Annales Academiæ Scientiarum Fennicæ Mathematica Volumen 22, 1997, 3–14

DEFORMING REAL PROJECTIVE STRUCTURES

Daniel M. Gallo

St. John’s University, Department of Mathematics 300 Howard Ave., Staten Island, N.Y. 10301, U.S.A.; gallod@simath.stjohns.edu

Abstract. Let S be a compact Riemann surface of hyperbolic type and T (S) the Teichmüller space of S. We show that the Maskit grafting map is a homeomorphism from T (S) onto T (S). It follows from this result that S has a projective structure which is different from the canonical Fuchsian uniformization, and has discrete, injective holonomy contained in PSL(2, R).

1. Introduction

We are concerned in this paper with the grafting operation along a simple, closed geodesic on a closed, hyperbolic Riemann surface (see Section 3 for definitions). This construction was first considered by Maskit [7] in the context of projective structures on surfaces; he used it to provide examples of distinct structures with identical holonomy maps. Later, Goldman [6] showed that, with the exception of the canonical Fuchsian structure, all real projective structures with discrete, injective holonomy are obtained by grafting.

It is natural to identify homotopy classes of simple, closed geodesics on distinct surfaces of the same genus via the marked equivalence classes of these surfaces in Teichmüller space. Using this identification, we will show (Theorem 3) that, given a hyperbolic Riemann surface S and a simple, closed geodesic on that surface, there exists a (unique) surface S′ such that S can be obtained from S′ by grafting along the corresponding geodesic in S′. As a corollary to this result (Corollary 4), it follows that every surface has a real projective structure which is distinct from the canonical Fuchsian uniformizing structure, and has discrete, injective holonomy contained in PSL(2, R); this solves a problem posed by Maskit in [7].

Section 2 contains basic definitions and theorems from the theories of Teichmüller spaces and projective structures. In Section 3, we define the Maskit grafting map for a simple closed geodesic and study the behavior of this map under the assumption that the geodesic has small hyperbolic length.

To analyze the grafting map for geodesics of arbitrarily large hyperbolic length, we need to consider the extremal length properties of families of curves homotopic to these geodesics; this is done in Section 4. The main result of the paper, Theorem 3, is proved in Section 3.

The constructions of Section 4 leading up to the proof of Lemma 4.2 are obtained with the use of an auxiliary metric; the author is grateful to J. Velling for pointing out the usefulness of this metric. We note that similar results have been obtained by H. Tanigawa (oral communication) using harmonic maps. Finally, the author would like to thank the referee for several helpful suggestions for bringing the paper into its final form.

1991 Mathematics Subject Classification: Primary 32G15; Secondary 30F10.

2. Teichmüller spaces and projective structures

2.1. Throughout, Γ will denote a Fuchsian covering group acting on the upper half plane U and covering a compact surface of genus g > 1. The space M(Γ) consists of all measurable functions μ: U → C, with ∥μ∥_{∞} < 1, satisfying μ(A(z))\overline{A′(z)} = μ(z)A′(z), for all A ∈ Γ.

For each μ ∈ M(Γ) there is a unique quasiconformal map w^{μ}: U → U which fixes 0, 1, ∞ and satisfies wμ ; \overline{z} = μw_{z}^{μ} (here the derivatives are taken in the generalized sense). The quantity (1 + ∥μ∥_{∞})/(1 − ∥μ∥_{∞}) ≥ 1 is called the dilatation of w^{μ}. Further, w^{μ} ∘ A = A^{μ} ∘ w^{μ} for some A^{μ} ∈ PSL(2, R). It follows that μ defines an isomorphism θ_{μ} from Γ onto a Fuchsian group Γ^{μ} with θ_{μ}(A) = A^{μ} (see [1]).

Two elements μ, ν ∈ M(Γ) are equivalent (denoted by μ ∼ ν) if they induce identical isomorphisms. The Teichmüller space T (Γ) is the space of equivalence classes (denoted by [μ] ∈ T (Γ)) of elements in M(Γ).

For μ ∈ M(Γ), let k = inf_{ν∼μ} ∥ν∥_{∞}, and K = (1 + k)/(1 − k). The Teichmüller distance from [0] to [μ] is given by

d_{Γ}([0], [μ]) = \frac{1}{2} log K.

The right translation mapping R_{μ}^{−1}: M(Γ) → M(Γ^{μ}) is defined by letting

R_{μ}^{−1}(ν) = \frac{(w^{ν} ∘ (w^{μ})^{−1})\overline{z}}{(w^{ν} ∘ (w^{μ})^{−1})_{z}}

(the Beltrami coefficient of w^{ν} ∘ (w^{μ})^{−1} ). Now, the Teichmüller metric on T (Γ) is defined by

(1)  d_{Γ}([μ], [ν]) = d_{Γ^{μ}}([0], [R_{μ}^{−1}(ν)]).

It is well known that T (Γ) has the natural structure of a real analytic manifold and that it is a cell of real dimension 6g − 6.

2.2. Let B_{2}(Γ) be the 6g − 6 real analytic space of holomorphic quadratic differentials defined on U for Γ. There exists a real analytic vector bundle Q(Γ) over T (Γ), with projection map p: Q(Γ) → T (Γ) and fibers equal to B_{2}(Γ^{μ}). Given ([μ], φ) ∈ Q(Γ) (here φ ∈ B_{2}(Γ^{μ})), let f_{φ} be a solution of the Schwarzian differential equation S(f) = φ. The differential operator S(f) is defined by

S(f) = (f′′/f′)′ − \frac{1}{2}(f′′/f′)^{2}.

The function f_{φ} determines a homomorphism Θ^{f_{φ}}: Γ^{μ} → PSL(2, C) (called the holonomy map), with f_{φ} ∘ γ^{μ} = Θ^{f_{φ}}(γ^{μ}) ∘ f_{φ}, for all γ^{μ} ∈ Γ^{μ}, and the pair (f_{φ}, Θ^{f_{φ}}) is called a projective structure on U/Γ^{μ}. We say that the projective structure has real holonomy if the image of Γ^{μ} under Θ^{f_{φ}} is contained in PSL(2, R).

There is also the associated homomorphism Θ_{f_{φ}}: Γ → PSL(2, C) defined by Θ_{f_{φ}}(γ) = Θ^{f_{φ}}(γ^{μ}). For T ∈ PSL(2, C), T ∘ f_{φ} is also a solution of S(f) = φ, and this determines the homomorphism γ^{μ} → T ∘ Θ^{f_{φ}}(γ^{μ}) ∘ T^{−1} . One is thus led to consider the equivalence class [Θ_{φ}] of Θ_{f_{φ}} under conjugation by elements of PSL(2, C).

Letting Hom(Γ, PSL(2, C)) = Hom Γ denote the space of equivalence classes of homomorphisms of Γ into PSL(2, C), one defines Φ: Q(Γ) → Hom Γ, the monodromy map, by

Φ([μ], φ) = [Θ_{φ}].

It is well known that there exists a real analytic manifold Hom′ Γ ⊂ Hom Γ such that Φ(Q(Γ)) ⊂ Hom′ Γ and that Φ: Q(Γ) → Hom′ Γ is an analytic local diffeomorphism (see [3]). The manifold Hom′(Γ, PSL(2, R)) ⊂ Hom′ Γ is defined similarly.

The following is proved in [4, p. 261]:

Theorem 1 (Faltings). The space R(Γ) = Φ^{−1}(Hom′(Γ, PSL(2, R))) is a real analytic submanifold of Q(Γ) of dimension 6g − 6. Moreover, R(Γ) is transverse to the fibers of Q(Γ). The projection map p: Q(Γ) → T (Γ) is a local diffeomorphism when restricted to R(Γ).

3. The Maskit grafting construction

3.1. We normalize Γ so that the positive imaginary axis iR^{+} projects onto a simple, closed geodesic l on U/Γ. Let γ ∈ Γ, with γ(z) = r_{∗}z, r_{∗} > 1, be a generator for the cyclic subgroup of Γ which stabilizes iR^{+} . (The group Γ will remain fixed from now on, with this normalization in force.) Choose 0 < α < \frac{1}{2}π so that

B_{α} = \{z : \frac{1}{2}π + α > arg z > \frac{1}{2}π − α\}

projects onto a collar about l in U/Γ (see [2]).

Define I_{α} = [\frac{1}{2}π −α, \frac{1}{2}π +α], I_{α+2π} = [\frac{1}{2}π −α, \frac{5}{2}π +α] and let v: I_{α} → I_{α+2π} be a C^{1} homeomorphism, with v(\frac{1}{2}π − α) = \frac{1}{2}π − α and v(\frac{1}{2}π + α) = \frac{5}{2}π + α. We assume that v has derivative equal to 1 at both endpoints and that v′(θ) ≥ 1 for all θ ∈ I_{α}. The map v will be called an allowable map.

A local homeomorphism f_{α,v}: U → Ĉ is defined as follows: Let ⟨γ⟩ be the cyclic subgroup generated by γ and set D_{α} = ⋃_{A∈Γ/⟨γ⟩} A(B_{α}). Let f_{α,v}(z) = z for z ∈ U − D_{α}, and f_{α,v}(z) = re^{iv(θ)} , for z = re^{iθ} ∈ B_{α}. For w = A(z), with z ∈ B_{α} and A ∈ Γ, let f_{α,v}(w) = A(f_{α,v}(z)).

One computes easily that

μ_{α,v}(z) = \frac{z}{\overline{z}} \frac{1 − v′(θ)}{1 + v′(θ)},

for z = re^{iθ} ∈ B_{α}; hence ∥μ_{α,v}∥_{∞} < 1. Clearly, f_{α,v} ∘ A = A ∘ f_{α,v} for all A ∈ Γ, and consequently,

(2)  μ_{α,v} = (f_{α,v})\overline{z}/(f_{α,v})_{z} ∈ M(Γ).

Denote the quasiconformal map w^{μ_{α,v}} (defined as in Section 2) by w^{α,v}. Then f_{α,v} ∘ (w^{α,v})^{−1} = g_{α,v} is meromorphic in U and φ = S(g_{α,v}) ∈ B_{2}(Γ^{μ_{α,v}}). Note that (g_{α,v}, Θ^{f_{φ}}) defines a projective structure on Γ^{μ_{α,v}}, where Θ^{f_{φ}}: Γ^{μ_{α,v}} → Γ is an isomorphism for the solution f_{φ} = g_{α,v}, and Θ^{f_{φ}}(γ^{μ_{α,v}}) = γ.

Lemma 3.1. The element ([μ_{α,v}], S(g_{α,v})) ∈ Q(Γ) is independent of the choices of α and v.

Proof. Fix 0 < α < \frac{1}{2}π and let u, v be allowable maps. Choose f_{t}, 0 ≤ t ≤ 1, a C^{1} homotopy of allowable maps with f_{0} = u, f_{1} = v and define h: [0, 1] → Q(Γ) by h(t) = ([μ_{α,f_{t}}], S(g_{α,f_{t}})). (Note that convex combinations of allowable maps are allowable so that f_{t} = tv + (1 − t)u could be used.)

We first show that h is continuous. Suppose t_{n} → t ∈ [0, 1]. One verifies easily that f_{α,f_{tn}} → f_{α,f_{t}} and μ_{α,f_{tn}} → μ_{α,f_{t}} uniformly on compact subsets of U. From basic properties of the solution to the Beltrami equation (see [1]), one has that w^{α,f_{tn}} → w^{α,f_{t}}. (Note that ∥μ_{α,f_{tn}} ∥_{∞} is uniformly bounded away from 1.) Hence, g_{α,f_{tn}} → g_{α,f_{t}} and S(g_{α,f_{tn}}) → S(g_{α,f_{t}}).

By its construction, Φ ∘ h(t) is constant and, since Φ is locally injective, it follows that h(t) is constant.

To show that [μ_{α,v}] is independent of the choice of α, let 0 < α′ < α and choose v: I_{α′} → I_{α′+2π} an allowable map. Extend v to an allowable map v̂: I_{α} → I_{α+2π}, letting

v̂(θ) = \{ \begin{matrix}θ, & θ ∈ [\frac{1}{2}π − α, \frac{1}{2}π − α′], \\ v(θ), & θ ∈ [\frac{1}{2}π − α′, \frac{5}{2}π + α′], \\ θ + 2π, & θ ∈ [\frac{5}{2}π + α′, \frac{5}{2}π + α]. \end{matrix}

Observe that for α′ < s < α, v̂_{s} = v̂ | I_{s} is allowable. As in the preceding argument, the map h(s) = ([μ_{s,v̂_{s}}], S(g_{s,v̂_{s}})) ∈ Q(Γ) is continuous, and by the same reasoning, it must be constant. The result follows.

3.2. We will now write [μ^{l}] = [μ_{α,v}] ∈ T (Γ) and φ_{l} = S(g_{μ_{α,v}}) ∈ B_{2}(Γ^{μ^{l}}).

Lemma 3.2. The Teichmüller distance

d_{Γ}([0], [μ^{l}]) ≤ \frac{1}{2} log(1 + \frac{π}{\frac{1}{2}π − tan^{−1}(sinh(l̂/2))}),

where l̂ is the hyperbolic length of l.

Proof. Fix 0 < α < \frac{1}{2}π and, for an allowable map v: I_{α} → I_{α+2π}, let r_{v} = sup_{x∈I_{α}} v′(x) < ∞. Then inf{r_{v} : v allowable} = M_{α}, where M_{α} = 1 + (π/α) is the slope of the linear map L: I_{α} → I_{α+2π}, with L(\frac{1}{2}π − α) = \frac{1}{2}π − α, L(\frac{1}{2}π + α) = \frac{5}{2}π + α.

It follows that for ε > 0 we may choose v_{ε} an allowable map such that

(3)  ∥μ_{α,v_{ε}}∥_{∞} = ∥\frac{1 − v′_{ε}}{1 + v′_{ε}} ∥_{∞} ≤ k_{ε},

where M_{α} + ε = (k_{ε} + 1)/(1 − k_{ε}). Here, ∥(1 − v′_{ε})/(1 + v′_{ε})∥_{∞} is taken over I_{α} and the equality in (3) is obtained by a straightforward computation from (2).

Applying Lemma 3.1, we have d_{Γ}([0], [μ^{l}]) ≤ \frac{1}{2} log(M_{α}+ε), and letting ε → 0,

(4)  d_{Γ}([0], [μ^{l}]) ≤ \frac{1}{2} log M_{α}.

There is a collar lemma (see Buser [2]) which guarantees that B_{α_{l}} projects onto a collar about l, where α_{l} ≥ \frac{1}{2}π − tan^{−1}(sinh(l̂/2)). Applying Lemma 3.1 and the inequality (4), we have

d_{Γ}([0], [μ^{l}]) ≤ \frac{1}{2} log M_{α_{l}} ≤ \frac{1}{2} log(1 + \frac{π}{\frac{1}{2}π − tan^{−1}(sinh(l̂/2))}).

3.3. We will now allow [ν] to vary over T (Γ) and perform Maskit’s grafting construction on U/Γ^{ν} in the following manner: The element γ^{ν} stabilizes iR^{+} , and iR^{+} projects onto a simple, closed geodesic l_{ν} in U/Γ^{ν}. Using the previous notation, we obtain [μ^{l_{ν}}] ∈ T (Γ^{ν}) and ([μ^{l_{ν}}], φ_{l_{ν}}) ∈ Q(Γ^{ν}).

Define the grafting map Ψ: T (Γ) → T (Γ) by Ψ([ν]) = [ν′], where ν′ is the Beltrami coefficient of w^{μ_{lν}} ∘w^{ν}. Note that w^{ν′} ∘Γ∘(w^{ν′})^{−1} = w^{μ_{lν}} ∘Γ^{ν} ∘(w^{μ_{lν}})^{−1} and that φ_{l_{ν}} represents a projective structure on U/Γ^{ν′}. Here, Θ^{f_{φ_{lν}}}: Γ^{ν′} → Γ^{ν}, and θ_{ν} = Θ_{f_{φ_{lν}}} = Θ^{f_{φ_{lν}}} ∘ θ_{ν′}: Γ → Γ^{ν} are isomorphisms.

We also define the map Ψ̂: T (Γ) → Q(Γ) given by Ψ̂([ν]) = ([ν′], φ_{l_{ν}}).

Lemma 3.3. Ψ̂ is a homeomorphism onto its image in R(Γ) ⊂ Q(Γ).

Proof. We first show that the map Ψ̂ is continuous. Suppose [ν_{j}] → [ν]. It is well known that l̂_{ν_{j}} → l̂_{ν}. Hence, we may choose 0 < α < \frac{1}{2}π such that B_{α} projects onto a collar about l_{ν_{j}} in U/Γ^{ν_{j}}, for all j sufficiently large. Let v: I_{α} → I_{α+2π} be an allowable map and define (using the group Γ^{ν_{j}}), for each j, the local homeomorphism f_{α,v}^{ν_{j}}: U → Ĉ, as in Section 3.1. Now, for A ∈ Γ, the corresponding elements A^{ν_{j}} ∈ Γ^{ν_{j}} converge to A^{ν} ∈ Γ^{ν}. Hence, f_{α,v}^{ν_{j}} → f_{α,v}^{ν} uniformly on compact subsets of U.

Let μ_{j} (respectively μ_{ν}) be the Beltrami coefficient of f_{α,v}^{ν_{j}} (respectively f_{α,v}^{ν}). From elementary properties of the solutions w^{μ_{j}}, we have that w^{μ_{j}} → w^{μ_{ν}}. It follows that [ν′_{j}] → [ν′], g_{μ_{j}} → g_{μ_{ν}}, and S(g_{μ_{j}}) = φ_{l_{ν_{j}}} → S(g_{μ_{ν}}) = φ_{l_{ν}}. Hence,

Ψ̂ is continuous.

Now let [ν_{1}], [ν_{2}] ∈ T (Γ) be distinct elements and suppose [ν′_{1}] = [ν′_{2}], so that Ψ̂([ν_{1}]) = ([ν′_{1}], φ_{1}), Ψ̂([ν_{2}]) = ([ν′_{2}], φ_{2}), where φ_{1} = φ_{l_{ν_{1}}} , φ_{2} = φ_{l_{ν_{2}}} are quadratic differentials for the same group Γ^{ν′_{1}} = Γ^{ν′_{2}} . As we have noted, there are solutions, f_{φ_{1}} and f_{φ_{2}} , with S(f_{φ_{1}}) = φ_{1} , S(f_{φ_{2}}) = φ_{2} , so that Θ^{f_{φ_{1}}} : Γ^{ν′_{1}} → Γ^{ν_{1}} and Θ^{f_{φ_{2}}} : Γ^{ν′_{1}} → Γ^{ν_{2}} . Moreover, the Teichmüller isomorphisms θ_{ν_{1}} = Θ_{f_{φ_{1}}} = Θ^{f_{φ_{1}}} ∘ θ_{ν′_{1}}: Γ → Γ^{ν_{1}} and θ_{ν_{2}} = Θ_{f_{φ_{2}}} = Θ^{f_{φ_{2}}} ∘ θ_{ν′_{1}}: Γ → Γ^{ν_{2}} are not conjugate in PSL(2, C), and it follows that Θ^{f_{φ_{1}}} and Θ^{f_{φ_{2}}} are not conjugate, either. Hence, the quadratic differentials φ_{1} , φ_{2} are distinct, and Ψ̂ is an injective map.

Corollary 2. The map Ψ is a local homeomorphism.

Proof. Since Ψ̂(T (Γ)) ⊂ R(Γ), we now apply Lemma 3.3 and Theorem 1 to deduce that p ∘ Ψ̂ = Ψ is locally injective. But T (Γ) is a finite dimensional space, and it follows that Ψ is a local homeomorphism.

Remark. It is easy to see that Ψ̂(T (Γ)) is a closed subset of R(Γ); hence, it also follows from Lemma 3.3 that Ψ̂(T (Γ)) is a connected component of R(Γ).

Lemma 3.4. Let [ν_{n}] ∈ T (Γ), with d_{Γ}([0], [ν_{n}]) → ∞, and let l be a simple, closed geodesic in U/Γ; suppose l̂_{ν_{n}} ≤ N for some positive constant N. Then d_{Γ}([0], Ψ([ν_{n}])) → ∞, also.

Proof. We obtain, applying Lemma 3.2,

d_{Γ^{νn}} ([0], [μ^{l_{νn}}]) ≤ \frac{1}{2} log(1 + \frac{π}{\frac{1}{2}π − tan^{−1}(sinh(l̂_{ν_{n}}/2))})

≤ \frac{1}{2} log(1 + \frac{π}{\frac{1}{2}π − tan^{−1}(sinh(N/2))}) = c_{0}.

Using the triangle inequality and the definition (1), we have

d_{Γ}([0], [ν′_{n}]) ≥ d_{Γ}([0], [ν_{n}]) − d_{Γ^{νn}} ([0], [μ^{l_{νn}}]) ≥ d_{Γ}([0], [ν_{n}]) − c_{0}.
Hence, d_{Γ}([0], [ν′_{n}]) → ∞.

Lemma 3.5. Let [ν_{n}] ∈ T (Γ), with d_{Γ}([0], [ν_{n}]) → ∞, and let l be a simple, closed geodesic in U/Γ; suppose l̂_{ν_{n}} → ∞. Then d_{Γ}([0], Ψ([ν_{n}])) → ∞, also.

The proof of Lemma 3.5 will be given in Section 4.

Theorem 3. Ψ: T (Γ) → T (Γ) is a surjective homeomorphism.

Proof. Suppose that Ψ([ν_{n}]) → [μ] ∈ T (Γ). By Lemmas 3.4 and 3.5, it follows that there exist [ν] ∈ T (Γ) and a subsequence [ν_{n}] → [ν]; by continuity, Ψ([ν]) = [μ]. Hence, Ψ(T (Γ)) is closed. Since Ψ is an open map (Lemma 3.3), we must have Ψ(T (Γ)) = T (Γ).

Now let [ν] ∈ T (Γ). In view of Corollary 2 and Lemmas 3.4, 3.5, we conclude that Ψ^{−1}([ν]) is finite. Hence, Ψ is a covering map; but T (Γ) is simply connected, and it must be a homeomorphism.

Since φ_{l_{ν}} defines a quadratic differential for the group Γ^{ν′}, where Ψ([ν]) = [ν′], the proof of the following corollary is clear.

Corollary 4. Every compact, hyperbolic Riemann surface has a projective structure where the holonomy is real, injective, discrete, and different from the canonical Fuchsian uniformizing structure.

4. Extremal length properties

Consider a family ℱ of simple, closed curves^{1} in U/Γ. The extremal length Λ(ℱ) is defined by Λ(ℱ) = sup_{ρ}(L(ρ)^{2}/A(ρ)), where ρ|dz| is a measurable, nonnegative metric on U/Γ; L(ρ) = inf ∫_{β} ρ|dz|, where the infimum is over all β in ℱ (here, ∫_{β} ρ|dz| = ∞ if ρ is not measurable over β); and A(ρ) = ∫∫_{U/Γ}ρ^{2} dx dy ≠ 0, ∞. (See [1] or [5].)

Let s be a simple, closed geodesic in U/Γ. An annulus A ⊂ U/Γ is homotopic to s, if it contains a nontrivial, simple, closed curve homotopic (in U/Γ) to s. Denote by ℱ_{s} the family of all simple, closed curves in U/Γ which are homotopic to s. We have the basic observation:

Lemma 4.1. Let s be a simple, closed geodesic in U/Γ. Then Λ(ℱ_{s}) < ∞.

Proof. Choose A ⊂ U/Γ an annulus homotopic to s, and let ℱ_{A} ⊂ ℱ_{s} be the set of all curves in ℱ_{s} which are contained in A. It is well known that Λ(ℱ_{A}) < ∞ (see [1]). (The same proof given in [1] remains valid when the area integral taken over C is replaced by an area integral over U/Γ.) Since ℱ_{A} ⊂ ℱ_{s} , we also have Λ(ℱ_{s}) ≤ Λ(ℱ_{A}) (see [1]).

The idea of the proof of Lemma 3.5 is to show that Λ(ℱ_{l_{ν′_{n}}}) → ∞. This will imply that [ν′_{n}] cannot have a convergent subsequence. First, an inequality relating Λ(ℱ_{l_{ν′_{n}}}) and l̂_{ν_{n}} is needed (Lemma 4.2).

1 In order to integrate, we assume in this section that all curves are piecewise smooth.

We continue using the notations of Section 3 and assume that B_{α} projects onto a collar about the geodesic l in U/Γ. Let v: I_{α} → I_{α+2π} be an allowable map. In order to get an estimate for Λ(ℱ_{l_{να,v}}) we will construct a non-negative metric on U/Γ^{μ_{α,v}}. For this, we need to establish complex local coordinates on U/Γ^{μ_{α,v}}.

Let G = {(z, f_{α,v}(z)) : z ∈ U} ⊂ U × Ĉ be the graph of f_{α,v}: U → Ĉ, and let p_{2}: G → Ĉ be the projection on the second factor. Since p_{2} is a local homeomorphism, we use it to obtain complex local coordinates on G.

The group Γ acts discontinuously on G as a group of biholomorphic self maps via the action γ(z, f_{α,v}(z)) = (γ(z), γ ∘ f_{α,v}(z)). Thus, G/Γ has a natural complex structure which it inherits from G. The map f̂: U/Γ → G/Γ, given by f̂([z]) = [(z, f_{α,v}(z))], lifts to a quasiconformal homeomorphism f: U → G, with Beltrami coefficient μ_{α,v}; hence, G/Γ is conformally equivalent to U/Γ^{μ_{α,v}} via the conformal map w^{μ_{α,v}} ∘ f̂^{−1} . (Here w^{μ_{α,v}} is the homeomorphism from U/Γ onto U/Γ^{μ_{α,v}} induced by w^{μ_{α,v}}.)

Let θ_{0} = v^{−1}(\frac{1}{2}π) and θ_{1} = v^{−1}(\frac{5}{2}π), and define

\begin{aligned}B′_{α} & = \{re^{iθ} ∈ B_{α} : θ ∈ [θ_{0}, θ_{1}]\}, \\ D′_{α} & = \{z ∈ D_{α} : z = γ(w), w ∈ B′_{α}, γ ∈ Γ\}. \end{aligned}

Set

S = {[(re^{iθ}, re^{iv(θ)})] ∈ G/Γ : re^{iθ} ∈ B′_{α}} ⊂ G/Γ.

One sees easily that (U/Γ) − l is conformally equivalent to (G/Γ) − S. Indeed, the map g: (U/Γ) − l ⟶ (G/Γ) − S, given by

g([re^{iθ}]) = [(re^{iθ}, re^{iθ})],  for re^{iθ} ∈ U − D_{α};
g([re^{iθ}]) = [(re^{iv^{−1}(θ)}, re^{iθ})],  for re^{iθ} ∈ B_{α}, with θ < \frac{1}{2}π;
g([re^{iθ}]) = [(re^{iv^{−1}(θ+2π)}, re^{i(θ+2π)})],  for re^{iθ} ∈ B_{α}, with θ > \frac{1}{2}π,

is a conformal homeomorphism from (U/Γ) − l onto (G/Γ) − S.

We will now construct a non-negative, continuous metric on G/Γ, and use it to obtain a lower bound on Λ(ℱ_{s}), where s is the unique geodesic in G/Γ homotopic to f̂(l) = [(re^{iπ/2}, re^{iv(π/2)})] ⊂ G/Γ.

We consider the Poincaré metric on U/Γ. When restricted to (G/Γ)−S, this metric is given locally by

ρ(re^{iϑ})|dre^{iϑ}| = \frac{\sqrt{dr^{2} + r^{2}dϑ^{2}}}{r sin ϑ} ,

for the local coordinates re^{iϑ} = f_{α,v}(se^{iθ}) on [(se^{iθ}, f_{α,v}(se^{iθ})] ∈ (G/Γ) − S, se^{iθ} ∈ U − D′_{α}.

We extend ρ(re^{iϑ})|dre^{iϑ}| to a metric ρ(re^{iϑ})|dre^{iϑ}| on all of G/Γ, defining

ρ(re^{iϑ})|dre^{iϑ}| = \frac{\sqrt{dr^{2} + r^{2}dϑ^{2}}}{r} ,

for the local coordinates re^{iϑ} = re^{iv(θ)} on [(re^{iθ}, re^{iv(θ)})] ∈ S, re^{iθ} ∈ B′_{α}. Note that ρ > 0 yields a continuous metric on G/Γ.

For β(t), 0 ≤ t ≤ 1, a simple, closed curve in ℱ_{s}, define the simple, closed curve on U/Γ, β̂(t) = f̂^{−1} ∘ β(t). Denote the Poincaré metric on U/Γ by ρ_{1}(se^{iθ})|dse^{iθ}|, and observe that, since β̂ is homotopic to l,

∫β̂ ρ_{1}(se^{iθ}) |dse^{iθ}| ≥ l̂.

Define c_{1} = sin(\frac{1}{2}π + α) < 1. Now, ∫_{β} ρ(re^{iϑ}) |dre^{iϑ}| = ∫_{0}^{1} R(t) dt, and ∫β̂ ρ_{1}(se^{iθ}) |dse^{iθ}| = ∫_{0}^{1} S(t) dt,

where R(t) and S(t) are defined as follows:

For t ∈ [0, 1], with

β(t) = [(r(t)e^{iv^{−1}(ϑ(t))}, r(t)e^{iϑ(t)})] ∈ S,

r(t)e^{iv^{−1}(ϑ(t))} ∈ B′_{α}, we have β̂(t) = [r(t)e^{iv^{−1}(ϑ(t))}], and

R(t) = \frac{\sqrt{r′(t)^{2} + r(t)^{2}ϑ′(t)^{2}}}{r(t)} ≥ c_{1} \frac{\sqrt{r′(t)^{2} + r(t)^{2}(v^{−1′}(ϑ(t))^{2}ϑ′(t))^{2}}}{r(t) sin v^{−1}(ϑ(t))} = c_{1}S(t)

(recall that v′ ≥ 1, and c_{1} < sin v^{−1}(ϑ(t)).

For t ∈ [0, 1], with

β(t) = [(r(t)e^{iv^{−1}(ϑ(t))}, r(t)e^{iϑ(t)})] ∈ (G/Γ) − S,

and r(t)e^{iv^{−1}(ϑ(t))} ∈ B_{α} − B′_{α}, we have β̂(t) = [r(t)e^{iv^{−1}(ϑ(t))}], and

R(t) = \frac{\sqrt{r′(t)^{2} + r(t)^{2}ϑ′(t)^{2}}}{r(t) sin ϑ} ≥ c_{1} \frac{\sqrt{r′(t)^{2} + r(t)^{2}(v^{−1′}(ϑ(t))^{2}ϑ′(t))^{2}}}{r(t) sin v^{−1}(ϑ(t))} = c_{1}S(t)

(again, we have v′ ≥ 1, and c_{1} sin(ϑ(t)) < sin v^{−1}(ϑ(t))).

For t ∈ [0, 1], with

β(t) = [(r(t)e^{iθ(t)}, r(t)e^{iϑ(t)})] ∈ (G/Γ) − S,

r(t)e^{iθ(t)} ∈ U − D_{α}, we have θ(t) = ϑ(t), and β̂(t) = [r(t)e^{iϑ(t)}], so that R(t) = S(t).

Hence, in general, for t ∈ [0, 1], R(t) ≥ c_{1}S(t), and it follows that

(5)  ∫_{β} ρ |dre^{iϑ}| = ∫_{0}^{1} R(t) dt ≥ c_{1} ∫_{0}^{1} S(t) dt = c_{1} ∫β̂ ρ_{1} |dse^{iθ}| ≥ c_{1}l̂.

Since β was an arbitrary curve in Λ(ℱ_{s}), we conclude from (5) that

(6)  L(ρ) ≥ c_{1}l̂.

On the other hand,

A(ρ) = ∫ ∫_{(G/Γ)−S} ρ^{2} rd rdϑ + ∫_{0}^{2π} ∫_{1}^{r_{∗}} \frac{rdrdϑ}{r^{2}}
= 2π(2g − 2) + 2π log r_{∗} = 2π(2g − 2) + 2πl̂.

(Recall that γ(z) = r_{∗}z generates the stabilizer of B_{α} in Γ.)

Hence,

(7)  Λ(ℱ_{s}) ≥ \frac{L(ρ)^{2}}{A(ρ)} = \frac{c_{1}^{2}l̂^{2}}{2π(2g − 2) + 2πl̂} .

We can now prove the following:

Lemma 4.2. For the simple, closed geodesic l in U/Γ and [ν] ∈ T (Γ), we have

(8)  Λ(ℱ_{l_{ν′}}) ≥ \frac{l̂_{ν}^{2}}{2π(2g − 2) + 2πl̂_{ν}} .

Proof. Choose 0 < α < \frac{1}{2}π so that B_{α} projects onto a collar about l_{ν} in U/Γ^{ν} and let v be an allowable map. Define f_{α,v}: U → Ĉ as in Section 3.1 and let G be the graph of f_{α,v}. Define f̂: U/Γ^{ν} → G/Γ^{ν} by f̂([re^{iθ}]) = [(re^{iθ}, f_{α,v}(re^{iθ}))]. Hence, G/Γ^{ν} is conformally equivalent to U/Γ^{ν′} via the map

ĥ = w^{μ_{α,v}} ∘ f̂^{−1}: G/Γ^{ν} → U/Γ^{ν′}.

Let s be the unique geodesic in G/Γ^{ν} homotopic to f̂(l_{ν}). It follows from the inequality (7) that

(9)  Λ(ℱ_{s}) ≥ \frac{c_{1}^{2}l̂_{ν}^{2}}{2π(2g − 2) + 2πl̂_{ν}} .

Clearly, ĥ(ℱ_{s}) = ℱ_{l_{ν′}} , and since ĥ is holomorphic, Λ(ℱ_{s}) = Λ(ℱ_{l_{ν′}}) (see [1]). Hence,

(10)  Λ(ℱ_{l_{ν′}}) ≥ \frac{c_{1}^{2}l̂_{ν}^{2}}{2π(2g − 2) + 2πl̂_{ν}} .

Observe, however, that c_{1} depends on the choice of α, while ℱ_{l_{ν′}} does not (Lemma 3.1). Thus, choosing α arbitrarily small, and noting that c_{1} → 1 as α → 0, we obtain from (10)

(11)  Λ(ℱ_{l_{ν′}}) ≥ \frac{l̂_{ν}^{2}}{2π(2g − 2) + 2πl̂_{ν}} ,

which is the desired result.

Proof of Lemma 3.5. Suppose that there exists a convergent subsequence Ψ([ν_{n}]) = [ν′_{n}] → [ν] ∈ T (Γ).

We can choose a constant K and quasiconformal maps w_{n} of dilatation K_{n} ≤ K (where K_{n} ≥ 1) such that w_{n}(U/Γ^{ν′_{n}}) = U/Γ^{ν}, with w_{n}(l_{ν′_{n}}) homotopic to l_{ν}. Further, we can assume w_{n} is C^{1} on the complement of a finite set (the minimal Teichmüller map has this property [5]), so that w_{n}(ℱ_{l_{ν′_{n}}}) = ℱ_{l_{ν}}. It is shown in [5] (see also [1]) that, in this situation,

(12)  Λ(ℱ_{l_{ν}}) ≥ \frac{1}{K_{n}} Λ(ℱ_{l_{ν′_{n}}}) ≥ \frac{1}{K} Λ(ℱ_{l_{ν′_{n}}}).

Applying Lemma 4.2, we have Λ(ℱ_{l_{ν′_{n}}}) ≥ l̂_{ν_{n}}^{2}/(2π(2g − 2) + 2πl̂_{ν_{n}}), and,

since l̂_{ν_{n}} → ∞, we deduce Λ(ℱ_{l_{ν′_{n}}}) → ∞.

From the inequality (12), we we now obtain Λ(ℱ_{l_{ν}}) = ∞; this contradicts Lemma 4.1, and we conclude that

d_{Γ}([0], Ψ([ν_{n}])) = d_{Γ}([0], [ν′_{n}]) → ∞.

Remarks. It is clear from the construction of the grafting map that allowable maps of the form v: I_{α} → I_{α+2πn} = [\frac{1}{2}π − α, \frac{1}{2}π + 2πn + α], n ∈ Z^{+} , could have been used. Thus, for every such n, there is a map Ψ_{(l,n)}: T (Γ) → T (Γ). The proof of the corresponding version of Theorem 2 remains valid, and Ψ_{(l,n)} is a homeomorphism onto T (Γ). Similarly, the restriction to one simple, closed geodesic was unnecessary; any finite collection l_{1}, …, l_{m} of simple, closed, disjoint geodesics could have been used.

14 Daniel M. Gallo

References

[1] Ahlfors, L.V.: Lectures on Quasiconformal Mappings. - Van Nostrand, Princeton, NJ, 1966.

[2] Buser, P.: Geometry and Spectra of Compact Riemann Surfaces. - Birkhäuser, Boston, MA, 1992.

[3] Earle, C.J.: On variation of projective structures. - In: Riemann Surfaces and Related Topics. - Ann. of Math. Stud. 97, Princeton University Press, Princeton, N.J., 1981, 87–99.

[4] Faltings, G.: Real projective structures on compact surfaces. - Compositio Math. 48, 1983, 223–269.

[5] Gardiner, F.P.: Teichmüller Theory and Quadratic Differentials. - John Wiley & Sons, New York, N.Y., 1987.

[6] Goldman, W.M.: Fuchsian structures with Fuchsian holonomy. - J. Differential Geom. 25, 1987, 297–326.

[7] Maskit, B.: On a class of Kleinian groups. - Ann. Acad. Sci. Fenn. Ser. A I Math. 442, 1969, 1–8.

Received 22 November 1994