THE ACTION OF THE MAPPING CLASS GROUP ON MAXIMAL REPRESENTATIONS

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1 THE ACTION OF THE MAPPING CLASS GROUP ON MAXIMAL REPRESENTATIONS ANNA WIENHARD Abstract. Let Γ g be the fundamental group of a closed oriented Riemann surface Σ g, g 2, and let G be a simple Lie group of Hermitian type. The Toledo invariant defines the subset of maximal representations Rep max (Γ g, G) in the representation variety Rep(Γ g, G). Rep max (Γ g, G) is a union of connected components with similar properties as Teichmüller space T (Σ g ) = Rep max (Γ g, PSL(2, R)). We prove that the mapping class group Mod Σg acts properly on Rep max (Γ g, G) when G = Sp(2n, R), SU(n, n), SO (4n), Spin(2, n). Contents 1. Introduction 1 2. Maximal Representations and Translation Lengths 3 3. Maximal Representations into the Symplectic Group 6 References Introduction Let Γ g be the fundamental group of a closed oriented surface Σ g of genus g 2. Let G be a connected semisimple Lie group and Hom(Γ g, G) the space of homomorphisms ρ : Γ g G. The automorphism groups of Γ g and G act on Hom(Γ g, G) by Aut(Γ g ) Aut(G) Hom(Γ g, G) Hom(Γ g, G) (ψ, α, ρ) α ρ ψ 1 : ( γ α(ρ(ψ 1 γ)) ). Date: 13th April Key words and phrases. Mapping class group, Modular group, Representation variety, Maximal representations, Toledo invariant, Teichmüller space. The author was partially supported by the Schweizer Nationalfond under PP and by the National Science Foundation under agreement No. DMS

2 2 A. WIENHARD Considering homomorphisms only up to conjugation in G defines the representation variety Rep(Γ g, G) := Hom(Γ g, G)/ Inn(G). The above action induces an action of the group of outer automorphisms Out(Γ g ) := Aut(Γ g )/ Inn(Γ g ) of Γ g on Rep(Γ g, G): Out(Γ g ) Rep(Γ g, G) Rep(Γ g, G) (ψ, [ρ]) [ψρ] := [( γ ρ(ψ 1 γ) )]. Recall that Out(Γ g ) is isomorphic to π 0 (Diff(Σ g )). The mapping class group Mod Σg is the subgroup of Out(Γ g ) corresponding to orientation preserving diffeomorphisms of Σ g. We refer to [16, 10] for a general introduction to mapping class groups and to [11] for a recent survey on dynamical properties of the action of Out(Γ g ) on representation varieties Rep(Γ g, G). This note is concerned with the action of the mapping class group on special connected components of Rep(Γ g, G) when G is of Hermitian type. Recall that a connected semisimple Lie group G with finite center is said to be of Hermitian type if its associated symmetric space X is a Hermitian symmetric space. When G is of Hermitian type there exists a bounded continuous integer valued function T : Rep(Γ g, G) Z called the Toledo invariant. The level set of the maximal possible modulus of T is the set of maximal representations Rep max (Γ g, G) Rep(Γ g, G), which is studied in [12, 13, 23, 15, 1, 14, 6, 3, 4, 21]. Since the Toledo invariant is locally constant, its level sets are unions of connected components. Results of [12, 13, 6, 4] suggest that maximal representations provide a meaningful generalization of Teichmüller space when G is of Hermitian type [25]. This note supports this similarity by proving the following theorem Theorem 1.1. Let G = Sp(2n, R), SU(n, n), SO (4n), Spin(2, n). Then the action of Mod Σg on Rep max (Γ g, G) is proper. The validity of Theorem 1.1 for all groups locally isomorphic to either Sp(2n, R), SU(n, n), SO (4n) or Spin(2, n) would follow from an affirmative answer to the following question:

3 ACTION OF MAPPING CLASS GROUP 3 Question. If G = Sp(2n, R), SU(n, n), SO (4n) or Spin(2, n), G the adjoint form of G, and ρ Rep max (Γ g, G), does there exist a lift of ρ to G? Remark 1.2. Note that maximal representations factor through maximal subgroups of tube type [6, 4]. Therefore the only case which is not covered by the above theorem is the exceptional group G = E 7( 25). We would like to remark that the study of maximal representations Rep max (Γ g, G) Rep(Γ g, G) when G is of Hermitian type is related to the study of the Hitchin component Rep H (Γ g, G) Rep(Γ g, G) for split real simple Lie groups G. François Labourie recently announced, as a consequence of his work on Anosov representations and crossratios [18, 20], that the mapping class group acts properly on Rep H (Γ g, SL(n, R)). After finishing this note, we learned that he also has a proof for maximal representations into Sp(2n, R) [19]. The author is indebted to Marc Burger for motivation, interesting discussions and for pointing out a mistake in a preliminary version of this paper. The author thanks Bill Goldman, Ursula Hamenstädt, Alessandra Iozzi and François Labourie for useful discussions, and the referee for detailed suggestions which helped to substantially improve the exposition of this note. 2. Maximal Representations and Translation Lengths 2.1. Maximal Representations. For an introduction and overview the reader is referred to [3, 4]. Let G be a connected semisimple Lie group with finite center. Denote by X = G/K, with K < G a maximal compact subgroup, its associated symmetric space. G is said to be of Hermitian type if there exists a G-invariant complex structure J on X. The composition of the Riemannian metric induced by the Killing form B on X with the complex structure defines a Kähler form ω X (v, w) := 1 B(v, Jw) 2 which is a G-invariant closed differential two-form on X. Given a representation ρ : Γ g G consider the associated flat bundle E ρ over Σ g defined by E ρ := Γ g \( Σ g X ), where Γ g acts diagonally by deck transformations on Σ g and via ρ on X. As X is contractible, there exists a smooth section f : Σ g E ρ which is unique up to homotopy. This section lifts to a smooth ρ-equivariant

4 4 A. WIENHARD map f : Σ g Σ g X X. The pull back of ω X via f is a Γ g -invariant two-form f ω X on Σ g which may be viewed as a two-form on the closed surface Σ g. The Toledo invariant of ρ is T(ρ) := 1 2π Σ g f ω X. The Toledo invariant is independent of the choice of the section f and defines a continuous function T : Hom(Γ g, G) Z. The map T is invariant under the action of Inn(G) and constant on connected components of the representation variety. The Toledo invariant satisfies a generalized Milnor-Wood inequality [8, 7] T p X rk X χ(σ g ), 2 where rk X is the real rank of X and p X N is explicitly computable in terms of the root system. Definition 2.1. A representation ρ : Γ g G is said to be maximal if T(ρ) = p X rk X χ(σ g ). 2 Remark 2.2. Changing the orientation of Σ g switches the sign of T. We will restrict our attention to the case when ρ is maximal with T(ρ) > 0. We define the set of maximal representations Rep max (Γ g, G) := {[ρ] Rep(Γ g, G) ρ is a maximal representation}, which is a union of connected components of Rep(Γ g, G). The set Rep max (Γ g, PSL(2, R)) is the union of the two Teichmüller components of Σ g [12]. The action of the group Out(Γ g ) := Aut(Γ g )/ Inn(Γ g ) of outer automorphism of Γ g on Rep(Γ g, G) given by Out(Γ g ) Rep(Γ g, G) Rep(Γ g, G) (ψ, [ρ]) [ψρ] := [(γ ρ(ψγ))]. preserves Rep max (Γ g, G). The mapping class group Mod Σg preserves, and hence acts on the components of Rep max (Γ g, G) where T > 0. Remark 2.3. Note that whereas Teichmüller space, the set of quasifuchsian representations and Hitchin components are always contractible

5 ACTION OF MAPPING CLASS GROUP 5 subsets of Rep(Γ g, G), certain components of the set of maximal representations might have nontrivial topology [14, 2] Translation Lengths. For a hyperbolization h : Γ g PSL(2, R) define the translation length tr h of γ Γ g as tr h (γ) := inf p d (p, γp). For a representation ρ : Γ g G define similarly the translation length tr ρ of γ Γ g as tr ρ (γ) := inf x X G d X (x, ρ(γ)x), where d X is any left-invariant distance on the symmetric space associated to G. Proposition 2.4. Fix a hyperbolization h of Γ g. Assume that for any maximal representation ρ : Γ g G there exists A, B > 0 such that (2.1)A 1 tr h (γ) B tr ρ (γ) A tr h (γ) + B for all γ Γ g. Then Mod Σg acts properly on Rep max (Γ g, G). The Proposition relies on the fact that Mod Σg acts properly discontinuous on Teichmüller space T (Γ g ), which is due to Fricke. Lemma 2.5. [9, Proposition 5] There exists a collection of simple closed curves {c 1, c 9g 9 } on Σ g such that the map T (Γ g ) R 9g 9 h (tr h (γ i )) i=1,,9g 9, where γ i is the element of Γ g corresponding to c i, is injective and proper. Remark 2.6. A family of such 9g 9 curves is given by 3g 3 curves α i giving a pants decomposition, 3g 3 curves β i representing seems of the pants decomposition and the 3g 3 curves given by the Dehn twists of β i along α i (see e.g. [10]). Proof of Proposition 2.4. We argue by contradiction. Suppose that the action of Mod Σg on Rep max (Γ g, G) is not proper. Then there exists a compact subset C Rep max (Γ g, G) such that #{ψ Mod Σg ψ(c) C} is infinite. Thus there exists an infinite sequence ψ n in Mod Σg and a representation ρ Rep max (Γ g, G) such that ψ n (ρ) converges to a representation ρ Rep max (Γ g, G). Since ψ n acts properly on Teichmüller

6 6 A. WIENHARD space T (Γ g ), the sequence of hyperbolizations ψ n h leaves every compact set of T (Γ g ). This implies that the sum of the translation lengths of the elements γ i, i = 1,, 9g 9 tends to : By assumption (2.1) hence 9g 9 i=1 A 1 tr h (ψ 1 n 9g 9 i=1 tr ψnh(γ i ) γ i) B tr ρ (ψ 1 γ i), tr ψnρ(γ i ). This contradicts lim n ψ n ρ = ρ, since, by (2.1), the sum 9g 9 i=1 tr ρ (γ i ) is bounded from above by A 9g 9 i=1 tr h(γ i ) + B. Note that the upper bound for the comparison of the translation lengths with respect to a hyperbolization h and a representation ρ is established quite easily Lemma 2.7. Fix a hyperbolization h. For every maximal representation ρ : Γ g G there exists A, B 0 such that tr ρ (γ) A tr h (γ) + B for all γ Γ g. Proof. Let X be the symmetric space associated to G. By [17, Proposition 2.6.1] there exists a ρ-equivariant (uniform) L-Lipschitz map f : D X. Let p 0 D such that tr h (γ) = d (p 0, γp 0 ), then tr ρ (γ) d X (f(p 0 ), ρ(γ)f(p 0 )) = d X (f(p 0 ), f(γp 0 )) L d (p 0, γp 0 ) = L tr h (γ). n 3. Maximal Representations into the Symplectic Group The main objective of this section is to establish the following Proposition 3.1. For any hyperbolization h of Γ g, there exist constants A, B 0 such that A 1 tr h (γ) B tr ρ (γ) A tr h (γ) + B for all ρ Rep max (Γ g, Sp(2n, R)) and all γ Γ g. Proposition 3.1 in combination with Proposition 2.4 gives

7 ACTION OF MAPPING CLASS GROUP 7 Corollary 3.2. The action of Mod Σg proper. on Rep max (Γ g, Sp(2n, R)) is That Theorem 1.1 can be deduced from Proposition 3.1 and Proposition 2.4 can be seen as follows - we refer the reader to [3, 5, 24] for more on tight homomorphisms and their properties. Satake [22, Ch. IV] investigated when a simple Lie group G of Hermitian type admits a homomorphism τ : G Sp(2m, R). such that the induced homomorphism of Lie algebras π : g sp(2m, R) is a so called (H 2 )-Lie algebra homomorphism. Examples of such are τ : SU(n, n) Sp(4n, R) τ : SO (4n) Sp(8n, R) τ : Spin(2, n) Sp(2m, R), where m depends on n mod 8. In [24, 5] we prove that any such (H 2 )-homomorphism τ is a tight homomorphism. This implies in particular that the composition of any maximal representation ρ : Γ g G for G = SU(n, n), SO (4n), Spin(2, n) with the homomorphism τ : G Sp(2m, R) is a maximal representation ρ τ := τ ρ : Γ g Sp(2m, R). By Proposition 3.1 the translation lengths tr h (γ) and tr ρτ (γ) are comparable. Since the embedding X G X Sp(2m, ), defined by τ, is totally geodesic and the image ρ τ (Γ g ) preserves X G, the same argument as in Lemma 3.9 below gives that tr ρτ (γ) = tr ρ (γ) for all γ Γ g The Symplectic Group. For a 2n-dimensional real vector space V with a nondegenerate skew-symmetric bilinear form,, the symplectic group Sp(V ) is defined as Sp(V ) := Aut(V,, ) := {g GL(V ) g, g =, }. The symmetric space associated to Sp(V ) is given by X Sp := {J GL(V ) J 2 = Id,, J >> 0}, where, J >> 0 indicates that, J is symmetric and positive definite. The action of Sp(V ) on X Sp is by conjugation g(j) = g 1 Jg. We specify a left invariant distance on X Sp as follows. Let J 1, J 2 X Sp, the symmetric positive definite forms, J i define a pair of Euclidean norms q i on V. Denoting by Id J1,J 2 the norm of the identity map from (V, q 1 ) to (V, q 2 ) we define a distance on X Sp by d Sp (J 1, J 2 ) := log Id J1,J 2 + log Id J2,J 1

8 8 A. WIENHARD 3.2. Transverse Lagrangians and Causal Structure. Let L(V ) := {L V dim(l) = n,, L = 0} be the space of Lagrangian subspaces of V. Two Lagrangian subspaces L +, L L(V ) are said to be transverse if L + L = {0}. Any two transverse Lagrangian subspaces L +, L L(V ) define a symmetric subspace Y L,L + X Sp by Y L,L + := {J X Sp J(L ± ) = L } X Sp. ( ) A B Writing an element g Sp(V ) in block decomposition g = C D with respect to the decomposition V = L L + defines a natural embedding GL(L ) Sp(V ) given by GL(L ) Sp(V ) ( ) A 0 A 0 A T 1, similarly for GL(L + ). The subgroup GL(L ) preserves the symmetric subspace Y L,L + and acts transitively on it. Remark 3.3. The space of Lagrangians L(V ) can be identified with the Shilov boundary ŠSp of X Sp, and realized inside the visual boundary as the G-orbit of a specific maximal singular direction. Two Lagrangians are transverse if and only if the two corresponding points in the visual boundary can be joined by a maximal singular geodesic γ L±. The symmetric subspace Y L,L + is the parallel set of γ L±, i.e. the set of points on flats containing the geodesic γ L± ; it is the noncompact symmetric space dual to L(V ) U(L )/O(L ). For J Y L,L + the restriction of, J to L is a positive definite symmetric bilinear form on L, and conversely, fixing L +, any positive definite symmetric bilinear form Z on L defines a complex structure J Y L,L +. Therefore, the space Y L,s := {Z Z positive bilinear form on L }. with the action of GL(L ) by A(Z) := A T ZA, where we choose a scalar product on L (i.e. a base point) and realize a bilinear form on L as a symmetric (n n) matrix Z, is GL(L )- equivariantly isomorphic to Y L,L +. We endow Y L,s with the leftinvariant distance induced by d Sp via this isomorphism.

9 ACTION OF MAPPING CLASS GROUP 9 The space Y L,s is endowed with a natural causal structure, given by the GL(L )-invariant family of proper open cones Ω Z := {Z Y L,s Z Z is positive definite }. Definition 3.4. A continuous map f : [0, 1] Y L,s is said to be causal if f(t 2 ) Ω f(t1 ) for all 0 t 1 < t 2 1. A consequence of the proof of Lemma 8.10 in [3] is the following Lemma 3.5. For all Z Y L,s, Z Ω Z and every causal curve f : [0, 1] Y L,s with f(0) = Z and f(1) = Z : where n = dim(l ). length(f) n d Sp (Z, Z ), The claim basically follows from the last inequality in the proof of Lemma 8.10 in [3]. However, for the reader s convenience we give a direct proof here. Proof. Since d Sp is left invariant it is enough to prove the statement for Z = Id n Y L,s. For any subdivision 0 = t 0 < t 1 < < t m = 1 let f(t i ) = B T i B i Y L,s, and note that by causality With n = dim(l ), we have det ( (B i B 1 i+1 )T (B i B 1 i+1 )) < 1. d Sp (f(t i ), f(t i+1 )) = d Sp (Bi T B i, Bi+1 T B i+1) = log [ ( λ max (Bi+1 B 1 i ) T (B i+1 B 1 i ) )] + log [ ( λ min (Bi+1 B 1 i ) T (B i+1 B 1 i ) )] log [ det ( (B i+1 B 1 i ) T (B i+1 B 1 i ) )] n log [ det ( (B i Bi+1 1 )T (B i Bi+1 1 ))] n log [ λ max (Bi+1B T i+1 ) ] n log [ λ min (Bi T B i) ] + n log [ λ min (Bi+1 T B i+1) ] n log [ λ max (B T i B i) ].

10 10 A. WIENHARD Summing over the subdivision we obtain m length(f) d Sp (f(t i ), f(t i+1 )) i+1 n [log λ max (f(1)) + log λ min (f(1))] n [log λ max (f(0)) log λ min (f(0))] = n [log λ max (f(1)) + log λ min (f(1))] = n d Sp (f(0), f(1)) as needed Quasi-isometric Embedding. Let ρ : Γ g Sp(V ) be a maximal representation. The choice of a hyperbolization h of Γ g defines a natural action of Γ g on S 1 = D. Lemma 3.6. [3, Corollary 6.3] There exists a ρ-equivariant continuous map ϕ : S 1 L(V ) such that distinct points x, y S 1 are mapped to transverse Lagrangians ϕ(x), ϕ(y) L(V ). A triple (L, L 0, L + ) L(V ) 3 of pairwise transverse Lagrangians gives rise to a complex structure ( ) 0 T + J L0 = 0 T0 0 on V = L L +, where T 0 ± : L ± L is the unique linear map such that L 0 = graph(t 0 ± ). A triple (L, L 0, L + ) of pairwise transverse Lagrangians is maximal if the symmetric bilinear form, J L0 is positive definite, that is if J L0 Y L,L + X Sp. We denote by L(V ) 3 + the space of maximal pairwise transverse triples in L(V ). Under the identification of the unit tangent bundle of the Poincarédisc T 1 D (S 1 ) 3 + with positively oriented triples in S 1, the map ϕ gives rise to a ρ-equivariant map (Equation (8.9) in [3]) J : T 1 D = (S 1 ) 3 + L(V ) 3 + X Sp u = (u, u 0, u + ) (ϕ(u ), ϕ(u 0 ), ϕ(u + )) J(u), where J(u) is the complex structure defined by the maximal triple (ϕ(u ), ϕ(u 0 ), ϕ(u + )) L(V ) 3 +. Let g t be the lift of the geodesic flow on T 1 Σ g to T 1 D. Then for all t the image of g t u = (u, u t, u + ) under J is contained in the symmetric subspace Y ϕ(u ),ϕ(u + ) X Sp associated to the two transverse Lagrangians ϕ(u ), ϕ(u + ).

11 ACTION OF MAPPING CLASS GROUP 11 Lemma 3.7. [3, Equation 8.8] Let γ Γ g \{Id} and p D a point. Denote by u T 1 D the unit tangent vector at p of the geodesic connecting p and γp. Then there exists a constant A > 0 such that A 1 d (γp, p) d Sp (J(u), ρ(γ)j(u)). Remark 3.8. Note that the statement of Lemma 3.7 implies that the action of Mod Σg on the connected components of maximal Toledo invariant in Hom(Γ g, G) is proper, but is not sufficient to deduce Theorem 1.1. The inequality in Lemma 3.7 is with respect to specific points in X Sp, but to compare the translation lengths we have to take infima on both sides of the inequality. There is in general no direct way to compare the translation length of ρ(γ) with the displacement length of ρ(γ) with respect to a specific point x X Sp. In our situation we will make use of the causal structure on Y L,s to compare the translation length of ρ(γ) with the displacement length d Sp (J(u), ρ(γ)j(u)) Translation Length and Displacement Length. Fix a hyperbolization h : Γ g PSL(2, R). Let γ Γ g \{Id}. Denote by γ +, γ S 1 the attracting, respectively repelling, fixed point of γ and by L ± L(V ) the images of γ ± under the ρ-equivariant boundary map ϕ : S 1 L(V ). Let Y L,L + X Sp be the symmetric subspace associated to L +, L. Lemma 3.9. The translation length of ρ(γ) is attained on Y L,L + : tr ρ (γ) = inf d Sp (J, ρ(γ)j ). J Y L,L + Proof. The symmetric subspace Y L,L + is a totally geodesic submanifold of X Sp. Denote by pr Y : X Sp Y L,L + the nearest point projection onto Y L,L +. The projection pr Y is distance decreasing. Since the element ρ(γ) stabilizes L ±, the projection pr Y onto Y L,L + is ρ(γ)- equivariant. Thus, for every J X Sp d Sp (J, ρ(γ)j) d Sp (pr Y (J), pr Y (ρ(γ)j)) = d Sp (pr Y (J), ρ(γ) pr Y (J)). In particular inf d Sp (J, ρ(g)j) = inf d Sp (J, ρ(g)j ). J X Sp J Y L,L + Lemma Let p 0 D be some point lying on the geodesic c γ connecting γ to γ +. Let u = (γ, u 0, γ + ) T 1 D be the unit vector tangent to c γ at p 0. Then (1) ρ(γ)j(u) Ω J(u).

12 12 A. WIENHARD (2) For any Z Y L,s there exists an N such that for all m N: ρ(γ) m J(u) Ω Z. Proof. (1) The map u J(u) is ρ-equivariant, thus ρ(γ)j(u) = J(v) Y L,s, with v = (γ, γu 0, γ + ). Since γ + is the attracting fixed point of γ, the triple (u 0, γu 0, γ + ) is positively oriented, and (ϕ(u 0 ), ϕ(γu 0 ), ϕ(γ + )) is a maximal triple. Then, by [3, Lemma 8.2], ρ(γ)j(u) J(u) = J(v) J(u) is positive definite, thus ρ(γ)j(u) Ω J(u). (2) Let µ be the maximal eigenvalue of Z Y L,s. It suffices to show that there exists some N such that the minimal eigenvalue of ρ(γ) N J(u) Y L,s is bigger than µ. Then ρ(γ) N J(u) Ω Z and, with statement (1), we have that ρ(γ) m J(u) Ω ρ(γ) N J(u) Ω Z for all m > N. Note that γ m u 0 γ + as m and, since ϕ is continuous, ρ(γ) m ϕ(u 0 ) ϕ(γ + ). Moreover ρ(γ) i+1 J(u) ρ(γ) i J(u) is positive definite for all i. Hence, the eigenvalues of ρ(γ) i J(u) grow monotonically towards. In particular, there exists N such that the minimal eigenvalue of ρ(γ) N J(u) is bigger than µ. Combining Lemma 3.10 with Lemma 3.5 we obtain Lemma There exist constants A, B > 0 such that for every Z Y L,s d Sp (J(u), ρ(γ)j(u)) A d Sp (Z, ρ(γ)z) + B. Proof. Fix Z Y L,s. Choose, by Lemma 3.10, N big enough such that Z := ρ(γ) N J(u) Ω Z. By Lemma 3.10 there are causal, distance realizing curves f Z from Z to Z and f i from ρ(γ) i Z to ρ(γ) i+1 Z for all 0 i. For every k 0 the concatenation f = f k 1 f 0 f Z is a causal curve from Z to ρ(γ) k Z = ρ(γ) N+k J(u). Thus applying Lemma 3.5 we get that for every k 0 d Sp (Z, Z ) + k d Sp (Z, ρ(γ)z ) k 1 ( = d Sp (Z, Z ) + d Sp ρ(γ) i Z, ρ(γ) i+1 Z ) = length(f) i=0 n d Sp (Z, ρ(γ) k Z ) n [ d Sp (Z, ρ(γ) k Z) + d Sp (ρ(γ) k Z, ρ(γ) k Z ) ] [ k 1 ] n d Sp (ρ(γ) i Z, ρ(γ) i+1 Z) + d Sp (Z, Z) i=0 = n d Sp (Z, Z) + nk d Sp (Z, ρ(γ)z).

13 In particular ACTION OF MAPPING CLASS GROUP 13 d Sp (Z, ρ(γ)z ) n d Sp (Z, ρ(γ)z) + n 1 d Sp (Z, Z). k Thus, for A = n and B > 0 fixed we can choose k big enough such d Sp (Z, Z) B to get that n 1 k d Sp (Z, ρ(γ)z ) A d Sp (Z, ρ(γ)z) + B. Since d Sp is left invariant, this implies d Sp (J(u), ρ(γ)j(u)) d Sp (ρ(γ) N J(u), ρ(γ) N+1 J(u)) hence the claim. = d Sp (Z, ρ(γ)z ) A d Sp (Z, ρ(γ)z) + B, Lemma There exist constants A, B > 0 depending only on dim(v ) such that for all u T 1 D and γ Γ g d Sp (J(u), ρ(γ)j(u)) A tr ρ (γ) + 2B. Proof. Fix some ɛ > 0. By Lemma 3.9 there exists Z 0 Y L,s such that d Sp (Z 0, ρ(γ)z 0 ) Therefore Lemma 3.11 implies inf d Sp (X, ρ(γ)x) + ɛ = tr ρ (γ) + ɛ. X Y L,s d Sp (J(u), ρ(γ)j(u)) A d Sp (Z 0, ρ(γ)z 0 ) + B A tr ρ (γ) + A ɛ + B. Since this holds for all ɛ > 0 we get that d Sp (J(u), ρ(γ)j(u)) A tr ρ (γ) + 2B. Proof of Proposition 3.1. Let ρ be a maximal representation of Γ g into Sp(V ) and let p D be such that tr h (γ) = d (p, γp);, then p lies on the unique geodesic c γ connecting γ to γ +. Let u = (γ, u 0, γ + ) T 1 D be the unit tangent vector to c γ at p and J(u) Y L,L + the image of u under the mapping J : T 1 D X Sp. By Lemma 3.7 there exists a constant A such that A 1 tr h (γ) = A 1 d (p, γp) d Sp (J(u), ρ(γ)j(u)). Applying Lemma 3.12 to this, there exist constants A, B > 0 such that tr h (γ) A d Sp (J(u), ρ(γ)j(u)) A A tr ρ (γ) + 2A B. This in combination with Lemma 2.7 finishes the proof.

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15 ACTION OF MAPPING CLASS GROUP , A generalisation of Teichmüller space in the Hermitian context, Séminaire de Théorie Spectrale et Géométrie Grenoble 22 (2004), address: wienhard@math.ias.edu School of Mathematics, Institute for Advanced Study, 1 Einstein Drive, Princeton NJ 08540, USA Department of Mathematics, University of Chicago, 5734 University Avenue, Chicago, IL , USA