projective transformation of IRP3 is determined by the action on five points in general position. As both A and B act as cycles of order three on this set, and their fixed points are disjoint, this is the action of the alternating group Alt(5). It can be shown that these five points are the vertices of a tessellation of IRP3 defined by the translates of ∆. The stabilizer of ∆ has 4 elements and the tessellation consists of 15 tetrahedra (each tetrahedron shares its vertices with two other tetrahedra of the tessellation). Furthermore, one can check that the edges invariant by A or B are adjacent to three simplices, and the edge invariant by C is adjacent to six simplices (and therefore the developing map has degree 2 = 12/6 around the edge of C ). The quotient of IRP3 by this action of A5 is the spherical orbifold J ×∗J J in [8]. Remark 6 By the discussion in the proof of Theorem 2 and Remark 5, the component of the parameter space described by (3.1) and (3.2) that contains (w, x, y, z) = (1, 1, 1, 1) corresponds to projective structures branched along a singular edge, as described by Ballas and Casella in [1]. Remark 7 The proof of Theorem 2 gives an interpretation of the coordinates in terms of traces. We wish to point out that there is an alternative interpretation in terms of cross ratios as follows. We look for cross ratios of points in the projective line l = he3 , e4 i that contains the edge e3 e4 , namely the fixed point set of B . We consider the projective plane Π = he1 , e2 , e3 i, so that both Π and A(Π) contain a face of the 3-simplex ∆. The first three points of l that we consider are the respective intersection of l with Π, A(Π) and A2 (Π). Their homogeneous coordinates are p3 = Π∩l = [0 : 0 : 1 : 0], p4 = A(Π)∩l = [0 : 0 : 0 : 1], and p5 = A2 (Π)∩l = [0 : 0 : 1 : 1]. Each coordinate x, y , z and w appears as a cross ratio of p3 , p4 and p5 with one of the following: px = AB −1 A(Π) ∩ l = [0 : 0 : 1 : 1 − x] py = ABA(Π) ∩ l = [0 : 0 : 1 : 1 − y], pz = A−1 B(Π) ∩ l = [0 : 0 : 1 − z : 1], pw = A−1 B −1 (Π) ∩ l = [0 : 0 : 1 − w : 1]. 8 Namely: x = (px , p3 ; p5 , p4 ), y = (py , p3 ; p5 , p4 ), z = (pz , p4 ; p5 , p3 ), w = (pw , p4 ; p5 , p3 ). Remark 8 Ballas and Casella [1] study the same example but, instead of properly con- vex projective structures, they consider structures equipped with peripheral flags. Their deformation space is just a point, in contrast with this setting, because for the generic structures the peripheral subgroups do not have invariant flags. References [1] Samuel A. Ballas and Alex Casella. Gluing equations for real projective structures on 3- manifolds. arXiv:1912.12508, 2019. [2] Samuel A. Ballas, Jeffrey Danciger, and Gye-Seon Lee. Convex projective structures on nonhyperbolic three-manifolds. Geom. Topol., 22(3):1593–1646, 2018. [3] Yves Benoist. Convexes divisibles. III. Ann. Sci. École Norm. Sup. (4), 38(5):793–832, 2005. [4] Suhyoung Choi. The convex real projective orbifolds with radial or totally geodesic ends: a survey of some partial results. In In the tradition of Ahlfors-Bers. VII, volume 696 of Contemp. Math., pages 51–85. Amer. Math. Soc., Providence, RI, 2017. [5] Daryl Cooper, Darren Long, and Stephan Tillmann. Deforming convex projective manifolds. Geom. Topol., 22(3):1349–1404, 2018. [6] Daryl Cooper, Darren D. Long, and Stephan Tillmann. On convex projective manifolds and cusps. Adv. Math., 277:181–251, 2015. [7] Daryl Cooper and Stephan Tillmann. The space of properly-convex structures. in prepara- tion, ≥ 2020. [8] William D. Dunbar. Nonfibering spherical 3-orbifolds. Trans. Amer. Math. Soc., 341(1):121– 142, 1994. Departament de Matemàtiques, Universitat Autònoma de Barcelona and CRM, 08193 Bellaterra, Spain School of Mathematics and Statistics, The University of Sydney, NSW 2006, Australia Email: porti@mat.uab.cat stephan.tillmann@sydney.edu.au 9