CROSS RATIOS ASSOCIATED WITH MAXIMAL REPRESENTATIONS

arXiv:0908.4101v1 [math.DG] 27 Aug 2009

CROSS RATIOS ASSOCIATED WITH MAXIMAL REPRESENTATIONS TOBIAS HARTNICK AND TOBIAS STRUBEL

Abstract. We define a generalization of the classical four-point cross ratio of hyperbolic geometry on the unit circle given by invariant functions on Shilov boundaries of arbitrary bounded symmetric domains of tube type. This generalization is functorial and well-behaved under products. In fact, these two properties determine our extension uniquely. Any maximal representation of a closed surface group can be used to pull back our generalized cross ratios to functions on the circle; these pullbacks turn out to be strict cross ratios in the sense of Labourie and can be used to estimate the translation length of an element under the corresponding representation. The corresponding estimates show that maximal representations are well-displacing. This implies in particular that the action of the mapping class group on the moduli space of maximal representations into a Hermitian Lie group is proper.

Contents 1.

Introduction

1

2.

The algebraic structure of bounded symmetric domains of tube type

5

3.

Normalized kernels of Euclidean Jordan algebras

14

4.

Generalized cross ratios on Shilov boundaries

24

5.

Maximal representations, limit curves and strict cross ratios

33

6.

Well-displacing and quasi-isometry property

37

References

46

1. Introduction Let Γ be the fundamental group of a closed surface Σ of genus g ≥ 2 and Γ ,→ P U (1, 1) a fixed hyperbolization of Σ, i.e. an isometric action of Γ on the Poincare disc D with Σ = Γ\D. Every γ ∈ Γ is hyperbolic, i.e. fixes a unique unit speed geodesic σγ . We have γ.σγ (t) = σγ (t + τD (γ)), where τD (γ) := inf d(x, γ.x) x∈D

Date: August 27, 2009. 1

2

TOBIAS HARTNICK AND TOBIAS STRUBEL

denotes the translation length of γ. The action of Γ on D extends to its boundary S 1 , and it turns out that the translation length of each γ ∈ Γ can be recovered from the boundary action: Indeed, denote by [a : b : c : d] :=

(a − d)(c − b) (c − d)(a − b)

the four point cross ration on CP1 and abbreviate γ ± := σγ (±∞) ∈ S 1 . Then for any ξ ∈ S 1 \ {γ ± } we have τD (γ) = τD∞ (γ) := log[γ − : ξ : γ + : γ.ξ]. In particular, τD∞ (γ) does not depend on the choice of ξ ∈ S 1 \ {γ ± }. We refer to τD∞ (γ) as the virtual translation length of γ. Now consider the case of a representation ρ : Γ → G into a semisimple Lie group (always assumed without compact factors and with finite center) with associated symmetric space X . Although ρ(Γ) need not consist of hyperbolic elements anymore, the translation length τX (ρ(γ)) := inf d(x, ρ(γ).x) x∈X

can still be defined for every γ ∈ Γ. We can thus ask the similar question, whether τX (ρ(γ)) can be computed from the action of ρ(Γ) on some suitable limit set in some boundary of X . In this article we deal with the case, where the associated symmetric space of G is isomorphic to a bounded symmetric domain of tube type and ρ : Γ → G is a representation with maximal Toledo invariant, or maximal representation for short. In this case we denote by Sˇ the Shilov boundary of the bounded symmetric domain D isomorphic to X . We then need to define a generalized cross ratio on Sˇ in order to define a virtual translation length for the Γ action. Such generalized cross ratios have been defined by various people in various degrees of generality. In the symplectic case the construction is quite classical (see e.g. [17, Subsec. 4.2.6]). Beyond that case there does not seem to be a consensus about the definition. Our key observation is that there is actually only one real-valued extension of the classical cross ratio to higher rank Shilov boundaries, provided one insists on the right kind of functoriality. (On the contrary, there are various natural generalizations to vector valued or operator valued cross ratios, see e.g. [19], [4] and [3]). This is the content of the following theorem: Theorem 1.1. For each bounded symmetric domain D of tube type with Shilov boundary Sˇ there exists a subset Sˇ(4+) of Sˇ4 (defined in Definition 4.1 below) and a function BSˇ : Sˇ(4+) → R× called the generalized cross ratio of D, such that the family of functions {BSˇ } is characterized uniquely by the following properties: (i) BSˇ is invariant under the group of orientation-preserving biholomorphic automorphisms of D. (ii) If f : D1 → D2 is an affine holomorphic map of symmetric spaces which is induced from a balanced Jordan algebra homomorphism (see Definition 3.11 below), then the corresponding generalized cross ratios BSˇ1 , BSˇ2 satisfy BSˇ2 (f¯(v1 ), . . . , f¯(v4 )) = BSˇ1 (v1 , . . . , v4 ), (4+) where (v1 , . . . , v4 ) ∈ Sˇ1 and f¯ is the boundary extension of f .

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(iii) If D = D1 × D2 is a direct product of symmetric spaces of ranks r1 , r2 with projections pj : D → Dj then BSˇ (v1 , . . . , v4 )r1 +r2 = BSˇ1 (p1 (v1 ), . . . , p1 (v4 ))r1 BSˇ2 (p2 (v1 ), . . . , p2 (v4 ))r2 . (iv) BS 1 is the restriction of the classical four point cross ratio. (Theorem 1.1 will be proved in Subsection 4.5 below.) Generalizing the classical cross ratio we will define BSˇ (a, b, c, d) :=

k(d, a)k(b, c) k(d, c)k(b, a)

for a kernel function k : Sˇ × Sˇ → C (see Section 3). Returning to our maximal representation ρ : Γ → G, the limit set of Γ in Sˇ is homeomorphic to a circle and can be parameterized by a continuous limit curve ˇ For γ ∈ Γ we may thus define the virtual translation length of ρ(γ) to ϕ : S 1 → S. be ∞ τD (ρ(γ)) := log BSˇ (ϕ(γ − ), ϕ(ξ), ϕ(γ + ), ρ(γ).ϕ(ξ)), where ξ ∈ S 1 \ {γ ± } is arbitrary. Again, this does not depend on the choice of the basepoint ξ. It is not quite true that the translation length and the virtual translation length of ρ(γ) always coincide. However, at least the virtual translation length can be used to bound the actual translation length from below. More precisely, if rk D denotes the rank of the bounded symmetric domain D and dimC D its complex dimension, then we obtain for all γ ∈ Γ the estimate (1)

τD (ρ(γ)) ≥

rk D · τ ∞ (ρ(γ)). dimC D D

In order to apply this fact we observe that the pullback of a generalized cross ratios BSˇ : Sˇ(4+) → R× to S 1 under a limit curve of a maximal representation happens to be a strict cross ratio in the sense of Labourie [17]. As a consequence, any two virtual translation length functions are at bounded distance. This together with the estimate (1) then implies the following results: Theorem 1.2. Let ρ : Γ → G be a maximal representation. (i) For any finite generating set S of Γ there exist A, B > 0 such that τD (ρ(γ)) ≥ A · lS (γ) − B for all γ ∈ Γ. (ii) For every x ∈ D and every finite generating set S of Γ the map (Γ, dS ) → (D, dD ),

γ 7→ γ.x

is a quasi-isometric embedding. (Theorem 1.2 will be proved in Theorem 6.20 and Theorem 6.21 below.) In the language of [14] the first part of the theorem says that maximal representations are well-displacing. This has the following well-known consequence:

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Corollary 1.3. Let G be a semisimple Lie group without compact factors and with finite center and assume that the symmetric space of G is a bounded symmetric space of tube type. Let Γ be the fundamental group of a closed surface Σg of genus g ≥ 2 and denote by Modg the corresponding mapping class group. Finally, write Mmax (Γ, G) for the space of maximal representations of Γ into G up to conjugation. Then the action of Modg on Mmax (Γ, G) is proper. (Corollary 1.3 will be proved in Subsection 6.6 below.) For symplectic, respectively, classical simple groups Corollary 1.3 was proved by Labourie [17] and Wienhard [20]. Since Mmax (Γ, PSL2 (R)) is canonically identified with the Teichm¨ uller space of Σ, we can think of the spaces Mmax (Γ, G) as higher Teichm¨ uller spaces. The quotients Modg \Mmax (Γ, G) should then be considered as higher analoga of the moduli space of hyperbolic structures on Σ. Our results imply that these spaces are orbifolds. Let us briefly summarize the structure of this article; for a more detailed overview over its content see also the introductions to the individual sections. The generalized cross ratios, whose existence is claimed in Theorem 1.1, are constructed in Section 4, where it is also proved that these functions have the desired properties. The preceding two sections contain preparatory material, which is needed for the construction of these cross ratios. We suggest to the hasty reader to start from Section 4 and to refer backwards as necessary. In order to support this approach let us briefly summarize the content of the preparatory sections: Section 2 contains mostly well-known material, albeit sometimes in a somewhat reorganized form. In the first subsection we relate bounded symmetric domains to Lie groups on the one hand and Jordan algebras on the other hand; in particular, we introduce notations for the various groups and spaces occurring in the context of bounded symmetric domains. We also provide a short dictionary on how to translate between the different pictures. The second subsection has the more specific purpose to relate maximal polydiscs to Jordan frames; the main result here is Proposition 2.10. The final subsection is devoted to orbits of pairwise transverse triples and quadruples. The Maslov index and the corresponding notion of maximality are introduced; these are important tools throughout. The most important result in this subsection is Proposition 2.15. The purpose of Section 3 is to establish the existence of suitable kernel functions on boundaries of bounded symmetric domains of tube types. These kernel functions are a crucial ingredient in our construction of the generalized cross ratios and their construction is probably one of the most technical parts in the whole article. Their main properties are summarized in Theorem 3.12 and Corollary 3.15. The reader who is willing to take these results on faith can skip the whole section on first reading. The Section is divided into four subsections, the first of which explains the relation between kernel functions and transversality. Based on this, the normalized kernel functions are then constructed in two steps, first for simple Jordan algebras and then in the general case. Finally, we compare our normalized kernel to the

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Bergman kernel of the underlying complex domain. The final two sections provide applications of the results of Section 4: In the first subsection of Section 5 it is explained how our generalized cross ratios can be used in order to associate a strict cross ratios in the sense of [17] to any maximal representation. The second subsection is devoted to the proof of the crucial Proposition 5.8, which allows one to compare to arbitrary strict cross ratios. This result is implicitly contained in [17], but unfortunately not stated in the present generality. For the convenience of the reader we included a full proof. Section 6 is finally devoted to the proof of Inequality (1) and its consequences. We first deal with a simple special case, which demonstrates the power of our functorial approach. The next two subsections establish our main inequality by first estimating the translation length from below and then estimating the virtual translation length from above. The final three subsections establish Theorem 1.2 and Corollary 1.3. In fact, both follow easily once (1) is established. Acknowledgements We would like to thank Marc Burger and Alessandra Iozzi for their interest in our work and for many useful conversations and Anna Wienhard for helpful comments. We would like to thank Kloster Mariastein for their hospitality.

2. The algebraic structure of bounded symmetric domains of tube type Bounded symmetric domains can be studied using geometric, Lie theoretic and Jordan algebraic methods. While our point of view in this article is mostly Jordan algebraic, we still need to be able to translate between the various approaches and in particular to express geometric notions in algebraic terms. The purpose of this section is to provide all the basic definitions and notations related to either of the three approaches and to sketch the relations between them. We will need this relations for example for the proof of Theorem 1.1 (ii). With the exception of Proposition 2.15 we do not present any new results here, but only collect and reformulate various foundational material for ease of later reference. We thus recommend readers familiar with the structure theory of bounded symmetric domains to skip this section and to return to it when required. In the first subsection we recall the equivalence of categories between Euclidean Jordan algebras and marked bounded symmetric domains. The main reference for this is [2]. We use this opportunity to fix our notations. The main result of this subsection is Proposition 2.2, which provides a linear action for the Levi factor of a certain maximal parabolic. The second subsection relates the geometric notion of a maximal polydisc to the algebraic notion of a Jordan frame. This correspondence can be deduced from the results in [15] quite easily; lacking a suitable reference, we decided to include the proofs to keep the exposition self-contained. In the last subsection we recall the classification of pairwise transverse triples in the Shilov boundary of a bounded symmetric domain of tube type due to Clerc and Ørsted [13]. Besides being important in their own right, these results finally allows us to prove Proposition 2.15, which provides sufficient conditions for quadruples to be

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contained in the boundary of a maximal polydisc, and is the main result of this section.

2.1. Algebraic categories related to bounded symmetric domains. A bounded symmetric domain (or bsd for short) is a connected bounded open subset D of a complex vector space together with a family of biholomorphic involutions {σx : D → D | x ∈ D} such that x is an isolated fix point of σx . We denote by G := G(D)0 denotes the group of orientation preserving biholomorphic automorphisms of D. It is well-known that G is transitive on D, in particular any bsd is homogeneous. A tube in Cn is a subset of the form Rn + iΩ, where Ω is an open convex cone in Rn and a bsd is of tube type if it is biholomorphic to a tube. A pointed bounded symmetric domain is a triple (D, {σx }, o), where (D, {σx }) is a bsd and o ∈ D is a fixed basepoint. The topological boundary D \ D of a bounded symmetric domain contains a unique closed G-orbit Sˇ referred to as the Shilov-boundary of D. A marked bounded symmetric domain is a quadruple (D, {σx }, o, ξ), where (D, {σx }, o) is a pointed bsd and ξ ∈ Sˇ is a point in the Shilov boundary. Together with the obvious morphisms, i.e. holomorphic maps preserving the pointwise involutions and fixing both base points, marked bsds form a category, which we denote by mBSD. The full subcategory of marked bsds of tube type is denoted mBSDT . The purpose of this subsection is to exhibit algebraic categories which are equivalent to mBSDT . This will allow us to reduce geometric statements on bounded symmetric domains to algebraic problems. e Lie algebra Let G be a connected semisimple Lie group with universal cover G, e → G e is called a Cartan involution if g and Killing form κ. An involution σ : G e σ) and (g, dσ) are called globally −κ(·, dσ(·)) is positive-definite. In this case (G, symmetric and infinitesimally symmetric pair respectively. Every globally symmete G e σ , which can be equipped with the ric pair defines a homogeneous space X = G/ e structure of a Riemannian symmetric space. If there exists a G-invariant complex e structure J on X , then the triple (G, σ, J ) is called a global Hermitian symmetric triple. In this case, a theorem of Harish-Chandra ensures that (X , J) can be reale σ, J ) is called of tube type, if D is. The action of G e ized as a bsd D; the triple (G, extends to the Shilov boundary Sˇ of D. This action is in fact transitive and the e A point stabilizers form a conjugacy class of maximal parabolic subgroups of G. maximal parabolic in this conjugacy class will be referred to a s a Shilov parabolic e A marked global Hermitian symmetric triple is a quadruple (G, e σ, J , Q g for G. +) consisting of a global Hermitian symmetric triple together with a Shilov parabolic. Morphisms of such triples are smooth group homomorphisms preserving the additional structure involved. We denote by mHST and mHSTT the categories of all marked global Hermitian symmetric triples respectively those of tube types. Every marked global Hermitian symmetric triple defines a marked infinitesimal Hermitian g symmetric triple (g, dσ, J, q+ ), where J = JeGeσ and q+ is the Lie algebra of Q +. We denote by mhstT the category of all marked infinitesimal Hermitian symmetric triples of tube type.

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Let V be a finite-dimensional (real or complex) vector space, equipped with a (realor complex-) bilinear mapping m : V × V → V,

(x, y) 7→ xy.

We call V (or rather the pair (V, m)) a (real or complex) Jordan algebra if the following three axioms are satisfied: (i) m is commutative, i.e. xy = yx (ii) There exists a unit element e ∈ V with ex = xe = x for all x ∈ V . (iii) For all x, y ∈ V we have x((xx)y) = (xx)(xy). A real Jordan algebra is called Euclidean if there exists a Euclidean inner product (·, ·) on V such that left multiplication by elements of V defines a self-adjoint operator, i.e. ∀x, y, z ∈ V : (xy, z) = (y, xz). In fact, the inner product can always be chosen to be (x, y) = tr(xy), where tr is the Jordan algebra trace in the sense of [15, p. 29]. Moreover [15, Prop. VIII.4.2], V is Euclidean iff it is formally real, i.e. ∀x, y ∈ V :

x2 + y 2 = 0 ⇒ x = y = 0.

A morphism of Jordan algebras is a (real or complex) linear map preserving both the Jordan product and the unit element. We denote by Jor+ the category of Euclidean Jordan algebras. Then we have: Proposition 2.1. There are equivalences of categories ∼ mHSTT ∼ mBSDT = = mhst ∼ = Jor+ . T

This follows immediately from the results in [2]. Since we need the explicit correspondences in the sequel, let us briefly recall them: Let us start from a Euclidean Jordan algebra V and denote by V C := V ⊗ C its complexification. The interior of the set {x2 | x ∈ V } in V is an open cone Ω and thus TΩ := V + iΩ ⊂ V C is a tube. We refer to Ω as the symmetric cone associated with V . Put D(c) := {w ∈ V C | e − w invertible},

D(p) := {z ∈ V C | z + ie invertible}.

Then the Cayley transform c : D(c) → D(p),

c(w) = i(e + w)(e − w)−1

is biholomorphic with inverse p : D(p) → D(c),

p(z) = (z − ie)(z + ie)−1 .

By [15, Theorem X.1.1], we have TΩ ⊂ D(p) and hence we obtain a bounded domain DV := p(TΩ ) of tube type. This bounded domain is in fact symmetric; since we do not need the symmetric structure {σx } explicitly, we refer the reader to [2, Thm. ˇ X.3.2] for a proof of this fact. A marking of DV is obtained by 0 ∈ DV and e ∈ S. Since 0 and e are preserved by Jordan algebra homomorphisms, the assignment Jor+ → mBSDT ,

V 7→ (DV , {σx }, 0, e)

is functorial. Now, given a marked bsd (D, {σx }, o, ξ) we define G = G(D)0 , K := Stabo (G), Q+ := Stabξ (G) and denote by g, k, q+ the respective Lie algebras. Denote by p a Killing-orthogonal complement of k in g. Then the Cartan decomposition g = k ⊕ p defines a Cartan involution dσ with ±1-eigenspaces k, p.

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Denoting by J the restriction of the complex structure of D to p = To D we obtain an assignment mBSD → mhst,

(D, {σx }, o, ξ) → (g, dσ, J, q+ ),

which clearly preserves tube type. For a proof that this assignment is functorial and defines in fact an equivalence of categories one uses [2, Thm. V.1.9]. On the other hand, the equivalence of categories mHSTT ∼ = mhstT is given simply by integration. It thus remains to prove the equivalence of categories mhstT ≡ Jor+ . Given any infinitesimal Hermitian symmetric triple (g, dσ, J) (not necessarily of tube type) we define a Lie triple product [·, ·, ·] : p → p by [X, Y, Z] := [[X, Y ], Z]

(X, Y, Z ∈ p).

The invariance of the complex structure J then yields [JX, Y, Z] = −[X, JY, Z], i.e. the triple (p, [·, ·, ·], J) is a twisted complex Lie triple system in the sense of [2, Def. III.2.1]. Its associated Hermitian Jordan triple system (p, {·, ·, ·}, J) is given by 1 {X, Y, Z} := ([X, Y, Z] − [JX, Y, JZ]). 2 Since the Lie triple product can be reconstructed from the Jordan triple product by the formula [X, Y, Z] = {X, Y, Z} − {Z, X, Y }, the categories of twisted complex Lie triple system and Hermitian Jordan triple products (morphisms being complex linear maps preserving the respective triple product) are equivalent. (This is a special case of Bertram’s Jordan-Lie functor, see [2, Prop. III.2.7]). Moreover, by [2, Thm. V.1.9] both categories are equivalent to the category hst of unmarked Hermitian symmetric triples. Now assume that the Jordan triple system (p, {·, ·, ·}, J) arises from a marked Hermitian symmetric triple system of tube type. Then there exists a unitary tripotent, i.e. an element an element e ∈ p such that {e, e, v} = v for all v ∈ p. Any such tripotent define a Jordan algebra structure on p via (2)

v · w := {v, e, w}.

Now the set of unitary tripotents in p can be canonically identified with the Shilov boundary Sˇ considered before. The marking therefore determines a unique unitary tripotent e by demanding q+ = Stabe (g). This choice of unitary tripotent then determines a specific Jordan algebra structure on p via (2). It remains to find the correct Euclidean real form V of the complex Jordan algebra p. For this we define an involution z 7→ z¯ of p by z¯ = {e, z, e}. Then V := {z ∈ p | z = z¯} is the desired real form and the assignment mhst → Jor+ ,

(g, dσ, J, q+ ) → V

provides the last functor in the circle Jor+ → mBSDT → mHSTT → mhstT → Jor+ . Given a Euclidean Jordan algebra V we will denote by DV the associated bounded symmetric domain. We also abbreviate G := G(DV )0 and Q+ = Stabe (G) as above. The explicit form of the equivalence mHSTT ∼ = Jor+ exhibits a close relation

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between the Shilov parabolic Q+ and the symmetric cone Ω associated with V : Write G(Ω) := {g ∈ GL(V ) | gΩ ⊂ Ω} for the group of linear transformations preserving Ω and denote by L := G(Ω)0 its identity component. (We warn the reader that in the literature on Hermitian Jordan algebras including [15] this group is often denoted G, which conflicts with our notation for G(D)0 .) L is reductive with one-dimensional center and the maximal compact subgroup of L is M := StabL (e). Now we have: Proposition 2.2. Denote by L(Q+ ) the Levi factor of Q+ . Then the Cayley transform c defines an isomorphism ∼ =

L(Q+ )0 − → L,

h 7→ c ◦ h ◦ p.

Proof. We denote by Q− the stabilizer of −e in G. Then Q± are opposite maximal parabolics and thus L(Q+ ) = Q+ ∩Q− . Now let g ∈ L and consider h := p◦g◦c ∈ G. Since g is linear, it fixes 0, whence h fixes p(0) = −e. Moreover, since g acts linearly we obtain h.e =

lim h.((1 − )e)

→0

2− · ie →0 →0 −1 2− · ig.e = lim g.e − e g.e + e = lim p →0 →0 2− 2−

=

lim (p ◦ g)((2 − )ie)(e)−1 ) = lim (p ◦ g)

=

(g.e)(g.e)−1 = e.

Thus h fixes ±e and since g ∈ L was arbitrary we obtain the inclusion p ◦ L ◦ c ⊂ (Q+ ∩ Q− )0 . It remains to show that both groups have the same real dimension. For this we observe that by [15, Thm. III.5.1] we have M = Aut(V )0 and by [15, ˇ whence Prop. X.3.1 and Thm. X.5.3] we have K/M = S, dim Sˇ = dim K − dim M. Since L/M = Ω is open in V and V is homeomorphic to an open subset of Sˇ we also have dim Sˇ = dim V = dim Ω = dim L − dim M, whence (3)

dim L = dim K.

On the other hand, since G/Q± = Sˇ and G/K = D we have dim G − dim K = dim D = 2 dim V = 2 dim Sˇ = 2(dim G − dim Q± ), i.e. dim Q± =

1 (dim G − dim K). 2

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Using (3) and the fact that Q+ Q− is open in G we obtain dim(Q+ ∩ Q− ) = dim Q+ + dim Q− − dim(G) = dim K = dim L, finishing the proof.

The first part of the proof of the Proposition above actually shows that (4)

c ◦ L(Q+ ) ◦ p ⊃ G(Ω).

Since L(Q+ ) need not be connected, the opposite inclusion is not obvious. However, since L(Q+ ) is algebraic, the possible failure of this opposite inclusion can easily be corrected: Corollary 2.3. There exists M ∈ N (depending only on G) such that for all g ∈ c ◦ L(Q+ ) ◦ p the power g M is contained in G(Ω). Proof. Since L(Q+ ) is algebraic and irreducible, π0 (L(Q+ )) is finite. Denote by M the order of this quotient. Now let g ∈ c◦L(Q+ )◦p, say g = c◦h◦p with h ∈ L(Q+ ). Then we have hM ∈ L(Q+ )0 and thus g M = c ◦ hM ◦ p ∈ L by Proposition 2.2. Another application of Proposition 2.2 concerns the following result: Corollary 2.4. Let α : V1 → V2 be a morphism of Euclidean Jordan algebras and αC : V1C → V2C its complexification. Then the following hold: (i) The corresponding bounded symmetric domains satisfy αC (D1 ) ⊂ D2 . (ii) The corresponding Shilov boundaries satisfy αC (Sˇ1 ) ⊂ Sˇ2 . (iii) The corresponding symmetric cones satisfy α(Ω1 ) ⊂ Ω2 . Proof. (i) is an immediate consequence of Proposition 2.1. (ii) follows from the fact that (5) Sˇ = {z ∈ V C | z invertible, z −1 = z¯}. (iii) Denote by α† : (g1 , σ1 , J1 , q+,1 ) → (g2 , σ2 , J2 , q+,2 ) the image of α under the equivalence of categories Jor+ → mhstT . Then α† maps the Levi factor of q+,1 to the Levi factor of q+,2 .Since Jordan algebra homomorphisms preserve the Cayley transform, we derive from Proposition 2.2 that α† maps the Lie algebra of L1 := G(Ω1 )0 to that of L2 . Since Ωj = Lj .e and α(e) = e the result follows. We will use the notations G, K, L, M, Q± throughout this article. If we want to stress the corresponding Euclidean Jordan algebra V then we will add a subscript −V , e.g. writing GV instead of G. We will use the corresponding small gothic letters to denote the associated Lie algebras. 2.2. Idempotents, Jordan frames and maximal polydiscs. Throughout this subsection V denotes a Euclidean Jordan algebra. A basic reference on such Jordan algebras suitable for our purposes is [15]. In view of the equivalence of categories provided by Proposition 2.1 all geometric properties of bsds of tube type are reflected by algebraic properties of the corresponding Euclidean Jordan algebras. Here we shall provide an algebraic description of maximal polydiscs. We start from the observation that a Jordan algebra while not associative in general, is always power-associative in the sense that the subalgebra of V generated by a single element x is associative (see [15, II.1.2]). In particular, the powers xn of an element are

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well-defined. An element x ∈ V C with x2 = x is called idempotent. The following definition is fundamental for the whole theory of Euclidean Jordan algebras: Definition 2.5. A Jordan frame of a Euclidean Jordan algebra V is a family (c1 , . . . , cr ) of non-zero idempotents satisfying the following conditions: (i) each ci is primitive, i.e. it cannot be written as the sum of two non-zero idempotents, (ii) P the idempotents are orthogonal, i.e. ci cj = 0 for i 6= j, r (iii) i=1 ci = e, where e is the unit of V . The cardinality r of a Jordan frame is independent of the choice of Jordan frame and called the rank of the Jordan algebra V ; it coincides with the rank of the associated bsd. The following well-known result is a version of the spectral theorem for Euclidean Jordan algebras: Proposition 2.6. ([15, III.1.2]) Suppose V has rank r. Then for x in V there exists a Jordan frame (c1 , . . . , cr ) and real numbers λ1 , . . . , λr such that x=

r X

λ i ci .

i=1

The following lemma will be used in Section 3 below: Lemma 2.7. Let d1 , . . . , dm be a collection of pairwise orthogonal idempotents in an Euclidean Jordan algebra V with d1 + · · · + dm = e. Then there exists a Jordan frame c1 , . . . , cr of V and numbers i1 < · · · < im < im+1 = r such that ij+1

dj =

X

cj .

ij +1

Proof. If each di is primitive, then we are done. Otherwise, there is at least one non-primitive idempotent among the dj , say d1 . We can then write d1 = f1 + f2 with f1 , f2 idempotents. Then f1 + f2 + d2 + · · · + dm = e. Moreover we have f1 + f2 = d1 = d21 = (f1 + f2 )2 = f12 + 2f1 f2 + f22 = f1 + f2 + 2f1 f2 , whence f1 f2 = 0. This implies in particular that d1 fj = (f1 + f2 )fj = fj2 = fj , whence f1 , f2 are in the 1-eigenspace of d1 , while d2 , . . . , dm are in the 0-eigenspace of d1 . We deduce from [5, Satz I.12.3 a)] that f1 , f2 , d2 , . . . , dm is a collection of pairwise orthogonal idempotents whose sum equals e. Proceeding inductively, we either end up with a Jordan frame or produce larger and larger systems of pairwise orthogonal idempotents. By finite-dimensionality of V , the latter is impossible. We now relate the notion of Jordan frame to the following geometric notion: Definition 2.8. Let D be a bounded symmetric domain of rank r. An isometrically embedded copy of Ds in D with s ≤ r is called polydisc. If r = s it is called maximal. A maximal polydisc is called centered, if it contains 0 ∈ V C .

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Since the embedding is isometric, every polydisc contains a flat of dimension r. In fact, every polydisc is the complexification of a maximal flat. There are lots of maximal polydiscs in D: Proposition 2.9. Let x, y ∈ D. Then there exists a maximal polydisc containing them. Proof. There exists a geodesic joining x and y. This geodesic is a flat and lies therefore in a maximal flat. The complexification of this maximal flat is a maximal polydisc. Now the relation between Jordan frames and polydiscs is provided by the following proposition: Proposition 2.10.

(i) If c = (c1 , . . . , cr ) is a Jordan frame, then the map ( Dr → D ϕc : Pr (λ1 , . . . , λr ) 7→ i=1 λi ci

defines a centered maximal polydisc in D. (ii) For any polydisc P and any Jordan frame (c1 , . . . , cr ) there exists g ∈ G such that X g·P ={ λi ci |λi ∈ D}. P Proof. (i) The map which maps (µ1 , . . . , µr ) ∈ Cr to µi ci is an injective Jordan algebra morphism α : Rr → V and thus induces α0 : Dr → D. The metric structures on both sides come from the Jordan algebra structures. Since they are equal on the image of this map, ϕ is an isometry. It is clearly injective, hence it defines an isometric copy of Dr in D. (ii) Let A ⊂ P be a maximal flat and σ a regular geodesic in A. Then A is the only flat containing σ. There exists h P∈ G(D) such that 0 = h · σ(0). The point h · σ(1) can be written in the form u · λi ci for some u ∈ K (see [15, P X.3.2]). Now put g := u−1 h. The geodesic g·σ is contained in the flats g·A and { λi ci |λi ∈ (−1, 1)}. But being regular, g · σ lies in only one unique flat, hence these two flats have to be equal. Therefore their complexifications have to be equal and we obtain (ii). Notice that the image of the Shilov boundary under the maximal polydisc embedding associated to the Jordan frame (c1 , . . . , cr ) is given by (6)

{

r X

λj cj | |λj | = 1}.

j=1

Proposition 2.11. Let z be in the Shilov boundary. Then there exists a Jordan frame (c1 , . . . , cr ) and complex numbers λi with |λi | = 1 such that z=

z X

λi ci .

i=1

Proof. Write z = x + iy with x, y ∈ V . By [15, Proposition X.2.3] we have (7)

x2 + y 2 = e

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13

and [L(x), L(y)] = 0. In view of [15, Lemma X.2.2] the latter yields the existence of a Jordan frame (c1 , . . . , cr ) and real numbers µ1 , . . . , µr , ν1 , . . . , νr such that x=

r X

µj cj ,

j=1

y=

r X

ν j cj

j=1

and hence z=

r X

λj cj ,

j=1

where λj = µj + iνj . Now (7) implies |λj | = µ2j + νj2 = 1. Hence z is contained in the Shilov boundary of the polydisc embedding ι : Dr → D associated with the Jordan frame (c1 , . . . , cr ). 2.3. Orbits of transverse triples and quadruples. Let V be a Euclidean Jordan algebra, D its bounded symmetric domain and Sˇ the corresponding Shilov boundary. We use the notations G, K, L, M, Q± for the associated groups as introduced in Subsection 2.1. Since Q+ is a maximal parabolic in G, the Shilov boundary Sˇ = G(D)0 /Q+ carries the structure of a generalized flag manifold. In ˇ particular, we can define transversality for points in S: Definition 2.12. gQ+ , hQ+ ∈ Sˇ are transverse, denoted z t w, if Q+ g −1 hQ+ coincides with the unique cell of maximal dimension in the Bruhat decomposition of Sˇ with respect to Q+ . For the definition of Bruhat decomposition see e.g. [16, Thm. 7.40]. We write Sˇ(n) := {(z1 , . . . , zn ) ∈ Sˇn | ∀i 6= j : zi t zj } ˇ Since the G-action preserves for the set of pairwise transversal n-tuples in S. (n) ˇ transversality, each S is a union of G-orbits. For n = 2 we see from the definition that Sˇ(2) is the unique G-orbit in Sˇ2 of maximal dimension This characterization can be used to identify Sˇ(2) in concrete examples. In the case of G = Sp(2n, R) the Shilov boundary is identified with the set L(R2n ) of Lagrangian subspaces of R2n . Classically, two Lagrangian subspaces V, W of R2n are called transverse if V ⊕ W = R2n . Clearly, L(R2n )(2) = {(V, W ) ∈ L(R2n )2 | V ⊕ W = R2n } is an open Sp(2n, R)-orbit, hence Sˇ(2) = L(R2n )(2) in this case. For n = 3 orbits in Sˇ(3) can be classified in terms of the Maslov index µSˇ of Sˇ as defined in [13]. Rather than repeating the definition, we explain a simple way to compute the Maslov index based on the following result: Proposition 2.13 (Clerc-Neeb,[12, Theorem 3.1]). Any triple (z1 , z2 , z3 ) ∈ Sˇ3 is contained in the Shilov boundary of an embedded maximal polydisc. Combining Proposition 2.13 and Proposition 2.10 we find for every (z1 , z2 , z3 ) ∈ Sˇ3 a Jordan frame (c1 , · · · , cr ) and g ∈ G(D) such that g · zi =

r X j=1

λij cj

(i = 1, 2, 3)

14


for some λij ∈ S 1 . If (z1 , z2 , z3 ) ∈ Sˇ(3) then all the triples (λ1j , λ2j , λ3j ) are pairwise transverse, hence contained in the domain of the orientation cocycle o : (S 1 )(3) → {±1}, which takes value ±1 depending on whether the given triple is positively or negatively oriented. The results of Clerc and Ørsted in [13] then imply that µSˇ (z1 , z2 , z3 ) = µSˇ (g · z1 , g · z2 , g · z3 ) =

r X

o(λ1j , λ2j , λ3j ).

j=1

From this description we see immediately that µ ˇ (Sˇ(3) ) = {−r, −r + 2, . . . , r − 2, r}. S

A triple (z1 , z2 , z3 ) ∈ Sˇ(3) with µSˇ (z1 , z2 , z3 ) = r is hence called maximal. As mentioned above, the Maslov index classifies triples of pairwise transverse points: Proposition 2.14 (Clerc-Ørsted, [13, Theorem 4.3]). Two triples (z1 , z2 , z3 ), (w1 , w2 , w3 ) ∈ Sˇ(3) lie in the same G(D)0 orbit iff µSˇ (z1 , z2 , z3 ) = µSˇ (w1 , w2 , w3 ). The classification of orbits in Sˇ3 is much more involved; see [12]. Here we are interested in generalized cross ratios, which are defined on a subset of Sˇ4 . Unfortunately, the analog of Proposition 2.13 fails in this case, i.e. not every orbit Sˇ(4) contains a representative in the boundary of a maximal polydisc. We shall use the following proposition as a substitute: Proposition 2.15. Let (z1 , . . . , z4 ) ∈ Sˇ(4) , and suppose (zi , zj , zk ) is maximal for some {i, j, k} ⊂ {1, . . . , 4}. Then z1 , . . . , z4 are contained in the boundary of a common maximal polydisc. Proof. We may assume w.l.o.g. that (z1 , z2 , z3 ) is maximal. Since µSˇ (z1 , z2 , z3 ) = r = µSˇ (−e, −ie, e), it follows from Proposition 2.14 that there exists g ∈ G with g.(z1 , z2 , z3 ) = (−e, −ie, e). Let z = g.z4 . By Proposition 2.11 there exists a Jordan frame (c1 , . . . , cr ) and λi ∈ C with |λi | = 1 such that z=

r X

λ i ci .

i=1

We deduce that g.(z1 , . . . , z4 ) = (

X X X X (−1) · cj , (−i) · cj , 1 · cj , λ j cj )

is contained in the Shilov boundary of the polydisc embedding ι : Dr → D associated with the Jordan frame (c1 , . . . , cr ). Correspondingly, (z1 , . . . , z4 ) is contained in the Shilov boundary of the embedded maximal polydisc g −1 ◦ ι. 3. Normalized kernels of Euclidean Jordan algebras After the preliminary results in Section 2 we now turn to the construction of generalized cross ratios. In view of the above equivalence of categories, we will work with Jordan algebras rather than bounded symmetric domains. Thus our problem is to construct a generalized cross ratio BV on 4-tuples on the Shilov boundary

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15

SˇV associated with a Euclidean Jordan algebra V . In view of the various cocycle relations expected from a generalized cross ratio, a natural ansatz is to define kV (d, a)kV (b, c) BV (a, b, c, d) := kV (d, c)kV (b, a) for some function kV : Sˇ × Sˇ → C. In order to obtain G-invariance this function should have some nice equivariance properties. If BV is supposed to be functorial, then also the normalization of kV has to be chosen carefully. Since this is a delicate issue we devote a whole section to the construction of the kernels kV and their basic properties. The actual study of the corresponding cross ratios will then be undertaken in the next section. The reader willing to take the main properties of the kernels kV on faith can skip this section except for the definition of balanced Jordan algebra homomorphisms (Definition 3.11) which will be used throughout. The first subsection introduces automorphy kernels and their relation to various notions of transversality. Based on this notion we construct in subsequent subsections the kernels kV first for simple Euclidean Jordan algebras, and then for general Euclidean Jordan algebras. Finally, we compare the our normalized kernels kV to the Bergman kernels of the corresponding bounded symmetric domains. 3.1. The automorphy kernel and transversality. Let V be a Euclidean Jordan algebra. Given z ∈ V C we denote by L(z) the left-multiplication by z. Then for all z, w ∈ V C the box operator and the quadratic representation are defined by zw := L(zw) + [L(z), L(w)], and P (z) := 2L(z)2 − L(z 2 ) respectively. Following [13] (see also [15] and [18]) we define the automorphy kernel K : V C × V C → End(V C ), by K(z, w) := I − 2zw + P (z)P (w). Example 3.1. Let V = R so that V C = RC = C. For x, w, z ∈ C we then have (zw)x = L(zw)x + [L(z), L(w)]x = (zw)x, whence zw is multiplication with zw. Similarly, P (z)x = (2L(z)2 − L(z 2 ))x = z 2 x, hence P (z) is multiplication by z 2 . Combining these two observations we can calculate K(z, w): We obtain K(z, w)x = x − 2z wx ¯ + z2w ¯ 2 = (1 − z w) ¯ 2x and thus K(z, w) is multiplication by (1 − z w) ¯ 2. We also use the quadratic representation to define the structure group of V C to be Str(V C ) := {g ∈ GL(V C ) | P (gx) = gP (x)g > }. If z, w ∈ D then K(z, w) ∈ Str(V C ) [13, p. 315]. Let us briefly describe the behavior of K under automorphisms. Since we do not want to digress to deeply into the structure theory of bounded symmetric domain, we refer the reader to [18,

16


Chapter II, Section 5] for the definition of the canonical automorphy factor J of a bounded symmetric domain D. All we have to know at this place is that J restricts to a map J : D × V C → Str(V C ) satisfying K(gz, gw) = J(g, z)K(z, w)J(g, w)∗

(8)

for g ∈ G(D)0 , z, w ∈ D. The automorphy kernel allows us to detect transversality ˇ on the Shilov boundary S: Proposition 3.2. Let V be a Euclidean Jordan algebra, D the associated bounded symmetric domain and Sˇ its Shilov boundary. Then z, w ∈ Sˇ are transverse iff one of the following conditions holds true: (i) (ii) (iii) (iv)

det(z − w) 6= 0. K(z, w) is invertible. K(z, w) ∈ Str(V C ). det K(z, w) 6= 0.

Proof. Let us first prove that z, w are transverse iff (iv) holds. Indeed, it follows from (8) that {(z, w) ∈ Sˇ | det K(z, w) 6= 0} is a G-orbit. By continuity of det K(·, ·) this orbit is open, hence coincides with Sˇ(2) . It thus remains to prove equivalence of (i)-(iv).The implication (i) ⇒ (ii) is provided in [13, Lemma 5.1]. The implication (ii) ⇒ (iii) follows from the fact that K(z0 , w0 ) ∈ Str(V C ) for z0 , w0 ∈ D together with the continuity of K and the fact that Str(V C ) is closed in GL(V C ). Finally, the implication (iii) ⇒ (iv) is obvious. Thus it remains to show (iv) ⇒ (i). Thus let w, z ∈ Sˇ be arbitrary and assume det K(z, w) 6= 0. We first claim that there exists g ∈ G such that e−g ·w and e−g ·z are invertible. Indeed, if D is a polydisc with w = (w1 , . . . , wr ) and z = (z1 , . . . , zr ), then one can clearly find an element g = (g1 , . . . , gr ) ∈ SO(2)r ⊂ G such that gi · wi and gi · zi are both not equal to 1. The general case is reduced to this case by applying Proposition 2.13 to the triple (e, w, z), thereby finishing the proof of the claim. Next observe that (8) implies det K(gz, gw) = det(J(g, z)) det K(z, w) det(J(g, w)), {z } | {z } | 6=0

6=0

and thus our assumption yields det K(gz, gw) 6= 0. Now note that with e − gw also e − gw = e − gw is invertible , hence [15, Lemma X.4.4 ii)] applies and yields K(gz, gw) = P (e − gz)P (c(gz) + c(gw))P (e − gw), whence det(P (c(gz) + c(gw))) 6= 0. A simple calculation shows that c(gw)) = −c(gw). Since gw ∈ Sˇ ∩ D(c), the image c(gw) is contained in V , whence c(gw) = c(gw). We thus obtain det(P (c(gz) − c(gw))) 6= 0. Using the definition of the Cayley transform and [15, p.190] we obtain c(gz) − c(gw) =i (e + gz)(e − zg)−1 − (e + gw)(e − gw)−1 =i − ie + 2i(e − gz)−1 + ie − 2i(e − gw)−1 = − 2 (e − gz)−1 − (e − gw)−1 .

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17

We thus obtain det P (−2((e − gz)−1 − (e − gw)−1 )) 6= 0.

(9)

Now we can apply Hua’s formula [15, Lemma X.4.4] to obtain P (−2((e − gz)−1 − (e − gw)−1 )) =P (e − gz)−1 P (−2((e − gz) − (e − gw)))P (e − gw)−1 =P (e − gz)−1 P (−2(gw − gz)))P (e − gw)−1 . Combinining this with (9) and using that P (e−gz)−1 and P (e−gw)−1 are invertible, we obtain det P (−2(gw − gz))) 6= 0. Thus P (−2(gw − gz)) is invertible. By [15, Prop. II.3.1] this implies that −2(gw − gz) and hence gz − gw is invertible. Thus det(gz − gw) 6= 0, which by [13, Prop. 3.2] implies det(z − w) 6= 0. This finishes the proof of Proposition 3.2. Proposition 3.2 implies immediately: Corollary 3.3. The image p(V ) of V under the inverse Cayley transform is preˇ which are transverse to e. cisely the subset of points in S, 3.2. The normalized kernel function I: The simple case. The aim of this subsection is to define a suitably normalized kernel functions on the bounded symmetric domain D of a given simple Euclidean Jordan algebra V , whose extension to the Shilov boundary reflects transversality. We have seen in Proposition 3.2 that transversality is characterized by non-singularity of the automorphy kernel (2) K : V C × V C → End(V C ), which when restricted to D := D ∪ Sˇ(2) takes values C in the structure group Str(V ). In order to obtain a numerical kernel function, one can compose K with an arbitrary character χ of Str(V C ). We denote the resulting function by (2)

kχ : D → C× , (z, w) 7→ χ(K(z, w)). Since we would like our kernel functions to be compatible under (certain) Jordan algebra homomorphisms, we will have to choose the character χ carefully. Recall our assumption that V is supposed to be simple and denote by n := dim V its 1 dimension. We then choose χ := det 2n and denote the associated kernel function 1 . Here, the fractional exponent has to be interpreted as follows: If by kV := k 2n det

kdet : D

(2)

→ C× denotes the kernel function det(K(z, w)), then kV is defined as (2)

the unique function D → C× satisfying kV2n = kdet and kV (0, 0) = 1. The kernel kV extends in fact continuously to all of Sˇ2 due to the following general lemma: Lemma 3.4. Let X be a manifold, f : X → C be a continuous function, X 0 := f −1 (C \ {0}). Let fe : X 0 → C \ {0} be any continuous function with fen = f |X 0 . Then fe extends continuously by 0 to all of X. Proof. Extend fe to all of X by 0. We show that this extension is continuous. For this let xk ∈ X 0 with xk → x, where x ∈ X \ X 0 . Then f (xk ) → f (x) = 0 by continuity of f , hence fe(xk )n → 0. This, however, implies already fe(xk ) → 0 = fe(x), which yields continuity of the extended function.

18


Applying this to the continuous function kdet we obtain the desired extension of kV to Sˇ2 . In view of Proposition 3.2 and (8) this extension has the following properties: Corollary 3.5. Let V be a simple Euclidean Jordan algebra. Then the normalized kernel kV : Sˇ2 → C satisfies kV (z, w) 6= 0 ⇔ z t w and kV (gz, gw) = jV (g, z)kV (z, w)jV (g, w), where jV := det

1 2n

ˇ (g ∈ G, z, w ∈ S),

◦J.

The factor 12 in the normalization exponent normalization:

1 2n

was added to achieve the following

Example 3.6. Let V = (R, ·) so that D = D is the unit disc. Since dim V = rk(V ) = 1 it follows from Example 3.1 that kdet (z, w) = (1 − zw)2 . Now our normalization is chosen in such a way that kR (z, w) = 1 − zw. This additional normalization factor does not affect the proof of the following desired invariance property: Lemma 3.7. Let α : V1 → V2 be an injective morphism of simple Jordan algebras. Denote by D1 and D2 respectively the corresponding bounded symmetric domains and by Sˇ1 and Sˇ2 the respective Shilov boundaries. Then for all z, w ∈ D1 ∪ Sˇ1 we have kV2 (αC (z), αC (w)) = kV1 (z, w).

(10)

Proof. By continuity it suffices to prove (10) for z, w ∈ D. We would like to apply [13, Prop. 6.2]; for this we have to translate this proposition into our language. Define a character χj of Str(VjC ) by det(gz) = χj (g) det(z)

(g ∈ Str(VjC ), z ∈ Dj ),

where det is the Jordan algebra determinant. Denote by rj and nj respectively the rank and dimension of Vj . Then the proposition of Clerc and Ørsted states that r2

kχ2 (αC (z), αC (w)) = kχ1 (z, w) r1 . On the other hand, since the Vj are simple, [15, Prop. III.4.3] applies and shows rj

χj = det nj . We thus obtain kV2 (αC (z), αC (w))

= = = = =

1

det 2n2 (K(αC (z), αC (w))) h i 2r1 r2 2 det n2 (K(αC (z), αC (w))) 1 kχ2 (αC (z), αC (w)) 2r2 1 h r2 i 2r 2 kχ1 (z, w) r1 h i 2r1 r1 1 det n1 K(z, w)

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19

1

=

det 2n1 K(z, w)

=

kV1 (z, w).

We have been slightly sloppy here by not specifying the arc of the various roots. Still, it is easy to make the above argument precise. Strictly speaking we have only shown that kV2 (αC (z), αC (w))2n1 = kV1 (z, w)2n1 rather than kV2 (αC (z), αC (w)) = kV1 (z, w). However, since kV2 (αC (0), αC (0)) = kV1 (0, 0) the former implies the latter by uniqueness. We will allow ourselves this kind of sloppyness regarding roots, whenever it is clear how to make the arguments precise.

3.3. The normalized kernel function II: The general case. We would like to define a normalized kernel with invariance properties generalizing those of Lemma 3.7 also in the case where the involved Jordan algebras are not simple. For this we first observe that any Euclidean Jordan algebra V is semisimple ([15, Prop. III.4.4]) and thus decomposes into simple ideals, say V = V1 ⊕ · · · ⊕ Vn . Accordingly we identify elements z ∈ V with vectors z = (z1 , . . . , zn )> with zj ∈ Vj and define kV : D2 → C× by the formula (11)

kV (z, w) =

n Y

rk Vi

(kVi (zi , wi )) rk V .

i=1 2 × More precisely, if kf V : D → C is given by the formula

kf V (z, w) :=

n Y

(kVi (zi , wi ))rk Vi .

i=1

then kV is defined to be the unique function satisfying kV (z, w)rk(V ) = kf V (z, w),

kV (0, 0) = 1.

As before Lemma 3.4 implies that kV extends to a function D2 ∪ Sˇ2 → C, which vanishes precisely on non-transversal pairs. Observe that for V simple the function kV coincides with the normalized kernel kV defined earlier so that there it is justified to use the same notation and to refer to kV in general as the normalized kernel function of V . Moreover, the following property follows right from the definition: Proposition 3.8. Let V be a Euclidean Jordan algebra. If V = V1 ⊕ V2 is the sum of two (not necessarily simple) ideals of ranks r1 , r2 with projections pj : V → Vj then kV (z, w)r1 +r2 = kV1 (p1 (z), p1 (w))r1 kV2 (p2 (z), p2 (w))r2 . Let us spell this out in the case of a polydisc:

20


Example 3.9. Let V = (R, ·)r so that D = Dr is the standard polydisc. In our sloppy notation the normalized kernel function kRr of the polydisc Dr is k ((z1 , . . . , zr ), (w1 , . . . , wr )) = Rr

r Y

1

(1 − zj wj ) r .

j=1

We can see already from the case of polydiscs that our normalized kernel is not invariant under arbitrary Jordan algebra homomorphisms: Example 3.10. Consider the Jordan algebra embedding α : R2 → R3 given by (λ1 , λ2 ) 7→ (λ1 , λ1 , λ2 ). Then kR2 (λ, µ)

1

1

2

1

=

(1 − λ1 µ1 ) 2 (1 − λ2 µ2 ) 2

6=

(1 − λ1 µ1 ) 3 (1 − λ2 µ2 ) 3 = kR3 (αC (λ), αC (µ)).

We want to exclude bad behavior as in the last example. Denote by trV the Jordan algebra trace of V . Thus if (c1 , . . . , cr ) is a Jordan frame for V then x=

r X

λj cj ⇒ trV (x) =

j=1

r X

λj .

j=1

Now we define: Definition 3.11. A Jordan algebra homomorphism αV : V → W is called balanced if for all v ∈ V 1 1 trV (v) = trW (α(v)). rk V rk W The notion is clearly invariant under complexification. Moreover, we have the following characterization of balanced Jordan algebra homomorphisms: Let (c1 , . . . , cr ) be a Jordan frame in V and α : V → W a Jordan algebra homomorphism. Then P α(c1 ), . . . , α(cr ) is a family of idempotents with α(ci )α(cj ) = 0 and α(ci ) = e. Thus Proposition 2.7 applies and there exists a Jordan frame (c11 , . . . , c1l1 , . . . , cr1 , . . . , crlr ) of W such that lj X α(cj ) = cjk . k=1

We have trW (α(cj )) = lj and thus α is balanced if and only if l1 = · · · = lr . Conversely, if the latter condition is true for any Jordan frame (c1 , . . . , cr ) of V , then α is balanced. Note that we obtain in particular rk W = lj · rk V

(j = 1, . . . , r),

so that rk W is divisible by rk V . In particular, the morphism in Example 3.10 is not balanced. Also note that a balanced Jordan algebra homomorphism is injective unless it is trivial. In our attempts to prove an invariance theorem we will restrict attention to balanced Jordan algebra homomorphisms. Even in this case we cannot quite obtain the same kind of invariance as in Lemma 3.7. In order to formulate our (slightly weaker) result, we introduce the following terminology: Two elements

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21

v1 , v2 are called co-diagonalizable if there exists a Jordan frame (c1 , . . . , cr ) and elements λj ∈ D, µj ∈ D such that v1 =

r X

λj cj ∈ DV ,

v2 =

j=1

r X

µj cj ∈ DV .

j=1

Then we have: Theorem 3.12. Let α : V → W be a non-trivial Jordan algebra homomorphism. If α is balanced, then for every pair of co-diagonalizable elements v1 , v2 ∈ D we have kW (αC (v1 ), αC (v2 )) = kV (v1 , v2 ).

(12)

Conversely, if (12) holds for all co-diagonalizable v1 , v2 ∈ DV , then α is balanced. For the proof we consider first the case, where V is a maximal polydisc in W . In this case we have the following version of Theorem 3.12, which is a slight extension of the results of Clerc and Ørsted in [13]: Lemma 3.13. If (c1 , . . . , cr ) is a Jordan frame in a Jordan algebra W and λj ∈ D, µj ∈ D, then (13)

r r r X X Y 1 kW ( λ j cj , µj cj ) = (1 − λj µj ) r . j=1

j=1

j=1

Proof. Assume first that W is simple. Then [13, Lemma 5.4] applies directly and we obtain r r r X X Y kχW ( λj cj , µj cj ) = (1 − λj µj )2 , j=1

j=1

j=1

whence kχW (

r X j=1

λj cj ,

r X

 µj cj )r = 

j=1

r Y

2r (1 − λj µj )

.

j=1

Since r Y

(1 − 0 · 0) = 1

j=1

this settles the simple case. For the general case, consider a decomposition W = W1 ⊕ · · · ⊕ Wn into simple ideals. Let rl := rk(Wl ) and (cl1 , . . . , clrl ) be a Jordan frame for Wl . Then (c11 , . . . , cnrn ) is a Jordan frame for W and, in fact, any Jordan frame for W is of this form (as follows e.g. from [15, Prop. X.3.2]). Let z :=

rl n X X

λlj clj ,

l=1 j=1

w :=

rl n X X

µlj clj .

l=1 j=1

By definition we have kW (z, w)rk W =

n Y (kWl (zl , wl ))rk Wl , l=1

22


where zl =

rl X

λlj cjl ,

wl :=

j=1

rl X

µlj clj .

j=1

By the simple case we have (kWl (zl , wl ))

rk Wl

=

rl Y

(1 − λlj µj ),

j=1

and thus rk W

kW (z, w)

=

rl n Y Y

(1 − λlj µj ).

l=1 j=1

From this the general case follows easily: Proof of Theorem 3.12. If α is balanced, then rV := rk(V ) and rW := rk(W ) are related by rW = µα rV for some constant µα . Given a Jordan frame (c1 , . . . , cr ) in V and elements r r X X v1 = λj cj ∈ DV , v2 = µj cj ∈ DV j=1

j=1

with λj ∈ D, µj ∈ D we have C

α (v1 ) =

r X

C

λj α(cj ),

α (v2 ) =

r X

µj α(cj ).

j=1

j=1

Now each α(cj ) decomposes as α(cj ) = dj1 + · · · + djµα , where the djl are primitive idempotents. Now we obtain kV (v1 , v2 )rV =

r Y

(1 − λj µj ),

j=1

whence

 kV (v1 , v2 )rW = 

r Y

µα (1 − λj µj )

=

j=1

r Y

(1 − λj µj )µα .

j=1

Similarly, kW (αC (v1 ), αC (v2 ))rW =

r Y

(1 − λj µj )µα .

j=1 C

C

rW

As kV (0, 0) = kW (α (0), α (0)) , this implies (12). On the other hand, if α is not balanced and v1 , v2 are as above, then r Y 1 kV (v1 , v2 ) = (1 − λj µj ) r V 6=

j=1 r Y

(1 − λj µj )

µα (cj ) rW

j=1

= kW (αC (v1 ), αC (v2 )).

CROSS RATIOS

23

Example 3.14. The following are examples of balanced Jordan algebra homomorphisms: • Jordan algebra homomorphisms α : V → W between simple Jordan algebras are balanced by Lemma 3.7. • If rk(V ) = rk(W ) then every injective Jordan algebra homomorphism α : V → W is balanced. • In particular, maximal polydisc embeddings are balanced. • Any Jordan algebra homomorphism α : R → W is balanced. • Compositions of balanced Jordan algebra homomorphisms are balanced. So far we have been working on the interior of a bounded symmetric domain. Of course, by continuity our discussion extends the Shilov boundary. In particular we have: ˇ 2 → C the Corollary 3.15. Let V be a Euclidean Jordan algebra and kV : (D ∪ S) associated normalized kernel. Then ˇ then kV (z, w) 6= 0 ⇔ z t w. (i) If z, w ∈ S, (ii) There exists a continuous function jV : G × V → C× such that kV (gz, gw) = jV (g, z)kV (z, w)jV (g, w),

ˇ (g ∈ G, z, w ∈ S).

(iii) If (c1 , . . . , cr ) is a Jordan frame for V and λ1 , . . . , λr , µ1 , . . . , µr ∈ D, then (14)

kV (

r X j=1

λj cj ,

r X

µj cj ) =

r Y

1

(1 − λj µj ) r .

j=1

j=1

3.4. Comparison to the Bergman kernel. Given a Euclidean Jordan algebra V with associated bounded symmetric domain D, we can forget about the algebraic structure of V and consider D ⊂ V simply as a bounded domain. From this data we can then define the Bergman kernel KD : D × D → C× as the reproducting kernel of the Bergman space H2 (D). It turns out that our normalized kernel kV is uniquely determined by KD . More precisely we have: Proposition 3.16. The restriction kV : D × D → C× of the normalized kernel and the Bergman kernel KD are related by the formula kV (z, w)r = C · KD (z, w)−1 , where C is some constant depending only on the domain D. Proof. By [15, Prop. X.4.5] the Bergman kernel on D is of the form KD (z, w) =

1 · det(K(z, w))−1 . Vol(D)

Let us first assume that V is simple of rank r. Then 1

1

kV (z, w) = det(K(z, w)) r = Vol(D) · KD (z, w)− r .

24


In the general case we decompose P V = V1 ⊕ · · · ⊕ Vn into simple ideals with ri := rk(Vi ) so that r := rk(V ) = ri . Now the Bergman kernel is multiplicative, i.e. we have n Y KD = KDi . i=1

Now the simple case yields −1 kVrii = Vol(Di )ri · KD , i

and thus by definition of kV we have kV

=

n Y

k

ri r Vi

i=1

=

=

n Y i=1 n Y

=

n Y

! r1 kVrii

i=1

! r1 −1 Vol(Di )ri · KD i

Vol(Di )

ri r

!

−1

· KD r .

i=1

Note that the relation to the Bergman kernel together with the normalization kV (0, 0) = 1 determines kV uniquely. Thus, while the definition of kV involves the Jordan algebra V and thus the structure of D as a marked bounded symmetric domain, the above proposition shows that the result depends only on D as a bounded domain. 4. Generalized cross ratios on Shilov boundaries In this section we use the normalized kernel functions defined in the last section in order to construct a family of generalized cross ratios on Shilov boundaries of bounded symmetric domains of tube type. In the first subsection we provide an axiomatic characterization of these generalized cross ratios. In the second subsection we give an explicit construction and show that this construction satisfies the desired axioms. The third subsection explains how to compute generalized cross ratios using maximal polydiscs. In the fourth subsection we deduce a number of cocycle properties of the generalized cross ratios. In the final subsection we express the generalized cross ratio in terms of Bergman kernels, thereby showing that it is independent of the marking of the bounded symmetric domain. 4.1. The axiomatic approach. Before we can define generalized cross ratios, we have to clarify, which properties these functions should have. We will thus collect in this subsection a number of axioms that our functions are supposed to satisfy. We will then see in the subsequent subsections that such functions exist and are in fact uniquely determined by the axioms. The object that we want to generalize is the classical four point cross ratio of hyperbolic geometry. Recall that the latter is defined on quadruples of mutually distinct points in CP1 by the formula

CROSS RATIOS

(15)

[a : b : c : d] :=

25

(a − d)(c − b) . (c − d)(a − b)

This cross ratio restricts to a real-valued function on (S 1 )(4) , which is invariant under P U (1, 1). We would like to generalize this cross ratio from S 1 to the Shilov boundary SˇV of an arbitrary Euclidean Jordan algebra, that is, we would like to define a family of functions (4) BV : SˇV → R× , which are invariant under the respective groups GV . Moreover, these generalized cross ratios should be related to each other. E.g. it would be nice to have (16)

BW (α(v1 ), α(v2 ), α(v3 ), α(v4 )) = BV (v1 , v2 , v3 , v4 )

for any Jordan algebra homomorphism α : V → W and v1 , v2 , v3 , v4 ∈ SˇV . This, however, is too much of wishful thinking. Firstly, we should not ask (16) for arbitrary Jordan algebra homomorphism α : V → W since these can be rather badly behaved, as we have seen before. Rather, as suggested by Theorem 3.12, we should restrict to balanced homomorphism. Secondly, if we insist that our generalized cross (4) ratio should be real-valued then we cannot define it on all of SˇV . A posteriori, the following domain turns out to be the correct one: Definition 4.1. Let V be a Euclidean Jordan algebra, D its bounded symmetric domain and Sˇ the associated Shilov boundary. A quadruple (a, b, c, d) ∈ Sˇ(4) is called extremal if any triple (x, y, z) ∈ Sˇ3 of pairwise distinct points with x, y, z ∈ {a, b, c, d} is either maximal or minimal. We denote the set of extremal quadruples in Sˇ4 by Sˇ(4+) . Note that an extremal quadruple is contained in the Shilov boundary of a maximal polydisc by Proposition 2.15. Now we can formulate the axioms, which the desired extension of the classical cross ratio should satisfy: Theorem 4.2. There exists a unique system of functions (4+) {BV : SˇV → R× | V Euclidean Jordan algebra}

with the following properties: (i) If DV is the bounded domain associated with V , then BV is G(DV )0 invariant. (ii) If α : V → W is a balanced Jordan algebra homomorphism, (v1 , . . . , v4 ) ∈ (4+) (4+) SˇV and (α(v1 ), . . . , α(v4 )) ∈ SˇW , then BW (α(v1 ), . . . , α(v4 )) = BV (v1 , . . . , v4 ). (iii) If V = V1 ⊕ V2 is the sum of two ideals of ranks r1 , r2 with projections pj : V → Vj then BV (v1 , . . . , v4 )r1 +r2 = BV1 (p1 (v1 ), . . . , p1 (v4 ))r1 BV2 (p2 (v1 ), . . . , p2 (v4 ))r2 . (iv) BR is the restriction of the classical four point cross ratio. Condition (iii) was added here, to rigidify the situation and to obtain uniqueness. It is unclear to us, whether it follows from the other three axioms.

26


4.2. Explicit construction of generalized cross ratios. The proof of Theorem 4.2 is constructive, i.e. we will construct the functions BV explicitly. The functions BV extend to all of Sˇ(4) and these extensions are given as follows: Definition 4.3. Let V be a Euclidean Jordan algebra. Then the cross ratio of V is the function BV : Sˇ(4) → C× given by kV (d, a)kV (b, c) BV (a, b, c, d) := . kV (d, c)kV (b, a) We claim that the cross ratios satisfy the conditions of Theorem 4.2. The proof of this fact will occupy the remainder of this section. We start by checking condition (iv): Example 4.4. Let V = (R, ·) so that Sˇ = S 1 . By Example 3.6 we have kR (a, b) = 1 − a¯b. Then for four mutually distinct points a, b, c, d ∈ S 1 we have (a − d)(c − b) (1 − d¯ a)(1 − b¯ c) = = [a : b : c : d] BR (a, b, c, d) = (1 − d¯ c)(1 − b¯ a) (c − d)(a − b) Condition (iii) follows directly from Proposition 3.8. For the proofs of the remaining properties it will be useful to extend BV to a continuous function eV : D(4) → C× , (a, b, c, d) 7→ kV (d, a)kV (b, c) , B (17) kV (d, c)kV (b, a) (4) eV is the unique where D := D4 ∪ Sˇ(4) . If V is simple then by definition B continuous function satisfying eV (a, b, c, d)2 dim V = BV,det (a, b, c, d), B eV (0, 0, 0, 0) = 1, (18) B

where BV,det : D

(4)

→ C× ,

(a, b, c, d) 7→

kdet (d, a)kdet (b, c) . kdet (d, c)kdet (b, a)

eV and hence the restriction BV in the simple case, and the This characterized B general case can be reduced to this situation using property (iii). Using this characterization we now prove: Proposition 4.5. Let V be a Euclidean Jordan algebra. Then the cross ratio BV is G-invariant, i.e. for all (a, b, c, d) ∈ Sˇ(4) we have BV (a, b, c, d) = BV (ga, gb, gc, gd)

(g ∈ G).

Proof. Let us first assume that V is simple. By Corollary 3.5 we have kdet (gz, gw) = jdet (g, z)kdet (z, w)jdet (g, w) (2)

for all g ∈ G, w, z ∈ D . From this we obtain for all g ∈ G and (a, b, c, d) ∈ D4 ∪ Sˇ(4) the following relation: kdet (gd, ga)kdet (gb, gc) BV,det (ga, gb, gc, gd) = kdet (gd, gc)kdet (gb, ga) =

jdet (g, d)kdet (d, a)jdet (g, a)jdet (g, b)kdet (b, c)jdet (g, c) jdet (g, d)kdet (d, c)jdet (g, c)jdet (g, b)kdet (b, a)jdet (g, a)

CROSS RATIOS

27

kdet (d, a)kdet (b, c) kdet (d, c)kdet (b, a) = BV,det (a, b, c, d).

=

This shows G-invariance of BV,det . As a consequence, we obtain for all g ∈ G eV (ga, gb, gc, gd)2 dim V = BV,det (ga, gb, gc, gd) = BV,det (a, b, c, d) B eV (g.0, g.0, g.0, g.0) = 1 and B eV is uniquely determined by (18), this implies Since B e e B(ga, gb, gc, gd) = B(a, b, c, d). Restricting to the Shilov boundary, we obtain the statement of the proposition for simple V . The general case is easily reduced to the simple case using Condition (iii). In order to deduce condition (ii) we make use of the equivalence of categories observed in Proposition 2.1: Given a Jordan algebra homomorphisms α : V → W , eV → G e W which is equivariant with there exists a group homomorphism α† : G C e e respect to α : DV → DW . (Here GV , GW denote the universal coverings of GV e V there exists h ∈ G e W such that for all v ∈ DV and GW .) In particular, given g ∈ G αC (gv) = hαC (v).

(19)

eV , G e W factor through the actions of GV and GW respectively Since the actions of G that for every g ∈ GV there exists h ∈ GW such that (19) holds for all v ∈ V . Now we can prove Condition (ii): If (v1 , . . . , v4 ) is extremal then by Proposition 2.15 we find g ∈ GV such that gv1 , . . . , gv4 are diagonalized by a common Jordan frame (c1 , . . . , cr ). Let h ∈ GW be an element such that (19) holds for all v ∈ V . Using Proposition 4.5 and Theorem 3.12 we now obtain BV (v1 , . . . , v4 )

= BV (gv1 , . . . , gv4 ) kV (gv4 , gv1 )kV (gv2 , gv3 ) = kV (gv4 , gv3 )kV (gv2 , gv1 ) kW (αC (gv4 ), αC (gv1 ))kW (αC (gv2 ), αC (gv3 )) = kW (αC (gv4 ), αC (gv3 ))kW (αC (gv2 ), αC (gv1 )) kW (hαC (v4 ), hαC (v1 ))kW (hαC (v2 ), hαC (v3 )) = kW (hαC (v4 ), hαC (v3 ))kW (hαC (v2 ), hαC (v1 )) = BW (hα(v1 ), . . . , hα(v4 )) = BW (α(v1 ), . . . , α(v4 )).

This shows that the cross ratios satisfy Conditions (i) - (iv) of Theorem 4.2. Conversely, these conditions determine the family {BV }: Any (a, b, c, d) ∈ Sˇ(4+) is contained in the boundary of a maximal polydisc by Proposition 2.15. Since the embedding of a maximal polydisc is balanced, the family {BV } is uniquely determined by the family {BRr }. Finally, condition (iii) implies that Y BRr (a, b, c, d)r = BR (aj , bj , cj , dj ). The right hand side is uniquely determined by (iv). If we denote by α : R → Rr the diagonal embedding, then α is balancing and therefore BRr (0, 0, 0, 0) = BRr (α(0), . . . , α(0)) = BR (0, 0, 0, 0),

28


and again the latter is determined by (iv). The last two equations determine {BRr } uniquely. The uniqueness of the whole system {BV } follows. Thus we have proved Theorem 4.2 up to the claim that the cross ratio is realvalued on Sˇ(4+) . This latter fact will be proved in the next subsection, where we also provide techniques to compute the cross ratio effectively. 4.3. Computation of generalized cross ratios. Let V be a Euclidean Jordan algebra and (a, b, c, d) ∈ Sˇ(4+) . The aim of this subsection is to provide an effective way to compute BV (a, b, c, d). As a byproduct we will see that BV (a, b, c, d) ∈ R, thereby finishing the proof of Theorem 4.2. As a first step we apply Proposition 2.15 in order to find a polydisc whose Shilov boundary contains (a, b, c, d). Actually, the proof of the proposition shows slightly more: Let us assume that (a, b, c) is maximal. Then we can find g ∈ G and a Jordan frame (c1 , . . . , cr ) of V such that g.(a, b, c, d) = (−e, −ie, e,

r X

λj cj ).

j=1

By Property (iii) of Theorem 4.2 we thus have BV (a, b, c, d) = BRr (−e, −ie, e, λ). We may therefore assume V = Rr and (a, b, c) = (−e, −ie, e) provided (a, b, c) is maximal. If (a, b, c) is minimal rather than maximal, then we may assume (a, b, c) = (e, −ie, −e) by the same argument. In the remainder of this section we will always assume (a, b, c) = (−e, −ie, e); the minimal case can be treated accordingly. Since (−e, −ie, e, λ) is assumed extremal, the possible values of λ are seriously restricted: Indeed, (−1, λj , 1) is positive iff λj is contained in the lower half-circle and negative, iff λj is contained in the upper half-circle. Since (−e, λ, e) is either maximal or minimal we see that either λj is contained in the lower half-circle for all j = 1, . . . , r or in the upper half-circle for all j = 1, . . . , r. Correspondingly, let us call λ positive or negative. In the positive case, all the λj are contained in a fixed quarter circle. This special position of λ allows us now to compute BV (−e, −ie, e, λ) as follows: ˜V as in (17). Given a ∈ Dr we write aj for its j-th component. Now let Let B (a, b, c, d) ∈ Dr (20)

(4)

. We have

r r r Y eV (a, b, c, d)r = kV (d, a) kV (b, c) = B [aj : bj : cj : dj ] kV (d, c)r kV (b, a)r j=1

and (21)

eV (0, 0, 0, 0) = 1. B

eV is uniquely determined by (20) and (21). From this we obtain the Moreover, B following result almost immediately: Lemma 4.6. If λ1 = · · · = λr , then BRr (−e, −ie, e, λ) = [−1 : −i : 1 : λ1 ]. Proof. The curve ϕ(t) := (−te, −ite, te, tλ) connects (0, 0, 0, 0) and (−e, −ie, e, λ). Since eV (ϕ(t))r = BR (−t : −it : t : λ1 t)r B

CROSS RATIOS

29

and both functions agree at 0, we see that eV (ϕ(1)) = BR (−1 : −i : 1 : λ1 ) = [−1 : −i : 1 : λ1 ]. BV (−e, −ie, e, λ) = B Although the situation of Lemma 4.6 is very special, the general case can now be reduced to it. Indeed we have: Proposition 4.7. The cross-ratio BRr is real-valued on ((S 1 )r )(4+) . More precisely, BRr (−e, −ie, e, λ) is positive/negative iff λ is positive/negative. Proof. Consider the function f : (S 1 \ {−1, −i, 1})r → S 1 given by f (λ) :=

BRr (−e, −ie, e, λ) . |BRr (−e, −ie, e, λ)|

We have f (λ)r ∈ R ∩ S 1 = {±1}, hence f takes values in the set R2r of 2rth roots of unity. Since R2r is discrete and f is continuous, f must be locally constant. In particular, if λ and µ are contained in the same connected component of (S 1 \{−1, −i, 1})r and BRr (−e, −ie, e, µ) is a positive/negative real number, then the same is true for BRr (−e, −ie, e, λ). Combining this with Lemma 4.6 we obtain the proposition. Since (20) admits precisely one positive/negative r-th real root, this determines the cross ratio. To summarize our discussion, let us call an extremal quadruple (a, b, c, d) positive/negative if it is conjugate to BRr (−e, −ie, e, λ) for some positive/negative λ. Then we obtain the following formula for BRr : Corollary 4.8. Suppose (a, b, c) is maximal and (a, b, c, d) ∈ ((S 1 )r )(4+) . Then v u r u Y u r BRr (a, b, c, d) = (a, b, c, d) · t [aj : bj : cj : dj ] , j=1 where

(a, b, c, d) =

+1 (a, b, c, d) positive −1 (a, b, c, d) negative

In the case, where (a, b, c, d) is positive, there are two possibilities for d: Either, each dj lies in between aj and bj or between bj and cj . This corresponds to the cases of (a, d, b) or (b, d, c) being maximal. These two cases can be distinguished by the cross ratio as follows: Lemma 4.9. If (a, b, c) and (a, d, b) are maximal, then 0 < BRr (a, b, c, d) < 1. If (a, b, c) and (b, d, c) are maximal, then BRr (a, b, c, d) > 1. Proof. The assumptions imply 0 < [aj : bj : cj : dj ] < 1, respectively [aj : bj : cj : dj ] > 1 for each j, hence the lemma follows from the explicit formula in Corollary 4.8. In particular we obtain: Corollary 4.10. For every Euclidean Jordan algebra V with associated Shilov boundary Sˇ we have BV (Sˇ(4+) ) = R \ {0, 1}.

30


Proof. Let (a, b, c, d) ∈ Sˇ(4+) . If (a, b, c) is maximal, then depending on d we have either BRr (a, b, c, d) < 0 (if (a, b, c, d) is negative) or BRr (a, b, c, d) < 1 (if (a, d, b) is maximal) or BRr (a, b, c, d) > 1 (if (b, d, c) is maximal). If (a, b, c) is minimal one may argue similarly (or reduce to the former case by means of the cocycle properties to be proved below). This shows the inclusion ⊂. For the converse inclusion, it suffices to see that BR is onto R \ {0, 1} and R has a balanced embedding into every Euclidean Jordan algebra. The reason that the cross ratio is not onto R is due to our choice of domain. In fact, we can extend BV continuously to the slightly larger domain Sˇ4∗ which is defined as follows: Sˇ4∗ contains Sˇ(4+) and (a, b, c, d) is contained in Sˇ4∗ \ Sˇ(4+) iff (a = c) ∨ (b = d) ∨ (b = a) ∨ (c = b) and moreover {a, b, c, d} contains a maximal triple. This implies in particular that at most two of a, b, c, d coincide and the non-coinciding pairs are transverse. Recall that the normalized kernel function kV extends to all of Sˇ and satisfies kV (a, b) 6= 0 iff a t b. Thus kV (d, a)kV (b, c) BV (a, b, c, d) := kV (d, c)kV (b, a) is well-defined as long as d t c and b t a. In particular, BV extends continuously to Sˇ4∗ . We denote this extension by the same letter. Then we have: Proposition 4.11. The cross ratio BV : Sˇ4∗ → R is onto. Moreover, x = z or y = t ⇔ BV (x, y, z, t) = 1 t = x or z = y

⇔ BV (x, y, z, t) = 0.

Proof. If x = z or y = t then enumerator and denominator coincide, whence BV (x, y, z, t) = 1. If y = x or z = t then one of the two terms in the enumerator vanishes. In any other case we have (x, y, z, t) ∈ Sˇ(4+) , hence BV (x, y, z, t) 6∈ {0, 1}. This proves the converse direction. 4.4. Cocycle properties. Generalized cross ratios satisfy various cocycle properties. The key observation for the proof of this fact is the following lemma: Lemma 4.12. If X is a set and k : X 2 → C∗ is an arbitrary function then ( X 4 → C∗ b: k(d,a)k(b,c) (a, b, c, d) 7→ k(d,c)k(b,a) has the following properties: (22)

b(a, b, c, d)

= b(c, d, a, b)

(23)

b(a, b, c, d)

= b(a, b, c, e)b(a, e, c, d)

(24)

b(a, b, c, d)

= b(a, b, e, d)b(e, b, c, d)

Proof. We have b(c, d, a, b)

= =

k(b, c)k(d, a) k(b, a)k(d, c) k(d, a)k(b, c) k(d, c)k(b, a)

CROSS RATIOS

=

31

b(a, b, c, d).

Moreover, b(a, b, c, e)b(a, e, c, d)

k(e, a)k(b, c) k(d, a)k(e, c) k(e, c)k(b, a) k(d, c)k(e, a) k(d, a)k(b, c) = k(d, c)k(b, a) = b(a, b, c, d) =

and b(a, b, e, d)b(e, b, c, d)

k(d, a)k(b, e) k(d, e)k(b, c) k(d, e)k(b, a) k(d, c)k(b, e) k(d, a)k(b, c) = k(d, c)k(b, a) = b(a, b, c, d). =

Since the normalized kernel k is only partially defined, this does not directly apply. Still we have: Corollary 4.13. The cross ratio BV : Sˇ4∗ → R satisfies (22)-(24) above, whenever both sides of the equation are well-defined. Proof. Again we can easily reduce to the case, where V is simple. In this case we can apply Lemma 4.12 to the function kV |D2 . By continuity, the relations extend to Sˇ(4) . We also have the following property: Corollary 4.14. For all (a, b, c, d) ∈ Sˇ(4∗) we have BV (a, b, c, d) = BV (b, a, d, c). Proof. We first do a number of reduction steps. (a, b, c, d) ∈ Sˇ(4∗) it suffices to show that

Since BV (a, b, c, d) ∈ R for

BV (a, b, c, d) = BV (b, a, d, c). eV and (a, b, c, d) ∈ D4 . The statement for We shall prove the latter statement for B (4+) ˇ will then follow by continuity. Using Property (iii) of Theorem (a, b, c, d) ∈ S 4.2 we may assume that V is simple. Moreover, since the desired symmetry is preserved by passing to a power, it suffices to show that BV,det (a, b, c, d) = BV,det (b, a, d, c)

((a, b, c, d) ∈ D4 ),

By [13] we have K ∗ (z, w) = K(w, z) and thus kdet (z, w) = kdet (w, z) for all w, z ∈ D. In particular, for (a, b, c, d) ∈ D4 we deduce BV,det (a, b, c, d)

=

kdet (d, a) · kdet (b, c) kdet (d, c) · kdet (b, a)

32


kdet (a, d)kdet (c, b) kdet (c, d)kdet (a, b) kdet (c, b)kdet (a, d) = kdet (c, d)kdet (a, b) = BV,det (a, b, c, d). =

4.5. The Bergman cross ratio and the proof of Theorem 1.1. Given any bounded symmetric domain we can find a Euclidean Jordan algebra V such that D = DV . We then obtain a generalized cross ratio BV on the Shilov boundary Sˇ of D. The Jordan algebra V , however, is not unique; it corresponds to a choice of a marking on the bounded symmetric domain D. In order to deduce Theorem 1.1 from Theorem 4.2 we need to show that BV is actually independent of this choice and depends only on D. Now for any complex domain C the corresponding Bergman kernel KC can be used in order to define a Bergman cross ratio BC : C 4 → C× ,

(x, y, z, t) 7→

KC (t, z)KC (y, x) KC (t, x)−1 KC (y, z)−1 = −1 −1 KC (t, z) KC (y, x) KC (t, x)KC (y, z);

in the case of a bounded symmetric domain D associated with a Euclidean Jordan rk V g g algebra V Proposition 3.16 yields B = BD , where B V V is the extension of BV 4 g as in (17). In particular, BD determines B on D , and thus by continuity also V BV . This shows that BV depends only on D and thus proves Theorem 1.1. We g can also use the above description of B V in terms of the Bergman kernel in order g to obtain the following invariance property of B V: Proposition 4.15. Let C be a complex domain and c : D → C be a biholomorphism. Then for all (x, y, z, t) ∈ D4 we have rk V g B = BC (c(x), c(y), c(z), c(t)). V (x, y, z, t) rk V g Proof. Since B = BD (x, y, z, t) it suffices to show that V (x, y, z, t)

BD (x, y, z, t) = BC (c(x), c(y), c(z), c(t)). For this we apply [15, Prop. IX.2.4] in order to relate the Bergman kernels on D and C. Denote by Jc the complex Jacobi matrix of c. Then KD (z, w) = KC (c(z), c(w)) det C (Jc (z))det C (Jc (w)). Hence BD (x, y, z, t)

= =

KD (t, z)KD (y, x) KD (t, x)KD (y, z) KC (c(t), c(z)) det C (Jc (t))det C (Jc (z)) KC (c(t), c(x)) det C (Jc (t))det C (Jc (x)) ·

KC (c(y), c(x)) det C (Jc (y))det C (Jc (x))

KC (c(y), c(z)) det C (Jc (y))det C (Jc (z)) KC (c(t), c(z))KC (c(y), c(x)) = KC (c(t), c(x))KC (c(y), c(z)) = BC (c(x), c(y), c(z), c(t)).

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5. Maximal representations, limit curves and strict cross ratios In this section we explain how the generalized cross ratio functions defined above can be used to associate with any maximal representation into a Hermitian group of tube type a strict cross ratio in the sense of [17]. We then prove that all strict cross ratios are equivalent in a sense made precise in Proposition 5.8. The latter result is essentially contained, but not stated in [17], where it is presented in the context of a more general theory of adapted flows. The proof we give here avoids this machinery and tries to be more elementary. 5.1. The cross ratio of a maximal representation. Returning to the problem of the introduction, let Σ be a closed, oriented surface of genus g ≥ 2 with fundamental group Γ. We fix a hyperbolization of Σ, i.e. a faithful homomorphism Γ → P U (1, 1) with discrete image so that Σ = Γ\D. We also fix a Euclidean Jordan ˇ respectively the associated bounded symmetric algebra V and denote by D and S, domain and Shilov boundary. The groups G, K, L, Q+ are defined as before. The aim of this section is to construct an invariant associated with a maximal representation % : Γ → G. Our basic references concerning maximal representations are [10] and [8]. Let us briefly recall the main definitions: Denote by ωD the K¨ahler form on D associated with the metric of minimal holomorphic sectional curvature −1. Given an arbitrary ρ-equivariant map f : D → D we define the Toledo invariant T% of ρ by Z 1 f ∗ ωD . T% := 2π Σ This does not depend on the choice of f . The Toledo invariant satisfies a generalized Milnor-Wood inequality, in the present normalization given by (25)

|T% | ≤ |χ(Σ)| · rk(V ).

Accordingly, the representation % is called maximal if T% = |χ(Σ)| · rk(V ). Every maximal representation % : Γ → G defines a canonical limit curve, i.e. an injective ˇ Since these limit curves play a crucial continuous %-equivariant map ϕ : S 1 → S. role in our construction and for lack of suitable references, we briefly digress to explain their construction: First consider a maximal representation % : Γ → G with Zariski-dense image. In this case, a %-equivariant injective map ϕ : S 1 → Sˇ is constructed in [7, Ch.7], and by [8] this map is continuous. We claim that ϕ is in fact the unique limit curve ϕ : S 1 → Sˇ for %. Indeed, S 1 is the closure of any of its Γ-orbit, and since ϕ is continuous and %-equivariant, this implies that ϕ(S 1 ) is a minimal closed Γˇ Now the first lemma in Section 3.6 of [1] applies to show that invariant subset of S. 1 ˇ This implies in ϕ(S ) = ΛΓ , where the latter denotes is the limit set of Γ in S. 1 ˇ particular, that any other limit curve S → S has to intersect ϕ. The uniqueness of ϕ then follows from the following lemma: Lemma 5.1. Let ϕ1 , ϕ2 : S 1 → Sˇ be two limit curves for the same maximal representation % with ϕ1 (S 1 ) ∩ ϕ2 (S 1 ) 6= ∅. Then ϕ1 = ϕ2 .

34


Proof. By equivariance of the ϕj the intersection contains a Γ-orbits, but since the Γ-action on S 1 is minimal this implies that this preimage is the full circle and thus ϕ1 (S 1 ) = ϕ2 (S 1 ). Every γ ∈ Γ has a unique attractive fixed point γ + in S 1 . By equivariance, this is mapped under both ϕj to the unique attractive fixed point of ρ(γ) in ϕ1 (S 1 ) = ϕ2 (S 1 ). We deduce ϕ1 (γ + ) = ϕ2 (γ + ) for all γ ∈ Γ and since {γ + | γ ∈ Γ} is dense in S 1 we deduce that ϕ1 = ϕ2 . Thus we have proved: Corollary 5.2. If % : Γ → G is a Zariski-dense maximal representation, then there exists a unique limit curve ϕ : S 1 → Sˇ for %. If we drop the assumption of Zariski-dense image, then Corollary 5.2 fails. Indeed, if ϕ is a limit curve for % and g ∈ CG (%(Γ)), then gϕ is again a limit curve for %. We therefore cannot hope for a unique limit curve in general. Still, there is an explicit construction for a canonical limit curve, which we now describe: For this let % : Γ → G be an arbitrary maximal representation, and denote by H the Zariski-closure of %(Γ) in G. It follows from [10, Thm. 5] that H is of tube type. Then % factors as % = ι ◦ %0 , where ι : H → G is the inclusion and %0 : Γ → H is a maximal representation with Zariski dense image. In particular, there is a unique limit curve ϕ0 : S 1 → SˇH for %0 . In order to extend this limit curve to a limit curve for % we observe that by the general structure theory of maximal representations developed in [10], the embedding ι is tight in the sense of [11]. Then [11, Thm 4.1] implies that a ι-equivariant map b ι : SˇH → SˇG can be constructed as follows: Denote by DH and DG the symmetric spaces of H and G respectively and choose a basepoint p ∈ DG . Then StabH (p) and hence ι(StabH (p)) are compact, whence ι(StabH (p)) is contained in a maximal compact subgroup of G. We thus find x ∈ G with StabG (x) ⊃ ιStabH (p). Given p and x we define a ι-equivariant continuous, totally geodesic map (26)

f : DH → DG ,

g · p 7→ ι(g)x.

Now every point ξ ∈ SˇH can be represented uniquely by a geodesic ray σ emanating from p. Then f ◦ σ is a geodesic ray in DG , which by the results in [11, Chapter 4] ends in some point b ι(ξ) ∈ SˇG . This yields the desired continuous ι-equivariant embedding b ι : SˇH → SˇG , and we can define the limit curve ϕ of % by ϕ := b ι ◦ ϕ0 . A priori the above construction of b ι depends on the choices of basepoints p and x. In order to obtain a canonical limit curve ϕ we have to show that b ι is a posteriori independent of these choices. We show the independence of x first: Suppose y ∈ DG is another base point with StabG (y) ⊃ ιStabH (p). Then for any g ∈ G we have d(ι(g)x, ι(g)y) = d(x, y). Since the geodesics through p in DH are of the form σ(t) = exp(tX)p this implies that the images of a geodesic in DH under the two maps fx and fy corresponding to x and y respectively are at bounded distance, hence define the same point in the Shilov boundary. This shows that b ι is independent of the choice of x once p is given. Now let us prove independence of p: For this we suppose that two points p, p0 ∈ DH are given. We then find h0 ∈ H such that p0 = h0 p. Now let x ∈ DG such that StabG (x) ⊃ ι(StabH (p)). Then p and x determine a map f : DH → DG

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35

by (26). Similarly, given x0 ∈ DG with StabG (x0 ) ⊃ ι(StabH (p0 )) we obtain a map f 0 : DH → DG by f 0 (hp0 ) := hx0 . We have to show that f and f 0 induce the same map b ι of Shilov boundaries. By the previous considerations we may choose x0 ∈ DG arbitrary subject to the condition StabG (x0 ) ⊃ ι(StabH (p0 )); in particular we may chose x0 = ρ(h0 ).x. But with this choice of x0 the maps f and f 0 coincide. This proves that b ι and consequently ϕ do not depend on any of the choices made to define them. We may thus refer to ϕ as the canonical limit curve of %. This canonical limit curve has the following key property: Proposition 5.3 (Burger-Iozzi-Wienhard, [9]). Let % : Γ → G be a maximal representation and ϕ : S 1 → Sˇ the canonical limit curve of %. Then for every positively/negatively oriented triple (x, y, z) ∈ (S 1 )(3) , the triple (ϕ(x), ϕ(y), ϕ(z)) is maximal/minimal. We deduce from the proposition that two distinct points x 6= y ∈ S 1 are mapped to transverse points under every limit curve. As another consequence of Proposition 5.3 we deduce that ϕ induces an injective map ϕ(4) : (S 1 )4∗ → SˇV4∗ , (x, y, z, t) 7→ (ϕ(x), ϕ(y), ϕ(z), ϕ(t)), where (S 1 )4∗ := {(x, y, z, t) ∈ (S 1 )4 | x 6= t, y 6= z}. Definition 5.4. Let ρ : Γ → G be a maximal representation and ϕ : S 1 → Sˇ the associated canonical limit curve. Then the function bρ := (ϕ(4) )∗ BV : (S 1 )4∗ → R,

(x, y, z, t) 7→ BV (ϕ(x), ϕ(y), ϕ(z), ϕ(t)).

is called the cross ratio of the maximal representation %. The main properties of cross ratios of maximal representations are collected in the following theorem: Theorem 5.5. The cross ratio bρ : (S 1 )4∗ → R is a continuous Γ-invariant function satisfying the following properties: (27)

bρ (x, y, z, t)

=

bρ (z, t, x, y)

(28)

bρ (x, y, z, t)

=

bρ (x, y, z, w)bρ (x, w, z, t)

(29)

bρ (x, y, z, t)

=

bρ (x, y, w, t)bρ (w, y, z, t)

(30)

x = z or y = t ⇔

bρ (x, y, z, t) = 1

(31)

t = x or z = y

⇔

bρ (x, y, z, t) = 0

Proof. Γ-invariance on (S 1 )(4) follows from G-invariance of BV on S (4+) (i.e. Property (i) of Theorem 4.2). By continuity we obtain Γ-invariance on all of (S 1 )4∗ . By a similar extension argument, Properties (27)-(29) follow from Corollary 4.13. Finally, Properties (30)-(31) follow from Proposition 4.11. In fact, it follows from Corollary 4.14 that bρ also satisfies bρ (x, y, z, t) = bρ (y, x, t, z). However, we are not going to use this property in the sequel. Instead let us focus on the properties mentioned in Theorem 5.5. In the language of [17] the theorem

36


says precisely that bρ is a strict cross ratio. In the next subsection we shall recall some general properties of such strict cross ratios. 5.2. Strict cross ratios. In this subsection we collect basic properties of strict cross ratios, i.e. continuous Γ-invariant functions on (S 1 )4∗ satisfying (27)-(31) above. The material is adapted from [17]. Let b : (S 1 )4∗ → R be any strict cross ratio. We will ocassionally use the following two cocycle identities: (32)

b(x, y, x, t)

=

b(x, y, z, t)b(z, y, x, t) = 1

(33)

log b(x, y, z, t)

=

− log b(z, y, x, t)

The former is an immediate consequence of (30) and (29) and the latter follows by applying the logarithm. We will usually consider x, y, z fixed and study g(t) = b(x, y, z, t) as a function of t. Let us assume that (x, y, z) is positively oriented. We then divide the circle into three open disjoint intervals I1 = (x, y), I2 = (y, z) and I3 = (z, x) so that S 1 = {x} ∪ I1 ∪ {y} ∪ I2 ∪ {z} ∪ I3 . The function g is then defined on S 1 \ {z}. By Axiom (31), x is the only zero of g. Since g(y) = 1 is positive, g is positive on Iu := I1 ∪ {y} ∪ I2 . Let a, b ∈ Iu such that g(a) = g(b). Then we have: 1 = g(a)g(b)−1 = b(x, y, z, a)b(x, y, z, b)−1 = b(−1, a, 1, b), but by Axiom (30) this can only be the case if a = b. Hence g|Iu is injective. Furthermore, by Axiom (31) and (32) we have lim g(t) = lim b(z, y, x, t)−1 = +∞.

t→z t∈Iu

t→z t∈Iu

Since g(x) = 0 the intermediate value theorem implies that g|Iu is surjective and therefore defines a homeomorphism between Iu and (0, ∞). Now consider the case t ∈ I3 . We claim that g(t) is negative on I3 . For this we first observe that as above lim g(t) ∈ {±∞}.

t→z t∈Iu

Again, since g(x) = 0 the intermediate value theorem implies that g maps I3 homeomorphically to either (0, ∞) or (−∞, 0). However, the former would imply the existence of t0 ∈ I3 with g(t0 ) = 1, which contradicts Axiom (30). Hence lim g(t) = −∞

t→z t∈Iu

and g maps I3 homeomorphically to (−∞, 0). We have proved: Proposition 5.6. If (x, y, z) is a positively oriented triple on S 1 and b : (S 1 )4∗ → R is a strict cross ratio, then g : S 1 \ {z} → R,

t 7→ b(x, y, z, t)

is a homeomorphism with g(x) = 0. Notice that as a homeomorphism S 1 \ {z} → R the function g is automatically monotonous. In the sequel we denote by (S 1 )3+ := {(x, y, z) ∈ (S 1 )(3) | (x, y, z) positively ordered} the set of positively ordered triples.

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37

Corollary 5.7. For every strict cross ratio there exists a Γ-equivariant continuous map ψ : R × (S 1 )3+ → S 1 such that log b(x, y, z, ψs (x, y, z)) = s. Proof. Put ψs (x, y, z) := (log g)−1 (s).

Proposition 5.8 (Labourie, [17]). Let b1 and b2 be strict cross ratios. Then there exists a C > 0 such that | log b1 | ≤C C −1 ≤ | log b2 | Proof. We adapt an argument of Labourie [17], Chapter 2: Let ψs1 and ψs2 be maps associated to b1 and b2 by means of Corollary 5.7. Define a function T : (S 1 )3+ → R by T (x, y, z) := b1 (x, y, z, ψ12 (x, y, z)) This map is positive and continuous. Since ψs2 is Γ-equivariant, T is Γ-invariant. Furthermore it satisfies ψ12 (x, y, z) = ψT1 (x,y,z) (x, y, z). Since (S 1 )3+ /Γ is compact, T has a global maximum A. For t ∈ S 1 there exists s, s˜ ∈ R such that t = ψs˜1 (x, y, z) = ψs2 (x, y, z). If |s| ∈ [n, n+1) for n ∈ N, then we have by definition of A that |˜ s| ∈ (0, A(n+1)) ⊂ (0, 2A|s|)., hence |˜ s| ≤ 2A|s|. We can calculate: | log b1 (x, y, z, t)| =| log b1 (x, y, z, ψs˜1 (x, y, z))| = |˜ s| ≤2A|s| = 2A| log b2 (x, y, z, ψs2 (x, y, z))| = 2A| log b2 (x, y, z, t)|. This proves the upper bound for (x, y, z) ∈ (S 1 )3+ . If (x, y, z) is negatively oriented, then (z, y, x) is in (S 1 )3+ . The upper bound for this case follows from the fact that: | log b(x, y, z, t)| = | log b(z, y, x, t)|. The lower bound is obtained by reversing the roles of b1 and b2 .

We will apply this in the following form: Corollary 5.9. Let ρ : Γ → G be a maximal representation with associated cross ratio bρ . Then there exists C > 0 such that for all (x, y, z, t) ∈ (S 1 )(4) , | log bρ (x, y, z, t)| ≥ C · | log[x : y : z : t]|. 6. Well-displacing and quasi-isometry property We keep the notation introduced in the last section. In particular, Σ denotes a closed, oriented surface of genus g ≥ 2 with fundamental group Γ and fixed hyperbolization Γ → P U (1, 1), while V denotes a Euclidean Jordan algebra with bounded symmetric domain D, Shilov boundary Sˇ and associated group G. We fix a maximal representation % : Γ → G and denote by ϕ : S 1 → Sˇ its canonical limit curve. Recall from the introduction that for γ ∈ Γ the translation length of ρ(γ) is defined by τD (ρ(γ)) = inf d(ρ(γ)x, x). x∈D

38


Since Σ is smooth and closed, γ cannot be elliptic or parabolic respectively, hence must be hyperbolic. We denote by γ + ∈ S 1 the attractive, and by γ − the repulsive fixed point of γ in S 1 . The virtual translation length of ρ(γ) is defined by ∞ τD (ρ(γ)) = log bρ (γ − , ξ, γ + , γξ)

for ξ ∈ S 1 \ {γ ± }. We will see below that this does not depend on the choice of ξ. The main result in this section is Theorem 6.17 which says that there exists a constant C > 0 such that (34)

∞ τD (%(γ)) ≥ CτD (%(γ)).

From the resulting inequality we deduce that ρ is well-displacing and that the corresponding action on D is quasi-isometric. This implies in particular the desired properness of the mapping class group on Hermitian higher Teichm¨ uller spaces. Before we prove Inequality (34) we discuss a special case, where we have equality and C = 1. 6.1. Fuchsian representations. A maximal representation % : Γ → G is called Fuchsian if it factors as % b t %:Γ− → P U (1, 1) → − G, where %b : Γ → P U (1, 1) is some fixed hyperbolization and t : P U (1, 1) → G is some embedding. It then follows automatically from maximality of % that the embedding t is tight in the sense of [11]. Now we claim: Proposition 6.1. If % : Γ → G is a Fuchsian representation then ∞ τD (%(γ)) = τD (%(γ)).

The proposition will be an immediate consequence of the following three lemmas: Lemma 6.2. For all γ ∈ Γ we have τD (γ) = τD∞ (γ). Proof. Observe that both sides of the claimed equality are invariant under P SL2 (R) and Cayley transform. Conjugating γ with the Cayley transform and some g ∈ P SL(2, R), we can transform the whole situation to the upper half plane model H and we may assume without loss of generality that γ acts by translation on the imaginary axis, i.e. the attractive fixpoint γ + is ∞, while the repulsive fix point γ − is 0. Then γ can be written as a 0 γ= 0 a−1 with a > 1 and we have τD (γ) = d(i, γi). Using the relation between the hyperbolic distance and the classical cross ratio on H we obtain γi − 0 ia2 τD (γ) = d(i, γi) = log[i : 0 : γi : ∞] = log = log = log a2 . i−0 i Now consider the geodesic σ through i which is orthogonal to the imaginary axis. It is a half circle with endpoints 1 and −1 on the real axis. The geodesic γ · σ goes

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39

through γi and intersects the real axis in the points γ(1) = a2 and γ(−1) = −a2 . Now we have a quadruple of points in R ∪ {∞} and we calculate their cross ratio: τD∞ (γ) = log[0 : 1 : ∞ : γ1] = log

a2 = log a2 = τD (γ). 1

Lemma 6.3. For all γ ∈ Γ we have (35)

∞ τD (%(γ)) = τD∞ (b %(γ))

Proof. The Lie group homomorphism t : P U (1, 1) → G corresponds to a morphism α : R → V of the associated Jordan algebras. By Example 3.14, this morphism is balanced. Now denote by ϕ and ϕ b the canonical limit curves of % and %b respectively. Then % = αC ◦ %b and thus Property (ii) of Theorem 4.2 implies that the generalized cross ratios associated to % and %b are the same. This in turn shows that the corresponding virtual translation lengths are the same. Lemma 6.4. For all γ ∈ Γ we have (36)

τD (%(γ)) = τD (b %(γ))

Proof. Let γ ∈ Γ. Then γ is hyperbolic and thus there exists a geodesic σ in D such that γ fixes σ and %b(γ).σ(s) = σ(s + τD (b %(γ)))

(s ∈ R).

Now there exists a t-equivariant totally geodesic embedding t† : D → D. If σ † denotes the image of σ under t† then %(γ) fixes σ † and %(γ).σ † (s) = σ † (s + τD (b %(γ))) By [6, Thm II.6.8] this implies the lemma.

(s ∈ R).

6.2. Bounding the translation length from below. In this subsection we return to the general case of an arbitrary maximal representation % : Γ → G of Γ into a Hermitian group of tube type. As a first step towards our main inequality we will bounded the translation length of ρ(γ) from below in terms of some auxiliary data. This auxiliary data will then be related to the virtual translation length in the next subsection. We will use the fact that for γ ∈ Γ the element %(γ) is contained in the Levi factor of a certain Shilov parabolic. We will transport the whole situation into the Levi factor L, which has a linear action on the symmetric cone Ω of the underlying Jordan algebra V . Using this linear action, we can estimate the dynamics of %(γ) using the reductive symmetric space of GL(V ). We introduce the notations g0 := ρ(γ) and g0± := ϕ(γ ± ). Since G is transitive on Sˇ(2) , there exists h1 ∈ G such that h1 g0± = ±e. Set g1 := h1 g0 h−1 1 . Since g0 fixes ˇ we see that g1 fixes ±e, whence Corollary 2.3 applies and we find M ∈ N g0± ∈ S, such that g2 := (c ◦ g1 ◦ c−1 )M ∈ G(Ω). We will use g2 to estimate τD (%(γ)); for this we need the following lemma: Lemma 6.5. Let X be a metric space and g ∈ Is(X) an isometry. Then for any M ∈ N: 1 τX (g) ≥ τX (g M ). M

40


Proof. We have: d(g M x, x) ≤

M X

d(g i x, g i−1 x) = M · d(gx, x).

i=1

Taking the infimum on both sides finishes the proof.

Since the Cayley transform is isometric we obtain τD (ρ(γ))

−1 = τD (g0 ) = inf d(g0 x, x) = inf d(g0 h−1 1 x, h1 x)

=

inf

x∈D

x∈D d(h1 g0 h−1 1 x, x)

x∈D

= τD (g1 ) = τTΩ (c ◦ g1 ◦ c−1 )

1 τT (g2 ). M Ω

≥ and thus

1 · τTΩ (g2 ). M Lemma 6.6. Let h ∈ L. Then for all z ∈ TΩ we have τD (ρ(γ)) ≥

(37)

d(z, hz) ≥ d(iIm(z), h(iIm(z))). Proof. Since TΩ is an open subset of V C , we can identify the tangent space of TΩ at any point z ∈ TΩ canonically with V C . Using this identification, the Hermitian metric H on TΩ admits the following description (see [15, Prop. X.1.3]): Let n := dim V , r := rk(V ). Then given z ∈ TΩ and a, b ∈ V C we have −1 ! z − z¯ 2n 2n −1 Hz (a, b) = P P (2Im(z)) a b = HIm(z) (a, b). a b = r i r In other words, translation in the direction of the real axis is isometric for H. We have H = g +iω, where g is the Riemannian metric on TΩ and ω is the K¨ahler form. In particular, since ω is skew-symmetric, we have for all z ∈ TΩ and all a ∈ V C the equality gz (a, a) = Hz (a, a)

=

Hz (Re(a), Re(a)) + Hz (iIm(a), iIm(a))

=

gz (Re(a), Re(a)) + gz (iIm(a), iIm(a)).

In particular, gz (a, a) ≥ gz (iIm(a), iIm(a)) = giIm(z) (iIm(a), iIm(a)). Thus, given any path σ : [0, 1] → TΩ with σ(0) = z, σ(1) = hz we have Z 1q l(σ) = gσ(t) (σ(t), ˙ σ(t))dt ˙ 0

Z ≥

1

q giIm(σ(t)) (iIm(σ(t)), ˙ iIm(σ(t)))dt ˙

0

= l(iIm(σ(t))). Observe that iIm(σ(t)) is a path joining iIm(z) and iIm(hz)) = h(iIm(z))). (The latter equality is the only place where we use the special form of h ∈ L.) Passing to the infimum over all σ, we obtain the lemma. Specializing to h = g2 and passing to the infimum over all z ∈ TΩ we deduce:

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41

Corollary 6.7. Let g2 ∈ G(Ω). Then τTΩ (g2 ) = τΩ (g2 ). Observe that Ω is the reductive symmetric space associated with L. In particular, the inclusion L → GL(V ) induces an isometric embedding of Ω into the reductive symmetric space P(V ) of GL(V ), which happens to be the space of all positive definite symmetric matrices of size dim(V ). A detailed exposition of P(V ) is provided in [6, Ch. II.10]. We will use the results described therein without further reference. We will need the following result concerning P(V ): Lemma 6.8. Let g ∈ GL(V ). Then τP(V ) (g) ≥ √

1 | log det(g)2 |. dim V

Proof. Let p ∈ P(V ) and c : [0, d(p, gp)] → P(V ) a unit speed geodesic joining p with gp. Then there exists h ∈ GL(V ) such that p = hh> and an element X of norm 1 in the Lie algebra of GL(V ) such that c(t) = h exp(tX)h> . Moreover, gp = ghh> g > . Since the d(p, gp) = gp we have h exp(d(p, gp) · X)h> = ghh> g > ⇒

det(h exp(d(p, gp) · X)h> ) = det(ghh> g > )

⇒

exp(d(p, gp) · tr(X)) = det(g)2

⇒

exp(d(p, gp) · tr(X)) = exp(log det(g)2 )

Since both d(p, gp) · tr(X) and log det(g)2 are real this implies d(p, gp) · tr(X) = log det(g)2 , whence d(p, gp) · | tr(X)| = | log det(g)2 |.

(38) Now observe that

| tr(X)| = |(X|1)| ≤ kXkk1k = 1 ·

√

√ dim V =

dim V .

Inserting into (38) we obtain d(p, gp) ≥ √

1 | log det(g)2 |. dim V

Passing to the infimum over all p ∈ P(V ) we obtain the lemma.

Remark 6.9. We want to apply Lemma 6.8 to the element g2 defined above. We recall that by construction g2 has 0 as a repulsive fixed point (since g1 has −e as a repulsive fixed point) at the boundary at infinity. This implies that det(g2 ) ≥ 1 and hence log det(g)2 ≥ 0. In particular, the estimate in Lemma 6.8 becomes τP(V ) (g2 ) ≥ √ when specialized to g2 .

1 log det(g2 )2 , dim V

42


Corollary 6.10. For all γ ∈ Γ, τD (ρ(γ)) ≥

1 √ · log det(g2 )2 . M · dim V

Proof. Combining (37), Corollary 6.7 and Lemma 6.8 (in combination with Remark 6.9) we obtain 1 1 τD (ρ(γ)) = τD (g0 ) ≥ τT (g2 ) = τΩ (g2 ) M Ω M 1 1 √ ≥ τP(V ) (g2 ) ≥ · log det(g2 )2 . M M · dim V 6.3. Bounding the virtual translation length from above. We keep the notations from the last subsection. In particular, given γ ∈ Γ the elements g0 , g1 , g2 , h1 , h2 are defined as before. In order to prove the desired inequality between translation ∞ length and virtual translation length (Theorem 6.17) we have to estimate τD (%(γ)) ± ± from above. As before we denote g0 := ϕ(γ ). ∞ Lemma 6.11. We have τD (ρ(γ)) > 0. ∞ Proof. By definition τD (%(γ)) = log b% (γ − , ξ, γ + , γξ), where ξ 6= γ ± . Since γ + is the attractive fixed point of γ, the point γξ is between ξ and γ + . More precisely: (γ − , ξ, γ + ) and (ξ, γξ, γ + ) are either both positive or both negative. In both cases ∞ (%(γ)) is positive. τD

Our next observation is that τ ∞ (%(γ)) does not depend on the choice of ξ. We shall prove a slightly more general statement: Given x, z ∈ Sˇ we define Sˇ(4) = {(y, t) ∈ Sˇ2 | (x, y, z, t) ∈ Sˇ(4) }. x,z

With this notation we have (4) (ϕ(ξ), ϕ(γξ)) = (ϕ(ξ), g0 .ϕ(ξ)) ∈ Sˇg+ ,g− 0

1

0

±

for all ξ ∈ S \ {γ }. Now we claim: (4) Lemma 6.12. Suppose that y ∈ Sˇ \ {g0± } and (y, g0 .y) ∈ Sˇg− ,g+ then 0

∞ τD (ρ(γ))

=

0

log BV (g0− , y, g0+ , g0 y)

Proof. Let F (y) := BV (g0− , y, g0+ , g0 y). We have to show that F (y) = F (z) for all (4) y, z ∈ Sˇ satisfying the conditions of the lemma. If (y, z) ∈ Sˇg+ ,g− . Then we can 0 0 compute F (z) = BV (g0− , z, g0+ , g0 z)

=

BV (g0− , z, g0+ , y) · BV (g0− , y, g0+ , g0 z)

=

BV (g0 g0− , g0 z, g0 g0+ , g0 y) · BV (g0− , y, g0+ , g0 z)

= BV (g0− , y, g0+ , g0 z) · BV (g0− , g0 z, g0+ , g0 y) = BV (g0− , y, g0+ , g0 y) = F (y). Otherwise we can find w ∈ Sˇ satisfying the conditions of the lemma such that both (4) (4) (y, w) ∈ Sˇ + − and (z, w) ∈ Sˇ + − . Then F (y) = F (w) = F (z), finishing the g0 ,g0

g0 ,g0

proof of the lemma in the general case.

CROSS RATIOS

43

(4) Corollary 6.13. Suppose that z ∈ Sˇ \ {±e} and that (z, g1 .z) ∈ Sˇ−e,e . Then ∞ τD (ρ(γ)) = log BV (−e, z, e, g1 z).

Proof. Apply Lemma 6.12 to y := h−1 1 z and use G-invariance of BV .

Recall that g2 = (c ◦ g1 ◦ p)M . We thus need the following lemma: Lemma 6.14. If g ∈ G and g ± are fixed points of g, then for all M ∈ N, log BV (g − , x, g + , g M x) = M · log BV (g − , x, g + , gx) Proof. We have log BV (g − , x, g + , g M x)

=

M X

log BV (g − , g i x, g + , g i+1 x)

i=1

= M · log BV (g − , x, g + , gx). For the next step we observe that by Proposition 3.16 and the transfomrmation behavior of the Bergman kernel under biholomophic maps [15, Prop. IX.2.4] that the Bergman kernel KTΩ of the tube TΩ extends continuously to transversal pairs on V . Denoting this extension by the same letter TΩ we prove: Proposition 6.15. Let w ∈ V such that 0, w and g2 w are pairwise transversal. Then KTΩ (w, 0) KTΩ (g2 w, c(xn )) 1 ∞ · log · lim , τD (ρ(γ)) = M · rk V KTΩ (g2 w, 0) n→∞ KTΩ (w, c(xn )) where xn is a sequence in D converging to e. Proof. Combinge Corollary 6.13 and Lemma 6.14 with Proposition 4.15 applied to the Cayley transform c : D → TΩ . Now the right hand side can be computed explicitly: Proposition 6.16. (39)

∞ τD (ρ(γ))

=

1 · log det(g2 )2 M · rk V

Proof. We first show that KTΩ (g2 w, c(xn )) = 1. n→∞ KTΩ (w, c(xn )) lim

Indeed, let λ ∈ [0, 1). Then c(λ · e) = −

1+λ e. 1−λ

Using [15, X.1.3] we obtain KTΩ (g2 w, c(xn )) lim n→∞ KTΩ (w, c(xn ))

=

lim

λ→1

1+λ 1−λ e) 1+λ 1−λ e)

det(g2 w − det(w −

!− 2n r

44


=

lim

det( 1−λ 1+λ g2 w − e)

λ→1

!− 2n r = 1.

det( 1−λ 1+λ w − e) K

(w,0)

In view of Proposition 6.15 it now suffices to show that KTTΩ(g2 w,0) = det(g2 )2 . Ω Since g2 : TΩ → TΩ is biholomorphic, we see from [15, Prop. IX.2.4] that KTΩ (w, 0) = KTΩ (g2 w, g2 0) det C Jg2 (w)det C Jg2 (0), where Jg2 denotes the complex Jacobi matrix of g2 . Note that g2 is a real matrix, because it is in G(Ω)0 ⊂ GL(V ). Since it is linear, we have Jg2 ≡ g2 and g2 0 = 0, whence KTΩ (w, 0) = KTΩ (g2 w, g2 0) det C Jg2 (w)det C Jg2 (0) = KTΩ (g2 w, 0) det(g2 )2 . Dividing both sides by KTΩ (g2 w, 0) the proposition follows.

Comparing with (6.10) we ultimately obtain: Theorem 6.17. For all γ ∈ Γ and all ξ ∈ S 1 \ {γ ± } the inequality rk V ∞ · τD (ρ(γ)) τD (ρ(γ)) ≥ √ dim V holds. Proof. By Corollary 6.10 and Proposition 6.16 we have 1 √ log det(g2 )2 τD (ρ(γ)) ≥ M · dim V rk V 1 2 = √ · log det(g2 ) · M · rk V dim V rk V ∞ = √ · τD (ρ(γ)). dim V 6.4. Well-displacing. Before we can deduce well-displacing of maximal representations from our main inequality, we need one more ingredient: If Γ is a finitely generated group and S a finite set of generators, then we denote by k · kS the word length with respect to S. We then define the word metric dS by dS (γ1 , γ2 ) = lS (γ2−1 γ1 ), where lS (γ1 ) := inf kηγη −1 kS . η∈γ

ˇ Then the Svarc-Milnor lemma reads as follows: ˇ Lemma 6.18 (Svarc-Milnor, [6, Prop. I.8.19]). Let (X, d) be a length space. If a group Γ acts properly and cocompactly by isometries on X, then Γ is finitely generated and for every finite generating set S with associated word metric lS on Γ and every basepoint x0 ∈ X the map (Γ, dS ) → (X, d), is a quasi-isometry.

γ 7→ γ.x0

CROSS RATIOS

45

Note that the constants appearing in the quasi-isometry inequality may depend on x0 . Nevertheless, applying this to the Γ action on the disc chosen in the introduction, we obtain: Corollary 6.19. Let S be an arbitrary finite generating set for Γ and lS the associated word length. Then there exist constants A, B > 0 such that for every γ ∈ Γ τD (γ) ≥ A · lS (γ) − B. Proof. We fix a compact fundamental domain F for the Γ-action on D. We know that every γ ∈ Γ is hyperbolic, i.e. there exists a geodesic σ on which γ acts by translation and we have γ · σ(t) = σ(t + τD (γ)) for all t. There exists η ∈ Γ such that ησ intersects F , say y := ησ(t0 ) ∈ F . Then we have for any x ∈ F : d(x, ηγη −1 x) ≤ d(x, y)+d(y, ηγη −1 y)+d(ηγη −1 y, ηγη −1 x) ≤ 2diam(F )+τD (ηγη −1 ). ˇ Now we fix x ∈ F and apply the Svarc-Milnor lemma with x = x0 to find positive constants A, B 0 satisfying d(x, γx) = d(ex, γx) ≥ A · dS (e, γ) − B 0 = A · lS (γ) − B 0 for all γ ∈ Γ. We deduce τD (γ)

= τD (ηγη −1 ) ≥ d(x, ηγη −1 x) − 2diam(F ) ≥ A · lS (ηγη −1 ) − B 0 − 2diam(F ) = A · lS (γ) − (B 0 + 2diam(F )).

Now we can finally prove: Theorem 6.20 (Well-displacing). For any finite generating set S of Γ there exist A, B > 0 such that τD (ρ(γ)) ≥ A · lS (γ) − B for all γ ∈ Γ. Proof. Using Corollary 5.9, Theorem 6.17, Lemma 6.2 and Corollary 6.19 we find positive constants C1 , · · · , C4 such that τD (ρ(γ))

∞ ≥ C1 · τD (ρ(γ))

= C1 · log bρ (γ − , ξ, γ + , γξ) ≥ C2 · log[γ − : ξ : γ + : γξ] = C2 · τD∞ (γ) = C2 · τD (γ) = C2 C3 `S (γ) − C2 C4 . 6.5. Quasi-isometry property. As a simple consequence of Theorem 6.20 we obtain the following result: Theorem 6.21 (Quasi-isometry). Let ρ : Γ → G be a maximal representation. Then for every x ∈ D and every finite generating set S of Γ the map (Γ, dS ) → (D, dD ), is a quasi-isometric embedding.

γ 7→ γ.x

46


Proof. Since for γ1 , γ2 ∈ Γ we have both dS (γ1 , γ2 ) = lS (γ2−1 γ1 ) and dD (γ1 .x, γ2 .x) = d(γ2−1 γ1 x, x) it suffices to show that there exist A, B > 0 such that for all γ ∈ Γ A−1 · lS (γ) − B ≤ dD (γx, x) ≤ A · lS (γ) + B. The left inequality follows immediately from Theorem 6.20: Indeed, we have dD (γx, x) ≥ τD (γ)≥A · lS (γ) − B for some constants A, B > 0 independent of γ. On the other hand, the right inequality is elementary: Write γ = s1 · · · slS (γ) with si ∈ S. Then d(x, γx) ≤d(x, s1 x) + d(s1 x, s1 s2 x) + . . . + d(s1 · · · slS (γ)−1 x, γx) =d(x, s1 x) + d(x, s2 x) + . . . + d(x, slS (γ) x) ≤ max d(x, sx) · lS (γ). s∈S

6.6. Properness of the mapping class group. It is now easy to deduce Corollary 1.3 from the introduction. Indeed, it suffices to establish Inequality (2.1) of [20, Prop. 2.4]. Now the upper bound is already established in [20, Lemma 2.7], and the lower bound was established within the proof of Theorem 6.20. References [1] Y. Benoist. Propri´ et´ es asymptotiques des groupes lin´ eaires. Geom. Funct. Anal., 7(1):1–47, 1997. [2] W. Bertram. The geometry of Jordan and Lie structures, volume 1754 of Lecture Notes in Mathematics. Springer-Verlag, Berlin, 2000. [3] D. Biallas. Verallgemeinerte Doppelverh¨ altnisse und Endomorphismen von Vektorr¨ aumen. Abh. Math. Sem. Univ. Hamburg, 29:263–291, 1966. [4] H. Braun. Doppelverh¨ altnisse in Jordan-Algebren. Abh. Math. Sem. Univ. Hamburg, 32:25– 51, 1968. [5] H. Braun and M. Koecher. Jordan-Algebren. Die Grundlehren der mathematischen Wissenschaften in Einzeldarstellungen mit besonderer Ber¨ ucksichtigung der Anwendungsgebiete, Band 128. Springer-Verlag, Berlin, 1966. [6] M. R. Bridson and A. Haefliger. Metric spaces of non-positive curvature, volume 319 of Grundlehren der Mathematischen Wissenschaften. Springer-Verlag, Berlin, 1999. [7] M. Burger and A. Iozzi. Bounded K¨ ahler class rigidity of actions on Hermitian symmetric ´ spaces. Ann. Sci. Ecole Norm. Sup. (4), 37(1):77–103, 2004. [8] M. Burger, A. Iozzi, F. Labourie, and A. Wienhard. Maximal representations of surface groups: symplectic Anosov structures. Pure Appl. Math. Q., 1(3, part 2):543–590, 2005. [9] M. Burger, A. Iozzi, and A. Wienhard. Maximal representations and Anosov structures. Preprint, 2009. [10] M. Burger, A. Iozzi, and A. Wienhard. Surface group representations with maximal Toledo invariant. Ann. of Math., to appear. [11] M. Burger, A. Iozzi, and A. Wienhard. Tight homomorphisms and Hermitian symmetric spaces. GAFA, to appear. [12] J.-L. Clerc and K.-H. Neeb. Orbits of triples in the Shilov boundary of a bounded symmetric domain. Transform. Groups, 11(3):387–426, 2006. [13] J. L. Clerc and B. Ørsted. The Maslov index revisited. Transform. Groups, 6(4):303–320, 2001. [14] T. Delzant, O. Guichard, F. Labourie, and S. Mozes. Well displacing representations and orbit maps. The Zimmer Festschrift, to appear. [15] J. Faraut and A. Kor´ anyi. Analysis on symmetric cones. Oxford Mathematical Monographs. The Clarendon Press Oxford University Press, New York, 1994. Oxford Science Publications.

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[16] A. W. Knapp. Lie groups beyond an introduction. 2nd ed. Progress in Mathematics 140, Birkh¨ auser, Boston, 2002. [17] F. Labourie. Cross ratios, Anosov representations and the energy functional on Teichm¨ uller ´ Norm. Sup´ space. Ann. Sci. Ec. er. (4), 41(3):437–469, 2008. [18] I. Satake. Algebraic structures of symmetric domains, volume 4 of Kanˆ o Memorial Lectures. Iwanami Shoten, Tokyo, 1980. [19] C. L. Siegel. Symplectic geometry. Amer. J. Math., 65:1–86, 1943. [20] A. Wienhard. The action of the mapping class group on maximal representations. Geom. Dedicata, 120:179–191, 2006.

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