On continuous extension of grafting maps

Kentaro Ito

November 6, 2004

Abstract The definition of the grafting operation for quasifuchsian groups is extended by Bromberg [Br] to all b-groups. Although the grafting maps are not necessarily continuous at boundary groups, in this paper, we show that the grafting maps take every “standard” convergent sequence to a convergent sequence. As a consequence of this result, we extend Goldman’s grafting theorem for quasifuchsian groups to all boundary b-groups.

1 Introduction

In this paper we are interested in the behavior of the holonomy map from projective structures to representations. Especially, we study the continuity and the noncontinuity of the local inverse of the holonomy map at the boundary of the space of discrete faithful representations.

Let S be an oriented closed surface of genus g > 1. A projective structure on S is a (G, X)-structure where X is a Riemann sphere ℂ̂ and G = PSL_{2}(ℂ) is the group of projective automorphism of ℂ̂. Let P (S) denote the space of projective structures on S and R(S) the space of conjugacy classes of representations ρ : π_{1}(S) → PSL_{2}(ℂ). The holonomy map hol : P (S) → R(S) takes a projective structure to its holonomy representation.

We are interested in the quasifuchsian space QF(S) ⊂ R(S) of faithful representations with quasifuchsian images. Although QF = QF(S) is a connected, contractible, open submanifold of R(S), the topology of its closure \overline{QF} in R(S) is very complicated; for instance McMullen [Mc] showed that QF self-bumps, i.e., there exists ρ ∈ ∂QF such that U ∩QF is disconnected for any sufficiently small neighborhood U of ρ, by using projective structures and ideas of Anderson and Canary [AC]. In fact, since the holonomy map hol : P (S) → R(S) is a local homeomorphism, some kind of complexity of the topology of QF ⊂ R(S) can be observed by studying the inverse image Q(S) = hol^{−1}(QF) of QF in P (S). By Goldman’s grafting theorem [Go], the set of components of Q(S) is in one-to-one correspondence with the set ML_{ℕ} of integral points of measured laminations. For every λ ∈ ML_{ℕ}, the associated component Q_{λ} of Q(S) is taken by the holonomy map biholomorphically onto QF. We let Ψ_{λ} : QF → Q_{λ} denote the univalent local branch of hol^{−1} and call it the grafting map for λ ∈ ML_{ℕ}. The standard component Q_{0} is the component of Q(S) of projective structures with injective developing map. Then McMullen [Mc] actually showed that there exists a sequence in Q(S)−Q_{0} converging to ∂Q_{0}, which implies that QF self-bumps. Moreover, we showed in [It1, It3] that for any n ∈ ℕ, there exists n-components of Q(S) which bump simultaneously (i.e., having intersecting closures), that any component of Q(S) except for Q_{0} self-bumps, and that any two components of Q(S) bump. All these phenomena are obtained by using exotic convergent sequence in QF (see definition below) constructed by Anderson and Canary [AC].

Let B : T (S)×T (S) → QF be the parameterization of QF of Bers’ simultaneous uniformization and let ρ_{n} = B(X_{n}, Y_{n}) ∈ QF be a sequence converging to ρ ∈ ∂QF. Then we say that the convergence ρ_{n} → ρ_{∞} is standard if there exists a compact subset K of T (S) which contains all X_{n} or all Y_{n}; otherwise it is exotic. We let ∂^{±}QF ⊂ ∂QF denote the set of limits of standard convergence sequences, or the set of all boundary b-groups (see §2.1). Then by using Bromberg’s observation in [Br], the grafting map Ψ_{λ} is naturally extended to Ψ_{λ} : QF ⊔ ∂^{±}QF → P̂(S) for every λ ∈ ML_{ℕ}. Here P̂(S) = P (S) ∪ {∞} denotes the one-point compactification of P (S). As described above, Anderson and Canary [AC] showed that there exist sequences ρ_{n} ∈ QF converging exotically to ρ_{∞} ∈ ∂^{±}QF, which course the noncontinuity of the map Ψ_{λ} : QF → P (S) at ρ_{∞} ∈ ∂^{±}QF for all λ ∈ ML_{ℕ}.

On the contrary, we study the continuity of the grafting maps in this paper. The following theorem (Theorem 5.1) is the main result of the paper, which states that each grafting map Ψ_{λ} behaves as a continuous map for every standard convergent sequence.

Theorem 1.1. Let ρ_{n} ∈ QF be a sequence converging standardly to ρ_{∞} ∈ ∂^{±}QF. Then the sequence Ψ_{λ}(ρ_{n}) converges to Ψ_{λ}(ρ_{∞}) in P̂(S) for every λ ∈ ML_{ℕ}.

Let \overline{Q_{0}} denote the closure of Q_{0} in P (S), not in P̂(S). As we see in §3.3, a sequence Σ_{n} ∈ Q_{0} converges to Σ_{∞} ∈ ∂Q_{0} if and only if the sequence hol(Σ_{n}) ∈ QF converges standardly to hol(Σ_{∞}) ∈ ∂^{+}QF. Hence we have the next corollary (Theorem 6.10) of Theorem 1.1.

Corollary 1.2. The map Gr_{λ} := Ψ_{λ} ∘ Ψ_{0}^{−1} : Q_{0} → Q_{λ} is extended continuously to Gr_{λ} : \overline{Q_{0}} → P̂(S) for each λ ∈ ML_{ℕ}, where \overline{Q_{0}} is the closure of Q_{0} in P (S).

An important remark is that, since we do not know whether Q_{0} self-bumps or not, the statement of Theorem 1.1 (or Corollary 1.2) is not trivial and worth considering.

The following theorem plays an important roll in the proof of Theorem 1.1, which have some interest in its own light (see §4). A Bers slice B is a slice of QF of the type {B(X, Y)}_{Y ∈T (S)} or {B(X, Y)}_{X∈T (S)}. In addition, we let π : P (S) → T (S) denote the projection which takes a projective structure to its underling conformal structure.

Theorem 1.3. Let B be a Bers slice and {λ_{n}} a sequence of distinct elements of ML_{ℕ}. Then the sequence {π ∘ Ψ_{λ_{n}}(B)} eventually escapes any compact subset K of T (S); that is, π ∘ Ψ_{λ_{n}}(B) ∩ K = ∅ for all large enough n.

As a consequence of Theorem 1.1, we extend Goldman’s grafting theorem as follows (see Theorem 6.1):

Theorem 1.4. For every boundary b-groups ρ ∈ ∂^{±}QF, all projective structures with holonomy ρ are obtained by grafting of ρ; that is, hol^{−1}(ρ) = {Ψ_{λ}(ρ) : λ ∈ ML_{ℕ}, Ψ_{λ}(ρ) ≠ ∞}.

As mentioned above, any two components of Q(S) = hol^{−1}(QF) bump. On the contrary, we obtain the following theorem (Theorem 6.11), which reveals that only exotic convergent sequences cause the bumping of distinct components of Q(S).

Theorem 1.5. Set QF_{K} = {B(X, Y) ∈ QF : X ∈ K or Y ∈ K} for a compact subset K ⊂ T (S). Then the inverse image hol^{−1}(QF_{K}) of QF_{K} in P (S) is discrete; that is, every connected component Ψ_{λ}(QF_{K}) of hol^{−1}(QF_{K}) has an open neighborhood in P (S) which is disjoint from any other one.

Remark. It is conjectured by Bers, Sullivan and Thurston that the closure of QF is equal to the space AH(S) ⊂ R(S) of discrete, faithful representations. This conjecture is closely related to Thurston’s ending lamination conjecture, which was recently announced by Minsky to be solved affirmatively. But we do not make use of these deep results in this paper.

Acknowledgements. The author would like to thank Katsuhiko Matsuzaki for useful discussions on the topics of §4.

2 Preliminaries

2.1 Quasifuchsian space

A Kleinian group Γ is a discrete subgroup of PSL_{2}(ℂ), which acts on hyperbolic 3space ℍ^{3} as isometries and on the sphere at infinity ℂ̂ as conformal automorphisms. The union \overline{ℍ^{3}} = ℍ^{3} ∪ ℂ̂ is naturally topologized as a closed 3-ball so that PSL_{2}(ℂ) acts continuously on it. For a Kleinian group Γ, we let Ω_{Γ} denote the region of discontinuity and Λ_{Γ} the limit set. We associate to a Kleinian group Γ the following orbit spaces:

M_{Γ} = ℍ^{3}/Γ, \overline{M_{Γ}} = (ℍ^{3} ∪ Ω_{Γ})/Γ, ∂M_{Γ} = Ω_{Γ}/Γ,

where ∂M_{Γ} is called the conformal boundary of M_{Γ}. In general if \overline{M} is an oriented maniford with boundary ∂M, we orient ∂M by requiring that the frame (f, n) has positive orientation whenever f is a positively oriented frame on ∂M and n is an inward-pointing vector.

Let S be an oriented closed surface of genus g > 1. Let R(S) be the space of conjugacy classes [ρ] of representations ρ : π_{1}(S) → PSL_{2}(ℂ) whose images ρ(π_{1}(S)) are non-abelian. The space R(S) is equipped with the algebraic topology, the topology of convergence on generators up to conjugation. (By abuse of notation, we also denote [ρ] by ρ if there is no confusion.) It is known that R(S) is a (6g − 6)- dimensional complex manifold (see Theorem 4.21 in [MT]).

Let AH(S) be the subset of R(S) of discrete, faithful representations, which is known to be a closed subset of R(S) by Jørgensen [Jo]. Let ρ_{n} → ρ_{∞} be an algebraic convergence sequence in AH(S). Then it is known by Jørgensen and Marden [JM] that, by passing to a subsequence if necessary, the sequence Γ_{n} = ρ_{n}(π_{1}(S)) of Kleinian groups converges geometrically to some Kleinian group Γ̂, which contains the algebraic limit Γ_{∞} = ρ_{∞}(π_{1}(S)). Here the convergence Γ_{n} → Γ̂ is geometric if and only if for any γ̂ ∈ Γ̂, there exist γ_{n} ∈ Γ_{n} such that γ_{n} → γ̂ and if for every convergent sequence γ_{n_{j}} ∈ Γ_{n_{j}} (n_{j} → ∞), the limit is contained in Γ̂. A convergence ρ_{n} → ρ_{∞} in AH(S) is said to be strong if Γ_{n} = ρ_{n}(π_{1}(S)) converges geometrically to the algebraic limit Γ_{∞} = ρ_{∞}(π_{1}(S)).

For every ρ ∈ AH(S) with image Γ = ρ(π_{1}(S)), Bonahon’s theorem [Bo] guarantees that there exists an orientation preserving homeomorphism ψ : S × (−1, 1) → M_{ρ} = M_{Γ}, where (−1, 1) is an open interval. The conformal boundary ∂M_{ρ} (possibly empty) of M_{ρ} decomposes into two parts ∂^{+}M_{ρ} ⊔ ∂^{−}M_{ρ}, where ∂^{+}M_{ρ} (resp. ∂^{−}M_{ρ}) is the limit of ψ(S × \{t\}) in \overline{M_{ρ}} as t → −1 (resp. t → 1). Associated to this decomposition, Ω_{Γ} decomposes into Ω_{Γ}^{+} ⊔ Ω_{Γ}^{−} for which Ω_{Γ}^{+}/Γ = ∂^{+}M_{ρ} and Ω_{Γ}^{−}/Γ = ∂^{−}M_{ρ}.

Let Γ = ρ(π_{1}(S)) be the image of some ρ ∈ AH(S). Then Γ is called a b-group if exactly one of Ω_{Γ}^{+} or Ω_{Γ}^{−} is simply connected and Γ-invariant. For a b-group ρ, we denote by para(ρ) its parabolic locus, a collection of homotopy classes of disjoint simple closed curves c on S such that ρ(c) ∈ Γ is parabolic. Similarly Γ is called a quasifuchsian group if both Ω_{Γ}^{+} and Ω_{Γ}^{−} are simply connected and Γ-invariant. Quasifuchsian space QF = QF(S) is the subset of R(S) of faithful representations with quasifuchsian images. It is known by Marden [Mar] and Sullivan [Su] that QF equals the interior of AH(S). Hence QF is a (6g −6)-dimensional complex manifold in R(S). On the other hand, it is trivial that \overline{QF} ⊂ AH(S) and is conjectured that \overline{QF} = AH(S), which is so called the Bers-Sullivan-Thurston conjecture. We let ∂QF denote the relative boundary of QF in R(S), whose element is called a boundary group.

Now let ρ ∈ QF. Then there exists an orientation preserving homeomorphism ψ : S × [−1, 1] → \overline{M_{ρ}} which induces the representation ψ_{∗} = ρ : π_{1}(S) → PSL_{2}(ℂ). Moreover, it provides orientation preserving homeomorphisms ψ|_{S×{−1}} : S → ∂^{+}M_{ρ} and ψ|_{S×{1}} : S̄ → ∂^{−}M_{ρ}, where S̄ denotes S with its orientation reversed. Hence ρ ∈ QF determines a pair of marked Reimann surfaces (∂^{+}M_{ρ}, ∂^{−}M_{ρ}) ∈ T (S)×T (S̄) in the product of the Teichmüller spaces. On the other hand, Bers [Be] showed that each pair (X, Ȳ) ∈ T (S) × T (S̄) has the unique simultaneous uniformization ρ = B(X, Ȳ) ∈ QF. Therefore the map

B : T (S) × T (S̄) → QF

gives us a global parameterization of QF. We define vertical and horizontal Bers slices in QF by B_{X} = {B(X, Ȳ) : Ȳ ∈ T (S̄)} and BȲ = {B(X, Ȳ) : X ∈ T (S)}, respectively. It is known by Bers that both B_{X} and BȲ are precompact in R(S), whose frontiers are denoted by ∂B_{X} and ∂BȲ. A representation ρ ∈ AH(S) is called a Bers boundary group if it is contained in the frontier of some Bers slice.

2.2 Sequences in quasifuchsian space

We introduce the notion of “standard” and “exotic” convergence for a sequence ρ_{n} ∈ QF tending to a limit ρ_{∞} ∈ ∂QF. One of our main purpose of this paper is to show that every grafting map behaves as a continuous map for every standard convergent sequence (see Theorem 5.1). For a given subset K of T (S), we set K̄ = {X̄ ∈ T (S̄) : X ∈ K} ⊂ T (S̄), where X̄ ∈ T (S̄) denote the complex conjugation of X ∈ T (S).

Definition 2.1 (Standard and exotic convergence). Suppose that a sequence ρ_{n} = B(X_{n}, Ȳ_{n}) in QF converges to ρ_{∞} ∈ ∂QF. Then the sequence ρ_{n} is said to converge standardly to ρ_{∞} if there exists a compact subset K ⊂ T (S) such that (i) X_{n} ∈ K for all n, or (ii) Ȳ_{n} ∈ K̄ for all n. Otherwise, we say that ρ_{n} converges exotically to ρ_{∞}.

We let ∂^{+}QF and ∂^{−}QF denote the subsets of ∂QF consisting of limits of standard convergent sequences of type (i) and (ii), respectively, and set ∂^{±}QF = ∂^{+}QF ⊔ ∂^{−}QF. It is easily seen that every Bers boundary group is an element of ∂^{±}QF and that every element in ∂^{±}QF is a boundary b-group, a b-group in ∂QF. On the contrary, it follows from the arguments in the paper of Brock, Bromberg, Evans and Souto [BBES] that every boundary b-group is a Bers boundary group (see Theorem 2.3 below). Hence we see that the set ∂^{±}QF equals the set of all boundary b-groups and that the following hold:

∂^{+}QF = ⨆_{X∈T (S)} ∂B_{X}, ∂^{−}QF = ⨆_{Ȳ ∈T (S̄)} ∂BȲ .

Moreover, Bromberg [Br] obtained the following result by developing the grafting operation for elements in ∂^{−}QF (see §3) and by combining with Minsky’s result [Mi].

Theorem 2.2 (Bromberg). Every b-group ρ ∈ AH(S) without parabolics is a Bers boundary group.

We now outline the proof of Theorem 2.3 by following the arguments in [BBES]. In the argument, we also obtain Corollary 2.4 below, which is required in §5.

Theorem 2.3 (Brock-Bromberg-Evans-Souto). Every boundary b-group is a Bers boundary group.

Corollary 2.4. Let ρ ∈ ∂^{±}QF with parabolic locus para(ρ). If ρ ∈ ∂^{+}QF (resp. ρ ∈ ∂^{−}QF), there exists a sequence ρ_{n} = B(X_{n}, Ȳ_{n}) ∈ QF which converges standardly to ρ and which satisfies lȲ_{n}(para(ρ)) → 0 (resp. l_{X_{n}}(para(ρ)) → 0) as n → ∞. Here lȲ_{n}(para(ρ)) denotes the total sum of hyperbolic lengths of components of para(ρ) on Ȳ_{n}.

Outline of the proof of Theorem 2.3. Let ρ ∈ ∂QF with para(ρ), which is possibly empty. We may assume that the positive part ∂_{c}^{+}M_{ρ} of the conformal boundary of M_{ρ} is homeomorphic to S, and hence determine a point X ∈ T (S). By Theorem 3.1 in [BBES], the drilling theorem plays an important role in its proof, there exists a strong convergent sequence ρ_{n} → ρ in \overline{QF} for which each ρ_{n} is a geometrically finite representation whose parabolic locus para(ρ_{n}) equals para(ρ). Since the convergence ρ_{n} → ρ is strong, X_{n} = ∂_{c}^{+}M_{ρ_{n}} converges to X in T (S). Moreover, for each n, there exists a strong convergent sequence ρ_{n,k} → ρ_{n} (k → ∞) of quasifuchsian representations all of whose positive parts of the conformal boundaries equal X_{n} ∈ T (S) (see Theorem 3.4 in [BBES]). Therefore ρ_{n,k} = B(X_{n}, Ȳ_{n,k}) is a sequence in the Bers slice B_{X_{n}} for each n. Since the convergence ρ_{n,k} → ρ_{n} is strong and since ρ_{n} is geometrically finite, the hyperbolic lengths lȲ_{n,k}(para(ρ)) of para(ρ) on Ȳ_{n,k} tend to 0 as k → ∞. By a diagonal argument, we can choose a sequence ρ′_{n} = B(X_{n}, Ȳ_{n}) from {ρ_{n,k}}_{n,k∈ℕ} so that ρ′_{n} → ρ and that lȲ_{n}(para(ρ)) → 0 as n → ∞, which satisfies the desired property in Corollary 2.4. Now let us consider new sequence ρ′′_{n} = B(X, Ȳ_{n}) in B_{X}. Then this sequence ρ′′_{n} ∈ B_{X} also converges to ρ because maximal dilatations of quasiconformal automorphism of ℂ̂ conjugating ρ′_{n} to ρ′′_{n} tend to 1 as n → ∞. This implies that ρ ∈ ∂B_{X}.

We remark that the set ∂QF − ∂^{±}QF is not empty; for instance, it contains limits of sequences which appear in Thurston’s double limit theorem. On the other hand, Anderson and Canary [AC] showed that there exists a sequence in QF which converges exotically to some point in ∂^{±}QF. All the known such sequences are basically obtained by their technique and here is a typical example: let c be a simple closed curve on S and let τ = τ_{c} be the Dehn twist along c. Then for a fixed pair (X, Ȳ) ∈ T (S) × T (S̄), the sequence

ρ_{n} = B(τ^{n} X, τ^{2n} Ȳ) (n ∈ ℤ)

in QF converges exotically to some ρ_{∞} ∈ ∂^{+}QF as |n| → ∞. We will make use of this sequence in §3.3 to explain how the grafting maps fail to be extended to a continuous map.

2.3 Space of projective structures

We only give a brief summary of projective structures and refer to [It1] and elsewhere for more details.

A projective structure on S is a (G, X)-structure where X is a Riemann sphere ℂ̂ and G = PSL_{2}(ℂ) is the group of projective automorphism of ℂ̂. Let P (S) be the space of marked projective structures on S. A projective structure Σ ∈ P (S) determine the underlying conformal structure π(Σ) ∈ T (S). It is known that P (S) is the holomorphic affine bundle over T (S) with the projection π : P (S) → T (S), and that P (S) is a (6g − 6)-dimensional complex manifold.

A projective structure Σ on S can be lifted to that Σ̃ on S̃, where S̃ → S is the universal cover on which π_{1}(S) acts as a covering group. Since Σ̃ is simply connected, we obtain a developing map f_{Σ} : S̃ → ℂ̂ by continuing charts of Σ̃ analytically, which induce a holonomy representation ρ_{Σ} : π_{1}(S) ≅ π_{1}(Σ) → PSL_{2}(ℂ) satisfying f_{Σ} ∘ γ = ρ_{Σ}(γ) ∘ f_{Σ} for every γ ∈ π_{1}(S). We remark that the pair (f_{Σ}, ρ_{Σ}) is determined uniquely up to PSL_{2}(ℂ). We now define the holonomy map

hol : P (S) → R(S)

by Σ ↦ [ρ_{Σ}]. Hejhal [He] showed that the map hol is a local homeomorphism and Earle [Ea] and Hubbard [Hu] independently showed that the map is holomorphic.

Theorem 2.5 (Hejhal, Earle and Hubbard). The holonomy map hol : P (S) → R(S) is a holomorphic local homeomorphism.

We denote by Q(S) = hol^{−1}(QF) the set of projective structures with quasifuchsian holonomy. An element of Q(S) is said to be standard if its developing map is injective; otherwise it is exotic. We denote by Q_{0} ⊂ Q(S) the subset of standard projective structures. Let ρ = B(X, Ȳ) ∈ QF with image Γ = ρ(π_{1}(S)). Then the quotient surface Σ = Ω_{Γ}^{+}/Γ can be regarded as a standard projective structure on S with bijective developing map f_{Σ} : S̃ → Ω_{Γ}^{+}, with holonomy representation ρ_{Σ} = ρ, and with underlying conformal structure X ∈ T (S). Let

Ψ_{0} : QF → Q_{0}

be the map defined by the correspondence ρ ↦ Ω_{Γ}^{+}/Γ as described above. Then the map Ψ_{0} turns out to be a univalent local branch of hol^{−1} onto the connected component Q_{0} of Q(S), which is called the standard component. It is known by Bers that every Bers slice B_{X} ⊂ QF is embedded by the map Ψ_{0} into a bounded domain Ψ_{0}(B_{X}) of a fiber π^{−1}(X) ⊂ P (S). Observe that hol|_{Q_{0}} : \overline{Q_{0}} → QF ⊔ ∂^{+}QF is bijective, where \overline{Q_{0}} is the closure of Q_{0} in P (S).

3 Grafting

3.1 Grafting maps on quasifuchsian space

We let ML_{ℕ} = ML_{ℕ}(S) denote the set of integral points of measured laminations on S. In other words, each element of λ ∈ ML_{ℕ} is a isotopy class of disjoint union ⊔_{i=1}^{l}k_{i}c_{i} of homotopically distinct simple closed curves c_{i} on S with positive integer k_{i} weights. We do not distinguish the isotopy class λ and its representative if there is no confusion. The “zero-lamination” 0 is also contained in ML_{ℕ}. In what follows, the parabolic locus para(ρ) of a b-group ρ is also regarded as an element of ML_{ℕ}.

For each non-zero λ ∈ ML_{ℕ}, we will explain haw to obtain the grafting map

Gr_{λ} : Q_{0} → P (S),

which satisfies hol ∘ Gr_{λ} ≡ hol on Q_{0}. We give here two equivalent definitions of grafting operation; the first one is as usual, and the second one is introduced by Bromberg in [Br] so that it also makes sense for elements of ∂^{−}QF. For a while, we assume that λ is a simple closed curve c of weight one for simplicity. In addition, we fix our notation as follows: for a given ρ ∈ QF, let Γ be the quasifuchsian image of ρ and Ω_{Γ} = Ω_{Γ}^{+} ⊔ Ω_{Γ}^{−} the region of discontinuity of Γ. We regard quotient surfaces Σ = Ω_{Γ}^{+}/Γ = Ψ_{0}(ρ_{0}) and Σ^{−} = Ω_{Γ}^{−}/Γ as projective structures on S and S̄, respectively. Let c^{+} ⊂ Σ and c^{−} ⊂ Σ^{−} be simple closed curves corresponding to c ⊂ S and suppose that γ ∈ Γ ≅ π_{1}(S) is a representative of the homotopy class of c. Let c̃^{+} ⊂ Ω_{Γ}^{+} and c̃^{−} ⊂ Ω_{Γ}^{−} be the ⟨γ⟩-invariant lifts of c^{+} ⊂ Σ and c^{−} ⊂ Σ^{−}, respectively.

Notational remark. Let A, B be open subsets of a topological space X such that A ∩ B = ∅ and that \overline{A} ∩ \overline{B} ≠ ∅. Then A ⨿ B denotes the interior of \overline{A ⊔ B}.

Definition 3.1 (Grafting). In the situation as described above, the grafting Gr_{λ}(Σ) of the standard projective structure Σ = Ψ_{0}(ρ) along c is a projective structure obtained by the following (equivalent) procedures:

I let A_{c} be a cylinder (ℂ̂ − c̃^{+})/⟨γ⟩ equipped with a projective structure induced from that of ℂ̂. We obtain Gr_{c}(Σ) by cutting Σ along c and inserting A_{c} at the cut locus without twisting; that is,

Gr_{c}(Σ) = (Σ − c) ⨿ A_{c}.

II Here we further assume that c separates S into two surfaces S_{1} and S_{2} with boundaries. (The non-separating case is described precisely in [Br].) Accordingly, Σ and Σ^{−} decompose into Σ − c^{+} = Σ_{1} ⊔ Σ_{2} and Σ^{−} − c^{−} = Σ_{1}^{−} ⊔ Σ_{2}^{−}, respectively. Let i denotes 1 or 2. Let Δ_{i} be the connected component of the inverse image of Σ_{i}^{−} ⊂ Σ^{−} in Ω_{Γ}^{−} which contains c̃^{−} in its closure \overline{Δ}_{i} ⊂ ℂ̂. Then the stabilizer subgroup Γ_{i} = Stab_{Γ}(Δ_{i}) of Γ ≅ π_{1}(S) is identified with π_{1}(S_{i}). Since Γ_{i} is a purely loxodromic free group with non-empty region of discontinuity, Maskit’s result [Mas] implies that Γ_{i} is a Schottky group. Note that the conformal boundary ∂M_{Γ_{i}} = Ω_{Γ_{i}}/Γ_{i} of M_{Γ_{i}} with natural projective structure containing both projective surfaces Σ_{i} and Σ_{i}^{−}. Then Gr_{c}(Σ) is obtained from projective surfaces Ω_{Γ_{1}}/Γ_{1} − Σ_{1}^{−} and Ω_{Γ_{2}}/Γ_{2} − Σ_{2}^{−} by gluing their boundaries without twisting (see Figure 1); that is,

Gr_{c}(Σ) = (Ω_{Γ_{1}}/Γ_{1} − Σ_{1}^{−}) ⨿ (Ω_{Γ_{2}}/Γ_{2} − Σ_{2}^{−}).

PSfrag replacements Σ_{1} Σ_{1}^{−} Σ_{2} Σ_{2}^{−}

Gr_{c}(Σ) M_{Γ_{1}} M_{Γ_{2}}

Figure 1: The grafting Gr_{c}(Σ) of Σ along c.

Observe that the Definitions I and II are equivalent. The grafting Gr_{λ}(Σ) of Σ along general λ = ⊔k_{i}c_{i} ∈ ML_{ℕ} is similarly obtained; for instance, Gr_{kc}(Σ) is obtained by inserting k-copies of A_{c} in Definition I. An important fact is that the grafting operation does not change the holonomy representation, that is, hol(Gr_{λ}(Σ)) = hol(Σ) is always satisfied.

Since the grafting map Gr_{λ} : Q_{0} → P (S) satisfies hol ∘ Gr_{λ} ≡ hol on Q_{0} and since hol|_{Q_{0}} : Q_{0} → QF is a biholomorphic map with its inverse Ψ_{0}, the map Gr_{λ} takes Q_{0} biholomorphically onto the image Gr_{λ}(Q_{0}), which is denoted by Q_{λ}. Hence we obtain a univalent local branch

Ψ_{λ} : QF → Q_{λ}

of hol^{−1} which is defined by Ψ_{λ} = Gr_{λ} ∘ Ψ_{0} and is also called the grafting map for λ (see the commutative diagram below). By Goldman’s grafting theorem [Go] below, PSfrag replacements

P (S) ⊃ Q_{0} R(S) ⊃ QF

Q_{λ} hol Ψ_{0} Ψ_{λ}

Gr_{λ}

we obtain the decomposition ⨆_{λ∈ML_{ℕ}} Q_{λ} of Q(S) into its connected components.

Theorem 3.2 (Goldmann [Go]). For every ρ ∈ QF, all projective structures with holonomy ρ are obtained by grafting of ρ; that is, we have hol^{−1}(ρ) = {Ψ_{λ}(ρ) : λ ∈ ML_{ℕ}}.

3.2 Extension of grafting maps

Let P̂(S) denotes the one-point compactification P (S) ∪ {∞} of P (S). We now extend the grafting map Ψ_{λ} : QF → Q_{λ} to Ψ : QF ⊔ ∂^{±}QF → P̂(S). (In fact Ψ_{λ}(ρ) is defined for all b-groups ρ ∈ AH(S) by the same manner.) Observe that Definition I works well even for ρ ∈ ∂^{+}QF whenever γ is loxodromic, because there still exists a ⟨γ⟩-invariant simple arc c̃^{+} in non-degenerate component Ω_{Γ}^{+}, for which (ℂ̂ − c̃^{+})/⟨γ⟩ is still an annulus. On the other hand, Definition II works well for ρ ∈ ∂^{−}QF whenever every connected component of para(ρ) intersects c essentially. In fact, on this assumption, Γ_{1} and Γ_{2} in Definition II are still Schottky groups, γ is still loxodromic, and there still exists a ⟨γ⟩-invariant simple arc c̃^{−} in non-degenerate component Ω_{Γ}^{−}. For general λ ∈ ML_{ℕ}, we can also define the grafting Ψ_{λ}(ρ) ∈ P (S) of ρ along λ if the pair (λ, ρ) satisfies the following condition:

Definition 3.3 (Admissible). The pair (λ, ρ) of λ ∈ ML_{ℕ} and ρ ∈ ∂^{±}QF is said to be admissible if

• ρ ∈ ∂^{+}QF and para(ρ) and λ have no parallel component in common, or

• ρ ∈ ∂^{−}QF and every component of para(ρ) intersects λ essentially.

If the pair (λ, ρ) is not admissible, we set Ψ_{λ}(ρ) = ∞ ∈ P̂(S). The extended grafting map Ψ_{λ} : QF ⊔ ∂^{±}QF → P̂(S) is also denoted by the same symbol Ψ_{λ}.

3.3 Non-continuity of grafting maps

We collect in Table 1 below the equivalent conditions with standard/exotic convergence of quasifuchsian representations (see [Mc, Appendix A] and [It1, §3]). The situation in which we consider is as follows: let ρ_{n} ∈ QF be a sequence converging to ρ_{∞} ∈ ∂^{+}QF. We may assume that the sequence Γ_{n} = ρ_{n}(π_{1}(S)) converges geometrically to a Kleinian group Γ̂, which contains the algebraic limit Γ_{∞} = ρ_{∞}(π_{1}(S)). Let Σ_{∞} be a unique projective structure in ∂Q_{0} such that hol(Σ_{∞}) = ρ_{∞} and let Φ : U → P (S) be a univalent local branch of hol^{−1} which is defined on a neighborhood U of ρ_{∞} and takes ρ_{∞} to Σ_{∞}. Then Σ_{n} = Φ(ρ_{n}) converges to Σ_{∞} = Φ(ρ_{∞}). In this situation, all conditions in the same line in Table 1 are equivalent.

Table 1: Equivalent conditions with standard/exotic convergence.

ρ_{n} → ρ_{∞} : standard  ρ_{n} → ρ_{∞} : exotic
Γ_{∞} is a component subgroup of Γ̂ Γ_{∞} is not a component subgroup of Γ̂
Ω_{Γ_{∞}}^{+} ∩ Λ(Γ̂) = ∅  Ω_{Γ_{∞}}^{+} ∩ Λ(Γ̂) ≠ ∅
Σ_{n} are standard (n ≫ 0)  Σ_{n} are exotic (n ≫ 0)

In the same setting as above, let ρ_{n} = B(τ^{n}X, τ^{2n}Ȳ) be the sequence introduced at the end of §2.1 which converges exotically to ρ_{∞}. Then Σ_{n} are exotic projective structure for all large enough |n|. Since Σ_{n} → Σ_{∞} ∈ ∂Q_{0}, the sequence Ψ_{0}(ρ_{n}) ∈ Q_{0} of standard projective structure can not accumulates on a point in ∂Q_{0} and hence diverges. This implies that the map Ψ_{0} : QF ⊔ ∂^{±}QF → P̂(S) is not continuous at ρ_{∞} ∈ ∂^{+}QF. In addition, we have shown in [It1] that Σ_{n} = Ψ_{c}(ρ_{n}) ∈ Q_{c} and that lim_{n→+∞} Ψ_{c}(ρ_{n}) = lim_{n→−∞} Ψ_{c}(ρ_{n}). On the other hand, we have shown in [It3] that lim_{n→+∞} Ψ_{c′}(ρ_{n}) ≠ lim_{n→−∞} Ψ_{c′}(ρ_{n}) for every simple closed curve c′ on S which intersects c essentially. This implies that the grafting map Ψ_{c′} : QF⊔∂^{±}QF → P̂(S) is not continuous at ρ_{∞} ∈ ∂^{+}QF. More general, we have the following

Theorem 3.4 ([It3]). For any λ ∈ ML_{ℕ}, the grafting map Ψ_{λ} : QF ⊔ ∂^{±}QF → P̂(S) is not continuous.

3.4 Pull-backs of region of discontinuities

Let Σ = (f_{Σ}, ρ_{Σ}) ∈ P (S) with Γ = ρ_{Σ}(π_{1}(S)) and let X be a subset of ℂ̂ which is Γ-invariant. Then we naturally obtain a subset f_{Σ}^{−1}(X)/π_{1}(S) of Σ since the inverse image f_{Σ}^{−1}(X) ⊂ Σ̃ of X is also invariant under the action of the covering group π_{1}(S) of the covering map Σ̃ → Σ. We call f_{Σ}^{−1}(X)/π_{1}(S) the pull-back of X in Σ.

For λ = ⊔_{i}k_{i}c_{i} ∈ ML_{ℕ}, a realization λ̂ of λ is a disjoint union of simple closed curves on S which realize each weighted simple closed curve k_{i}c_{i} by k_{i} parallel disjoint simple closed curves which are homotopic to c_{i}. Let N (λ̂) denotes a regular neighborhood of λ̂ in S.

Let ρ ∈ QF with Γ = ρ(π_{1}(S)), let Σ = Ψ_{λ}(ρ) be the grafting of ρ along λ ∈ ML_{ℕ}, and let s denotes either + or −. Then we denote by Ω_{Σ}^{s} the pull-back f_{Σ}^{−1}(Ω_{Γ}^{s})/π_{1}(S) ⊂ Σ of an invariant component Ω_{Γ}^{s} of Ω_{Γ}. From the definition of grafting, we have the following lemma (see also the proof of Theorem 4.2).

Lemma 3.5. Let Σ ∈ Q_{λ}. Then there is a automorphism of Σ which is homotopic to the identity and which takes Σ − N (λ̂) onto Ω_{Σ}^{+} and N (λ̂) onto Ω_{Σ}^{−}.

Therefore the topological type of the pull-backs Ω_{Σ}^{+} and Ω_{Σ}^{−} are constant for every elements Σ ∈ Q_{λ}. The following lemma states that the similar result holds for a grafting Σ = Ψ_{λ}(ρ) of ρ ∈ ∂^{±}QF and for the pull-back of the unique invariant component of Ω_{Γ} (see Figure 2). In fact, we use this fact in §5 to show that Σ = Ψ_{λ}(ρ) for ρ ∈ ∂^{±}QF is actually a limit of a sequence in Q_{λ}. Similar arguments can be found in [It1] and [It3].

Lemma 3.6. Let (λ, ρ) be an admissible pair of λ ∈ ML_{ℕ} and ρ ∈ ∂^{±}QF, and set Σ = Ψ_{λ}(ρ).

• If ρ ∈ ∂^{+}QF, there is a one-to-one correspondence between the set of connected components of Ω_{Σ}^{+} ⊂ Σ and that of Σ − N (λ̂) ⊂ Σ. Moreover, for each connected component of ω ⊂ Ω_{Σ}^{+}, there is a homeomorphism from ω onto the corresponding component of Σ − N (λ̂) which is isotopic to the identity.

• If ρ ∈ ∂^{−}QF, there is a one-to-one correspondence between the set of connected components of Ω_{Σ}^{−} ⊂ Σ and that of N (λ̂) ⊂ Σ. Moreover, for each connected component of ω ⊂ Ω_{Σ}^{−}, there is a homeomorphism from ω onto the corresponding component of N (λ̂) which is isotopic to the identity.

Proof. We discuss in the same setting of Definition 3.1 and §3.2, except for one change; here Σ denotes the grafting Ψ_{λ}(ρ) but not a standard projective structure. We first suppose that ρ ∈ ∂^{+}QF. Let us consider the quotient torus T_{c} = ℂ̂/⟨γ⟩, in which the arc c̃^{+} ⊂ Ω_{Γ}^{+} descends to an essential curve c^{+} in T_{c}. Then the grafting annulus A_{c} = (ℂ̂ − c̃^{+})/⟨γ⟩ is identified with the subset T_{c} − c^{+} of T_{c}. On the other hand, since the unique invariant component Ω_{Γ}^{+} of Ω_{Γ} is simply connected, Ω_{Γ}^{+}/⟨γ⟩ ⊂ T_{c} is also an annulus with core curve c^{+}. Hence the intersection A_{c}∩(Ω_{Γ}^{+}/⟨γ⟩) equals Ω_{Γ}^{+}/⟨γ⟩ − c^{+} in T_{c}, which is a disjoint union of two essential annuli A_{1} and A_{2}. We now regard that A_{1} ⊔ A_{2} ⊂ A_{c} ⊂ Σ. Then the pull-back Ω_{Σ}^{+} ⊂ Σ of Ω_{Γ}^{+} coincides with the union (Σ − A_{c}) ⨿ A_{1} ⨿ A_{2} ⊂ Σ, which is obtained by gluing those pieces along copies of c^{+}. Thus the result follows in this case.

Next we suppose that ρ_{∞} ∈ ∂^{−}QF. By the same argument as above, we see that the quotient Ω_{Γ}^{−}/⟨γ⟩ of the unique invariant component Ω_{Γ}^{−} of Ω_{Γ} is an annulus with core curve c^{−} = c̃^{−}/⟨γ⟩, and that the disjoint union Ω_{Γ}^{−} − c̃^{−} = Ω_{1}^{−} ⊔ Ω_{2}^{−} descends to that of essential annuli A_{1}^{−} ⊔ A_{2}^{−} in Ω_{Γ}^{−}/⟨γ⟩, where A_{i}^{−} = Ω_{i}^{−}/⟨γ⟩ for i = 1, 2. We may assume that Δ_{i} ⊂ Ω_{i}^{−} for i = 1, 2. Recall that Γ_{i} = Stab_{Γ}(Δ_{i}) is a Schottky group. Since Stab_{Γ_{1}}(Ω_{2}^{−}) = ⟨γ⟩, the image of Ω_{Γ}^{−} = Ω_{1}^{−} ⨿ Ω_{2}^{−} via the covering map Ω_{Γ_{1}} → Ω_{Γ_{1}}/Γ_{1} is Σ_{1}^{−} ⨿ A_{2}^{−} ⊂ Ω_{Γ_{1}}/Γ_{1}, which is obtained from Σ_{1}^{−} = Δ_{1}/Γ_{1} and A_{2}^{−} = Ω_{2}^{−}/⟨γ⟩ by gluing along c^{−}. Similarly, the image of Ω_{Γ}^{−} = Ω_{1}^{−} ⨿ Ω_{2}^{−} via the covering map Ω_{Γ_{2}} → Ω_{Γ_{2}}/Γ_{2} is Σ_{2}^{−} ⨿ A_{1}^{−} ⊂ Ω_{Γ_{2}}/Γ_{2}. Since Σ is obtained from Ω_{Γ_{1}}/Γ_{1} −Σ_{1}^{−} and Ω_{Γ_{2}}/Γ_{2} −Σ_{2}^{−} by gluing along c^{−}, we see that the pull-back Ω_{Σ}^{−} ⊂ Σ of Ω_{Γ}^{−} equals the annulus A_{1}^{−} ⨿ A_{2}^{−} = Ω_{1}^{−}/⟨γ⟩ ⨿ Ω_{2}^{−}/⟨γ⟩ = Ω_{Γ}^{−}/⟨γ⟩ ⊂ Σ with core curve c^{−}. Thus we have obtained the result also in this case.

PSfrag replacements Ω_{Σ}^{+} Ω_{Σ}^{−}

Figure 2: Pull-backs Ω_{Σ}^{+} ⊂ Σ for ρ ∈ ∂^{+}QF and Ω_{Σ}^{−} ⊂ Σ for ρ ∈ ∂^{−}QF.

Corollary 3.7. Let λ, μ ∈ ML_{ℕ}, λ ≠ μ and ρ ∈ ∂^{±}QF. If both pairs (λ, ρ) and (μ, ρ) are admissible then Ψ_{λ}(ρ) ≠ Ψ_{μ}(ρ).

4 Finiteness

The aim of this section is to prove Theorem 4.3, which guarantees some finiteness: suppose that a sequence ρ_{n} ∈ QF converges standardly to ρ ∈ ∂^{±}QF and that Φ : U → P (S) is a univalent local branch of hol^{−1} defined on a neighborhood U of ρ. Then the sequence Φ(ρ_{n}) is actually contained in some finite union of components of Q(S). This finiteness theorem plays an important role in the proof of our main theorem (Theorem 5.1), which guarantees some continuity of the grafting maps and which can be seen also as the uniqueness theorem.

4.1 Length-Modulus inequality

Recall that the modulus Mod(A) of a conformal annulus A is defined uniquely as the ratio of the height and the circumference of an Euclidian annulus which is conformally equivalent to A. The following inequality is a direct consequence of the geometric and analytic definitions of the extremal length.

Lemma 4.1. Let R be a complete hyperbolic surface of finite area Area(R) < ∞ and let A be an essential annular domain in R. Then we have

Mod(A) l_{R}(c)^{2} ≤ Area(R),

where l_{R}(c) is the hyperbolic length of the homotopy class c of a core curve of A.

Proof. Let E_{R}(c) denotes the extremal length of the homotopy class c in R. From the analytical definition of E_{R}(c), we have

E_{R}(c) := sup_{ρ} \frac{(inf_{c′} ∫_{c′} ρ(z) |dz|)^{2}}{∫∫ ρ(z)^{2} |dz|^{2}} ≥ \frac{(l_{R}(c))^{2}}{Area(R)},

where the supremum is taken over all metrics ρ consistent with the conformal structure of R and the infimum is taken over all closed curves c′ in the homotopy class c. On the other hand, from the geometrical definition of E_{R}(c), we have

E_{R}(c) := \frac{1}{sup_{A′⊂R} Mod(A′)} ≤ \frac{1}{Mod(A)},

where the supremum is taken over all annuli A′ ⊂ R whose core curve is in the homotopy class c. From the above two inequality, the desired inequality follows.

4.2 Quasiconformal deformations

We introduce the notion of a quasiconformal deformation of a projective structure with quasifuchsian holonomy, which was developed by Shiga and Tanigawa in [ST]. Let Σ ∈ Q(S) and let ρ_{Σ} ∈ QF be its holonomy. Suppose that ρ′ ∈ QF is a quasiconformal deformation of ρ_{Σ} induced by a quasiconformal automorphism q : ℂ̂ → ℂ̂, whose Beltrami differential is denoted by μ. Then we obtain a new projective structures Σ′ with holonomy ρ′ which is characterized as follows:

1. There is a quasiconformal map φ̃ : Σ̃ → Σ̃′ whose Beltrami differential is equal to the pull-back f_{Σ}^{∗}(μ) of μ via the developing map f_{Σ} : Σ̃ → ℂ̂. Moreover, the map φ̃ : Σ̃ → Σ̃′ descends to a quasiconformal map φ : Σ → Σ′, which is consistent with their markings.

2. The developing map of Σ′ is defined by f_{Σ′} = q ∘ f_{Σ} ∘ φ̃^{−1} : Σ̃′ → ℂ̂.

Here we say that a map between projective structures is quasiconformal if it is a quasiconformal map between their underling conformal structures. We call Σ′ as the quasiconformal deformation of Σ. We remark that every grafting map Ψ_{λ} : QF → Q_{λ} is obtained by quasiconformal deformations of some fixed Σ ∈ Q_{λ} and its holonomy ρ_{Σ} ∈ QF.

4.3 Length estimates

Theorem 4.2. Fix X ∈ T (S) arbitrarily. Suppose that λ ∈ ML_{ℕ} contains a weighted simple closed curve kc of weight k ∈ ℕ. Let ρ ∈ B_{X} ∪ BX̄ and let X′ ∈ T (S) denotes the underlying conformal structure of Ψ_{λ}(ρ). Then we have l_{X′}(c)^{2} ≤ 4(g − 1) l_{X}(c)/k, where g denotes the genus of S.

Remark. We make use of the inequality above mostly in the following form:

\frac{l_{X′}(c)}{l_{X}(c)} ≤ \sqrt{\frac{4(g − 1)}{k · l_{X}(c)}},

which ensures that if k·l_{X}(c) is large enough then the Teichmüller distance d_{T (S)}(X, X′) between X and X′ is also large.

Proof of Theorem 4.2. We first treat the case where λ is a simple closed curve c of weight one. Let ρ^{0} = B(X, X̄) ∈ QF be the Fuchsian representation with image Γ^{0} = ρ^{0}(π_{1}(S)). We normalize ρ^{0} so that Ω_{Γ^{0}}^{+} equals the upper half plane H = {x + iy ∈ ℂ : y > 0}, Ω_{Γ^{0}}^{−} equals the lower half plane L = {x + iy ∈ ℂ : y < 0}, and that the hyperbolic element γ = ρ^{0}(c) ∈ Γ^{0} fixes the positive imaginary axis iℝ_{+}. We let Σ^{0} denote the standard projective structure Ψ_{0}(ρ^{0}) = Ω_{Γ^{0}}^{+}/Γ^{0}. Recall that the grafting Ψ_{c}(ρ^{0}) = Gr_{c}(Σ^{0}) of Σ_{0}, which is denoted by Σ_{c}^{0}, is obtained from Σ^{0} by cutting along c and inserting the annulus A = (ℂ − iℝ_{+})/⟨γ⟩. We set H −iℝ_{+} = H_{L}⊔H_{R}. Then ℂ−iℝ_{+} = H_{R}⊔L⊔H_{L} and then A = A_{1}⨿A_{2}⨿A_{3}, where A_{1} = H_{R}/⟨γ⟩, A_{2} = L/⟨γ⟩ and H_{L}/⟨γ⟩. Here Mod(A_{1}) = Mod(A_{3}) = π/2l_{X}(c) and Mod(A_{2}) = π/l_{X}(c). From now on, we regard A = A_{1} ⨿ A_{2} ⨿ A_{3} as a subset of Σ_{c}^{0}. By definition of grafting, it hold that A ∩ Ω_{Σ_{c}^{0}}^{+} = A_{1} ⊔ A_{3} and that Ω_{Σ_{c}^{0}}^{−} = A_{2} on Σ_{c}^{0}, where Ω_{Σ_{c}^{0}}^{+} ⊂ Σ_{c}^{0} (resp. Ω_{Σ_{c}^{0}}^{−} ⊂ Σ_{c}^{0}) is the pull-back of the upper half plane H = Ω_{Γ^{0}}^{+} (resp. the lower half plane L = Ω_{Γ^{0}}^{−}).

Let ρ ∈ B_{X} ∪ BX̄ be a quasiconformal deformation of ρ^{0} = B(X, X̄) which is induced by a quasiconformal map q : ℂ̂ → ℂ̂. If ρ is contained in B_{X} (resp. BX̄) we may assume that the support of the Beltrami differential of q is contained in the lower half plane L = Ω_{Γ^{0}}^{−} (resp. the upper half plane H = Ω_{Γ^{0}}^{+}). Associated to the quasiconformal deformation ρ of ρ^{0}, we have a quasiconformal deformation Σ_{c} = Ψ_{c}(ρ) of Σ_{c}^{0} = Ψ_{c}(ρ^{0}) and a quasiconformal map φ : Σ_{c}^{0} → Σ_{c}. It is important to notice that the map φ takes Ω_{Σ_{c}^{0}}^{+} (resp. Ω_{Σ_{c}^{0}}^{−}) conformally into Σ_{c} if ρ ∈ B_{X} (resp. ρ ∈ BX̄). Here we claim that the annulus φ(A) ⊂ Σ_{c} always satisfies Mod(φ(A)) ≥ π/l_{X}(c). In fact, if ρ ∈ B_{X} then the the map φ takes A_{1} ⊔A_{3} ⊂ Ω_{Σ_{c}^{0}}^{+} conformally into φ(A) ⊂ Σ_{c} and thus Mod(φ(A)) ≥ Mod(φ(A_{1}))+Mod(φ(A_{3})) = π/l_{X}(c). Similarly, if ρ ∈ BX̄ then φ takes A_{2} ⊂ Ω_{Σ_{c}^{0}}^{−} conformally into φ(A) ⊂ Σ_{c} and thus Mod(φ(A)) ≥ Mod(φ(A_{2})) = π/l_{X}(c). Recall that X′ denote the underlying conformal structure of Σ_{c} = Ψ_{c}(ρ). By Lemma 4.1, we have Mod(φ(A)) l_{X′}(c)^{2} ≤ Area(X′) = 4π(g − 1). By combining this inequality with the inequality Mod(φ(A)) ≥ π/l_{X}(c), the result follows in the case where λ = c.

Next suppose that λ ∈ ML_{ℕ} contains a weighted simple closed curve kc with k ∈ ℕ. Then Σ_{λ} = Ψ_{λ}(ρ) contains an annulus Ã which is a union of succeeding k parallel annuli each of whose modulus ≥ π/l_{X}(c). Thus Mod(Ã) ≥ kπ/l_{X}(c) and hence we have obtained the desired inequality in general.

From Theorem 4.2, we see that the both π ∘ Ψ_{λ}(B_{X}) and π ∘ Ψ_{λ}(BX̄) are proper subsets of T (S) for every non-zero λ ∈ ML_{ℕ}. In fact if kc ⊂ λ, they are contained in the proper subset

\{X′ ∈ T (S) : l_{X′}(c) ≤ \sqrt{4(g − 1) l_{X}(c)/k}\}

of T (S). Compare this fact with a result which was independently obtained by Gallo [Ga] and Tanigawa [Ta]; they showed that the map π ∘ Ψ_{λ} takes the Fuchsian space F = {B(X, X̄) ∈ QF : X ∈ T (S)} ⊂ QF bijectively onto T (S) for every λ ∈ ML_{ℕ}. In addition, we remark that both π ∘Ψ_{λ}(B_{X}) and π ∘Ψ_{λ}(BX̄) are non-precompact in T (S) for every non-zero λ ∈ ML_{ℕ}. In fact if c ⊂ λ, we can make the length l_{X′}(c) arbitrarily small by varying ρ = B(X, Ȳ) in B_{X} so that lȲ (c) → 0 or ρ = B(Y, X̄) in BX̄ so that l_{Y} (c) → 0.

4.4 Finiteness theorem

As a consequence of Theorem 4.2, we obtain the following

Theorem 4.3 (Finiteness). Suppose that a sequence Σ_{n} ∈ Q(S) of quasifuchsian projective structures converges to Σ_{∞} ∈ P (S) and that the sequence ρ_{Σ_{n}} ∈ QF of their holonomies converges standardly to ρ_{Σ_{∞}} ∈ ∂^{±}QF. Then there is a finite union Q_{λ_{1}} ⊔ ⋯ ⊔ Q_{λ_{l}} of components of Q(S) in which Σ_{n} are contained for all n.

Instead proving this theorem directly, we prove the following theorem, which is a contraposition of Theorem 4.3.

Theorem 4.4. Suppose that a sequence ρ_{n} ∈ QF converges standardly to ρ_{∞} ∈ ∂^{±}QF and {λ_{n}} is a sequence of distinct element of ML_{ℕ}. Then the sequence Ψ_{λ_{n}}(ρ_{n}) converges to ∞ in P̂(S) as n → ∞, that is, the set {Ψ_{λ_{n}}(ρ_{n}) : n ∈ ℕ} has no accumulation point in P (S).

Proof. Set ρ_{n} = B(X_{n}, Ȳ_{n}). We first assume that ρ_{∞} ∈ ∂^{+}QF. Then ρ_{∞} ∈ ∂B_{X} for some X ∈ T (S) and X_{n} → X in T (S). One can choose a sequence of weighted simple closed curves k_{n}c_{n} ⊂ λ_{n} which satisfies k_{n} · l_{X}(c_{n}) → ∞ as n → ∞. Let X′_{n} ∈ T (S) denote the underlying conformal structure of Ψ_{λ_{n}}(ρ_{n}). Then it follows from Theorem 4.2 that

\frac{l_{X′_{n}}(c_{n})}{l_{X_{n}}(c_{n})} ≤ \sqrt{\frac{4(g − 1)}{k_{n} · l_{X_{n}}(c_{n})}}

and that the Teichmüller distance d_{T (S)}(X_{n}, X′_{n}) between X_{n} and X′_{n} tends to ∞ as n → ∞. Since X_{n} → X in T (S), d_{T (S)}(X, X′_{n}) also tends to ∞, which implies that Ψ_{λ_{n}}(ρ_{n}) → ∞ in P̂(S). The proof for ρ_{∞} ∈ ∂^{−}QF is completely parallel.

5 Continuity

The following is our main theorem in this paper.

Theorem 5.1 (Continuity). Let ρ_{n} ∈ QF be a sequence which converges standardly to ρ_{∞} ∈ ∂^{±}QF. Then the sequence Ψ_{λ}(ρ_{n}) converges to Ψ_{λ}(ρ_{∞}) in P̂(S) for every λ ∈ ML_{ℕ}.

We devote this section to the proof of the above theorem, which is divided into two cases; the case where Ψ_{λ}(ρ_{∞}) ∈ P (S) and that where Ψ_{λ}(ρ_{∞}) = ∞. Throughout this proof, we let s denote either + or − for which ρ_{∞} ∈ ∂^{s}QF.

5.1 Case where Ψ_{λ}(ρ_{∞}) ∈ P (S)

Let Φ : U → V, ρ_{∞} ↦ Ψ_{λ}(ρ_{∞}) be a univalent local branch of hol^{−1} from a neighborhood U of ρ_{∞} to that V of Ψ_{λ}(ρ_{∞}) and set Σ_{n} := Φ(ρ_{n}) and Σ_{∞} := Φ(ρ_{∞}) = Ψ_{λ}(ρ_{∞}). Then the sequence Σ_{n} converges to Σ_{∞} in P (S). We claim in Lemma 5.2 below that Σ_{n} ∈ Q_{λ} for all large enough n. Assuming this lemma, we have Σ_{n} = Ψ_{λ}(ρ_{n}) and thus obtain the desired convergence Ψ_{λ}(ρ_{n}) = Σ_{n} → Σ_{∞} = Ψ_{λ}(ρ_{∞}).

Lemma 5.2. Σ_{n} ∈ Q_{λ} for all large enough n.

The rest of this subsection is devoted to the proof of Lemma 5.2. The idea of the proof is as follows: by Lemma 3.6, we have already known the topological type of the set Ω_{Σ_{∞}}^{s} ⊂ Σ_{∞} = Ψ_{λ}(ρ_{∞}) in relation to λ. As we will see in Lemma 5.5, the set Ω_{Σ_{∞}}^{s} ⊂ Σ_{∞} coincides with the essential part of the Carathéodory limit of the sequence Ω_{Σ_{n}}^{s} ⊂ Σ_{n} as n → ∞. It is essential that the sequence Σ_{n} is contained in a finite union of components of Q(S) by Theorem 4.3. Then we will show that for any subsequence {Σ_{n_{j}}}_{j=1}^{∞} of {Σ_{n}}_{n=1}^{∞} contained in a component of Q(S), the topological types of Ω_{Σ_{n_{j}}}^{s} ⊂ Σ_{n_{j}} is the same to that of Ω_{Σ_{∞}}^{s} ⊂ Σ_{∞} for all j, which implies that Σ_{n} ∈ Q_{λ} for all n.

Now we fill in the details. We may assume that the sequence Γ_{n} = ρ_{n}(π_{1}(S)) of quasifuchsian groups converges geometrically to some Kleinian group Γ̂, which contains the algebraic limit Γ_{∞} = ρ_{∞}(π_{1}(S)). Then Kerckhoff and Thurston [KT, Corollary 2.2] showed that the sequence Ω_{Γ_{n}} of the regions of discontinuity of Γ_{n} converges to the region of discontinuity ΩΓ̂ of Γ̂ in the sense of Carathéodory. Here Ω_{Γ_{n}} → ΩΓ̂ in this sense if (i) for every compact K ⊂ ΩΓ̂, we have K ⊂ Ω_{Γ_{n}} for all large enough n and if (ii) for any open set U ⊂ Ω_{Γ_{n_{j}}} (n_{j} → ∞), we have U ⊂ ΩΓ̂. Moreover ΩΓ̂ decomposes into two Γ̂-invariant regions Ω+ ; Γ̂, Ω− ; Γ̂, each of which are the Carathéodory limits of Ω_{Γ_{n}}^{+}, Ω_{Γ_{n}}^{−}, respectively. Since the convergence ρ_{n} → ρ_{∞} ∈ ∂^{s}QF is standard, the argument in the proof of [KT, Proposition 2.3] reveals that Ω_{Γ_{∞}}^{s} is a component of Ωs ; Γ̂, that Ωs ; Γ̂ = ⋃_{δ∈Γ̂} δ(Ω_{Γ_{∞}}^{s}), and that δ ∈ Γ̂ fixes Ω_{Γ_{∞}}^{s} if and only if δ ∈ Γ_{∞}. We put them together in the following

Lemma 5.3 (Kerckhoff-Thurston). We have Ωs ; Γ̂ = ⨆_{δ∈Γ̂/Γ_{∞}} δ(Ω_{Γ_{∞}}^{s}).

Recall that Ω_{Σ_{n}}^{s} denotes the pull-back of Ω_{Γ_{n}}^{s} in Σ_{n}. Similarly Ω_{Σ_{∞}}^{s} is the pullback of Ω_{Γ_{∞}}^{s} in Σ_{∞}. In addition, we let Ω̂_{Σ_{∞}}^{s} denotes the pull-back of Ωs ; Γ̂ in Σ_{∞}. By the same argument of [It1, Lemma 3.3], we obtain the following

Lemma 5.4. The sequence Ω_{Σ_{n}}^{s} ⊂ Σ_{n} converges to Ω̂_{Σ_{∞}}^{s} ⊂ Σ_{∞} in the sense of Carathéodory. More precisely, there exist K_{n}-bi-Lipshiz diffeomorphisms ψ_{n} : Σ_{∞} → Σ_{n} between hyperbolic surfaces (consistent with their markings) such that K_{n} → 1 as n → ∞ and that the sequence ψ_{n}^{−1}(Ω_{Σ_{n}}^{s}) ⊂ Σ_{∞} converges to Ω̂_{Σ_{∞}}^{s} ⊂ Σ_{∞} in the sense of Carathéodory.

We claim in the following lemma that Ω_{Σ_{∞}}^{s} ⊂ Σ_{∞} carries the full fundamental group of Ω̂_{Σ_{∞}}^{s} ⊂ Σ_{∞}.

Lemma 5.5. Each connected component of Ω̂_{Σ_{∞}}^{s} − Ω_{Σ_{∞}}^{s} is homotopically trivial in Σ_{∞}.

Proof. Suppose for contradiction that there exists a connected component ω of Ω̂_{Σ_{∞}}^{s} − Ω_{Σ_{∞}}^{s} which contains an essential closed curve α. Let ω̃ ⊂ Σ̃_{∞} be a lift (i.e., a connected component of the inverse image) of ω ⊂ Σ_{∞} and α̃ ⊂ Σ̃_{∞} a lift of the curve α ⊂ ω which is contained in ω̃. Let Ω be the connected component of Ωs ; Γ̂ − Ω_{Γ_{∞}}^{s} which contains the developing image f_{Σ_{∞}}(ω̃) ⊂ ℂ̂. Let γ = ρ_{∞}(α) be an element of Γ_{∞} which fixes f_{Σ_{∞}}(α̃), and hence Ω.

Now we see that γ is loxodromic from Definition 3.3 and Lemma 3.5. In fact if ρ_{∞} ∈ ∂^{+}QF, then each connected component of Σ_{∞} − Ω_{Σ_{∞}}^{+} is an annulus whose core curve is homotopic to a component of λ. Hence the curve α in ω ⊂ Σ_{∞} − Ω_{Σ_{∞}}^{+} is homotopic into λ and hence is not homotopic into para(ρ_{∞}). On the other hand, if ρ_{∞} ∈ ∂^{−}QF then each component of para(ρ_{∞}) must traverse λ, and thus Ω_{Σ_{∞}}^{−}, in Σ_{∞}. Therefore the curve α in Σ_{∞} − Ω_{Σ_{∞}}^{−} is not homotopic into para(ρ_{∞}).

From Lemma 5.3, there exists δ ∈ Γ̂ − Γ_{∞} such that Ω = δ(Ω_{Γ_{∞}}^{s}). Since γ fixes the component Ω, δ^{−1}γδ fixes the component Ω_{Γ_{∞}}^{s}, which implies that δ^{−1}γδ ∈ Γ_{∞}. But this contradicts to the following lemma (Lemma 2.4 in [ACCS]).

Lemma 5.6 (Anderson-Canary-Culler-Shalen). Let G be a finitely generated group. Let ρ_{n} : G → Γ_{n} ⊂ PSL_{2}(ℂ) be a sequence of discrete faithful representations converging algebraically to ρ_{∞} : G → Γ_{∞}. Moreover, assume that Γ_{n} converges to Γ̂ geometrically. Then for any δ ∈ Γ̂ − Γ_{∞}, Γ_{∞} ∩ δ^{−1}Γ_{∞}δ is {id} or rank-one parabolic subgroup.

Let us combine the above observations to complete the proof of Lemma 5.2, which states that Σ_{n} ∈ Q_{λ} for all large enough n. Note that the sequence Σ_{n} is contained in a finite union of components of Q(S) by Theorem 4.3. Therefore to show Lemma 5.2, we only need to show that λ′ = λ by assuming that the sequence Σ_{n} is contained in a component Q_{λ′} of Q(S).

Let ρ_{0} ∈ QF be a fixed element and set Σ_{0} = Ψ_{λ′}(ρ_{0}). Then every Σ_{n} is a quasiconformal deformation of Σ_{0} in Q_{λ′}, which is induced by the quasiconformal deformation ρ_{n} of ρ_{0}, and which induces the quasiconformal map φ_{n} : Σ_{0} → Σ_{n}. Note that the map φ_{n} is consistent with their markings and takes Ω_{Σ_{0}}^{s} onto Ω_{Σ_{n}}^{s}. Since the convergence ρ_{n} → ρ_{∞} ∈ ∂^{s}QF is standard, the dilatation of φ_{n}|_{Ω_{Σ_{0}}^{s}} tends to 1, and hence we assume that φ_{n}|_{Ω_{Σ_{0}}^{s}} : Ω_{Σ_{0}}^{s} → Ω_{Σ_{n}}^{s} is conformal for every n for simplicity.

Now let C(Ω_{Σ_{0}}^{s}), C(Ω_{Σ_{∞}}^{s}) denote the set of connected components of Ω_{Σ_{0}}^{s}, Ω_{Σ_{∞}}^{s}, respectively. Then Lemma 3.6 tells us that λ = λ′ if and only if there exists a bijection

g : C(Ω_{Σ_{0}}^{s}) → C(Ω_{Σ_{∞}}^{s})

which satisfies π_{1}(ω) = π_{1}(g(ω)) for every ω ∈ C(Ω_{Σ_{0}}^{s}). Here we identify π_{1}(Σ_{0}) and π_{1}(Σ_{∞}) with π_{1}(S) via their markings. Note that ψ_{n}^{−1} ∘ φ_{n}(Ω_{Σ_{0}}^{s}) ⊂ Σ_{∞} converge to Ω̂_{Σ_{∞}}^{s} ⊂ Σ_{∞} is the sense of Lemma 5.4 and that π_{1}(ω) = π_{1}(ψ_{n}^{−1} ∘ φ_{n}(ω)) hold for all ω ∈ C(Ω_{Σ_{0}}^{s}) and n. We first define a map h : C(Ω_{Σ_{∞}}^{s}) → C(Ω_{Σ_{0}}^{s}), which will turn out to be the inverse of g. Let ω′ ∈ C(Ω_{Σ_{∞}}^{s}) and let K ⊂ ω′ be a compact subset such that π_{1}(K) = π_{1}(ω′). Then there exists ω ∈ C(Ω_{Σ_{0}}^{s}) such that K ⊂ ψ_{n}^{−1} ∘ φ_{n}(ω) for all large enough n, and hence that π_{1}(K) = π_{1}(ω′) ⊂ π_{1}(ω). Now we let h(ω′) = ω. Next we define the map g : C(Ω_{Σ_{0}}^{s}) → C(Ω_{Σ_{∞}}^{s}) as follows: let ω ∈ C(Ω_{Σ_{0}}^{s}). By passing to a subsequence if necessary, the sequence ψ_{n}^{−1} ∘ φ_{n}(ω) ⊂ Σ_{∞} converges to an open subset ω̂ of Ω̂_{Σ_{∞}}^{s} ⊂ Σ_{∞}. We claim that ω̂ contains exactly one essential component ω′ of Ω̂_{Σ_{∞}}^{s} ⊂ Σ_{∞}, which satisfies π_{1}(ω) ⊂ π_{1}(ω′). We postpone the proof of this claim and continue the argument. We have ω′ ∈ C(Ω_{Σ_{∞}}^{s}) by Lemma 5.5 and let g(ω) = ω′. Then it follows immediately that g is a bijective map with g^{−1} = h and hence that λ = λ′. We have completed the proof of Lemma 5.2, and hence that of Theorem 5.1 in the case where Ψ_{λ}(ρ_{∞}) ∈ P (S).

We only need to show the above claim. Suppose for contradiction that ω̂ ⊂ Σ_{∞} is trivial or contains more that two essential connected components. Since ψ_{n}^{−1}∘φ_{n}(ω) converge to ω̂, φ_{n}(ω) ⊂ Σ_{n} then become more and more constricted, that is, there exists an arc α_{n} ⊂ φ_{n}(ω) joining two distinct components of φ_{n}(∂ω) whose hyperbolic length tends to 0 as n → ∞. But since the map φ_{n}|_{Ω_{Σ_{0}}^{s}} : Ω_{Σ_{0}}^{s} → φ_{n}(Ω_{Σ_{0}}^{s}) is conformal, this contradicts to Lemma 5.7 below. Thus we have proved the claim.

Lemma 5.7 ([It1, Lemma 4.4]). Let R be a hyperbolic surface, c ⊂ R a simple closed curve and A ⊂ R is an annulus whose core curve is homotopic to c. Then there is a positive constant C > 0, depending only on l_{R}(c) and Mod(A), such that l_{R}(α) ≥ C holds for every arcs α ⊂ A joining points in distinct components of ∂A.

5.2 Case where Ψ_{λ}(ρ_{∞}) = ∞

We first prepare the following

Lemma 5.8. Let λ ∈ ML_{ℕ} and let ρ_{n} = B(X_{n}, Ȳ_{n}) ∈ QF be a sequence which converges standardly to ρ_{∞} ∈ ∂^{±}QF and which satisfies the property as in Corollary 2.4. Suppose that Ψ_{λ}(ρ_{∞}) = ∞. Then the sequence Ψ_{λ}(ρ_{n}) → ∞ in P̂(S).

Proof. We first suppose that ρ_{∞} ∈ ∂^{+}QF. Since (λ, ρ_{∞}) is not admissible, λ and para(ρ_{∞}) have parallel components in common and let c be one of such components. Since lȲ_{n}(c) → 0, there are annular neighborhoods A_{n} of c in Ȳ_{n} such that Mod(A_{n}) → ∞ as n → ∞. We set Σ_{n} = Ψ_{λ}(ρ_{n}). Then observe that the annulus A_{n} is conformally embedded in Ω_{Σ_{n}}^{−} ⊂ Σ_{n} for every n. Hence the underlying conformal structures π(Ψ_{λ}(ρ_{n})) of Ψ_{λ}(ρ_{n}) diverge in T (S), and hence Ψ_{λ}(ρ_{n}) → ∞ in P̂(S).

Next we suppose that ρ_{∞} ∈ ∂^{−}QF. Then there is a component c of para(ρ_{∞}) which do not intersect λ essentially. Since l_{X_{n}}(c) → 0, there are annular neighborhoods A_{n} of c in X_{n} such that Mod(A_{n}) → ∞ as n → ∞. Then observe that the annulus A_{n} is conformally embedded in Ω_{Σ_{n}}^{+} ⊂ Σ_{n} = Ψ_{λ}(ρ_{n}) for every n. By the same argument as above, we see that Ψ_{λ}(ρ_{n}) → ∞ in P̂(S).

Now we back to the proof of Theorem 5.1 in the case where Ψ_{λ}(ρ_{∞}) = ∞; we intend to show that Ψ_{λ}(ρ_{n}) → Ψ_{λ}(ρ_{∞}) = ∞ for every standard convergent sequence QF ∋ ρ_{n} → ρ_{∞} ∈ ∂^{±}QF. Set Σ_{n} = Ψ_{λ}(ρ_{n}). To obtain a contradiction, we suppose that the sequence Σ_{n} has a subsequence (which is denoted by the same symbols) converging to some Σ_{∞} ∈ P (S). In addition, let ρ′_{n} ∈ QF be a sequence converging to ρ_{∞} as in Lemma 5.8 and let Φ : U → P (S), ρ_{∞} ↦ Σ_{∞} be a univalent local branch of hol^{−1}. Then we have Σ_{n} = Φ(ρ_{n}) for all large enough n. Note that both Σ_{n} = Φ(ρ_{n}) and Σ′_{n} := Φ(ρ′_{n}) converge to Σ_{∞}. From Theorem 4.3, the sequence Σ′_{n} has a subsequence (also denoted by the same symbols) which is contained in a component Q_{μ} of Q(S) for some μ ∈ ML_{ℕ}. Then we have Σ′_{n} = Ψ_{μ}(ρ′_{n}), and thus Ψ_{μ}(ρ′_{n}) = Σ′_{n} → Σ_{∞} ≠ ∞, which implies that Ψ_{μ}(ρ_{∞}) ≠ ∞ by Lemma 5.8 and that λ ≠ μ. Since both ρ_{n}, ρ′_{n} converge standardly to ρ_{∞}, Theorem 5.1 for admissible pairs guarantees that both Ψ_{μ}(ρ_{n}), Ψ_{μ}(ρ′_{n}) convege to Ψ_{μ}(ρ_{∞}). On the other hand, since Ψ_{μ}(ρ′_{n}) = Σ′_{n} → Σ_{∞}, we have Ψ_{μ}(ρ_{n}) → Σ_{∞}. Now we obtain two sequences Ψ_{λ}(ρ_{n}), Ψ_{μ}(ρ_{n}) both of which converge to Σ_{∞}. But since λ ≠ μ, this contradicts to the fact that the map hol is a local homeomorphism. Now we have completed the proof of Theorem 5.1.

6 Applications

We collect in this section some results which are obtained as consequences of Theorem 5.1.

6.1 Grafting theorem for boundary b-groups

The following statement is conjectured by Bromberg in [Br].

Theorem 6.1. For every boundary b-groups ρ ∈ ∂^{±}QF, all projective structure with holonomy ρ are obtained by grafting of ρ, that is, we have

hol^{−1}(ρ) = {Ψ_{λ}(ρ) : λ ∈ ML_{ℕ}, Ψ_{λ}(ρ) ∈ P (S)}.

Proof. Let ρ ∈ ∂^{±}QF and Σ ∈ hol^{−1}(ρ). We only need to show that Σ = Ψ_{λ}(ρ) for some λ ∈ ML_{ℕ}. Let ρ_{n} ∈ QF be a sequence converging standardly to ρ. Since the map hol is a local homeomorphism, there exists a convergent sequence Σ_{n} → Σ of projective structures with hol(Σ_{n}) = ρ_{n} for all n. From Theorem 4.3, there exists a subsequence {Σ_{n_{j}}}_{j=1}^{∞} of the sequence {Σ_{n}}_{n=1}^{∞} which is contained in a component Q_{λ} of Q(S). Then we have Σ_{n_{j}} = Ψ_{λ}(ρ_{n_{j}}) for all j and thus Ψ_{λ}(ρ_{n_{j}}) = Σ_{n_{j}} → Σ as j → ∞. On the other hand, since the convergence ρ_{n} → ρ is standard, Theorem 5.1 implies that Ψ_{λ}(ρ_{n}) → Ψ_{λ}(ρ) in P̂(S). Hence Σ = Ψ_{λ}(ρ) and the result follows.

From Bromberg’s result (Theorem 2.2), we obtain the following

Corollary 6.2. Suppose that ρ ∈ AH(S) is a b-group with no parabolics. Then we have hol^{−1}(ρ) = {Ψ_{λ}(ρ) : λ ∈ ML_{ℕ}}.

6.2 Analytic continuations and lifts of paths

Theorem 6.3. Suppose that Ψ_{λ}(ρ) ∈ P (S) for λ ∈ ML_{ℕ} and ρ ∈ ∂^{±}QF. Then there exists a path α : [0, 1] → R(S) with α(0) ∈ QF and α(1) = Ψ_{λ}(ρ) along which the grafting map Ψ_{λ} : QF → P (S) is continued analytically to a univalent local branch Φ : U → P (S), ρ ↦ Ψ_{λ}(ρ) of hol^{−1} which is defined on some neighborhood U of ρ.

Proof. We may assume that ρ ∈ ∂^{+}QF, and hence that ρ ∈ ∂B_{X} for some X ∈ T (S). Let D ⊂ T (S) de a bounded domain with X ∈ D and set B_{D} = ⊔_{X′∈D}B_{X′}. We claim that there exists a neighborhood U of ρ such that Φ ≡ Ψ_{λ} holds on U ∩ B_{D}. For otherwise there is a sequence ρ_{n} ∈ B_{D} converging to ρ for which Φ(ρ_{n}) ≠ Ψ_{λ}(ρ_{n}) hold for all n. On the other hand, Theorem 5.1 implies that both sequences Φ(ρ_{n}), Ψ_{λ}(ρ_{n}) converge to Φ(ρ) = Ψ_{λ}(ρ), which contradicts to the fact that Φ is univalent. Now let α : [0, 1] → R(S) be an arc such that α(0) ∈ QF, α(1) = ρ and that α([0, 1]) ⊂ B_{D} ∪ U. Then Ψ_{λ} : QF → P (S) is continued analytically to Φ : U → P (S) along α.

Let α : [0, 1] → R(S) be a path as in Theorem 6.3. Then the above theorem implies that there exists the lift α̃ : [0, 1] → P (S) of α via the holonomy map with the starting point Ψ_{λ}(α(0)). On the other hand, we have the following

Theorem 6.4. Suppose that Ψ_{λ}(ρ) = ∞ for λ ∈ ML_{ℕ} and ρ ∈ ∂^{±}QF. Then there exists a path α : [0, 1] → R(S) of which there is no lift to P (S) with the starting point Ψ_{λ}(α(0)).

Proof. We may assume that ρ ∈ ∂^{+}QF, and hence that ρ ∈ ∂B_{X} for some X ∈ T (S). Let take a path α : [0, 1] → R(S) with α(0) ∈ QF and α(1) = ρ for which there exists a sequence 0 < t_{n} < 1, t_{n} → 1 such that α(t_{n}) ∈ B_{D} = ⊔_{X′∈D}B_{X′} for some bounded domain D ⊂ T (S) with X ∈ D. Then the sequence α(t_{n}) ∈ B_{D} converges standardly to ρ, and thus Ψ_{λ}(α(t_{n})) → ∞ by Theorem 5.1. Hence there is no lift of α to P (S) with the starting point Ψ_{λ}(α(0)).

6.3 Obstructions for hol to be a covering map

We first fix our terminology.

Definition 6.5. Let X , Y be topological spaces. A continuous map f : Y → X is said to be a covering map or a weak-covering map if for any x ∈ X there is a neighborhood U of x such that f|_{V} : V → U is a homeomorphism

• for any connected component V of f^{−1}(U), or

• for any y ∈ f^{−1}(x) and some neighborhood V of y,

respectively.

Although the holonomy map hol : P (S) → R(S) is a local homeomorphism and the map hol|_{Q(S)} : Q(S) → QF is a covering map, Hejhal showed the following

Theorem 6.6 ([He, Theorem 8]). The holonomy map hol : P (S) → R(S) is not a covering map onto its image.

We explain this fact in our context. Let ρ ∈ ∂^{±}QF with para(ρ) ≠ ∅. Then there are λ, μ ∈ ML_{ℕ} such that Ψ_{λ}(ρ) ∈ P (S) and Ψ_{μ}(ρ) = ∞. One can find a path α : [0, 1] → R(S) with α(0) ∈ QF and α(1) = ρ which satisfies both conditions in the proofs of Theorems 6.3 and 6.4. Then there is the lift of the path α with the starting point Ψ_{λ}(α(0)) but there is no lift with the starting point Ψ_{μ}(α(0)), which implies that the map hol is not a covering map. We remark that the idea used in the argument above is the same to that of Hejhal [He], but he made use of a path α : [0, 1] → R(S) of Shottky representations.

Moreover we claim in Corollary 6.8 below that the covering map hol|_{Q(S)} : Q(S) → QF is not extended any more even in the sense of weak-covering. The essential observation is the following

Theorem 6.7. Let ρ ∈ ∂^{±}QF. For every λ ∈ ML_{ℕ} with Ψ_{λ}(ρ) ∈ P (S), suppose that Φ_{λ} : U_{λ} → P (S), ρ ↦ Ψ_{λ}(ρ) is a univalent local branch of hol^{−1} defined on a neighborhood U_{λ} of ρ. Then ρ can not be an interior point of ⋂_{λ} U_{λ}, where the intersection is taken over all λ ∈ ML_{ℕ} with Ψ_{λ}(ρ) ∈ P (S).

Proof. We may assume that ρ ∈ ∂B_{X} for some X ∈ T (S). Then there exists a sequence ρ_{n} of maximal cusps converging to ρ in ∂B_{X} for which the parabolic locus λ_{n} = para(ρ_{n}) ∈ ML_{ℕ} has no parallel component in common with para(ρ) (which is possibly empty) for every n (see [It2, Theorem 5.5]). Since Ψ_{λ_{n}}(ρ) ≠ ∞, there exists a univalent local branch Φ_{λ_{n}} : U_{λ_{n}} → P (S), ρ ↦ Ψ_{λ_{n}}(ρ) for every n. On the other hand, since Ψ_{λ_{n}}(ρ_{n}) = ∞, ρ_{n} ∉ U_{λ_{n}} for every n. Therefore we have obtained the result.

Corollary 6.8. For any open neighborhood O of QF in R(S), the map hol|_{hol^{−1}(O)} : hol^{−1}(O) → O is not a weak-covering map.

Proof. For any ρ ∈ ∂QF and its neighborhood U contained in O, one can find a boundary b-group ρ′ ∈ ∂^{±}QF in U. Then the result follows from the above theorem.

6.4 Further continuity

We extend our definition of standard convergence to sequences in \overline{QF}. Let K be a compact subset of T (S) and K̄ ⊂ T (S̄) its complex conjugation. We set B_{K} = ⨆_{X∈K} B_{X} and BK̄ = ⨆_{X̄∈K̄} BX̄. Recall that a sequence ρ_{n} ∈ QF converges standardly to ρ_{∞} ∈ ∂QF if there is a compact subset K ⊂ T (S) such that ρ_{n} ∈ B_{K} ∪ BK̄ for all n. Similarly we say that a sequence ρ_{n} ∈ \overline{QF} converges standardly to ρ_{∞} ∈ ∂QF if there is a compact subset K ⊂ T (S) such that all ρ_{n} are contained in the closure of B_{K} ∪ BK̄ in R(S). Note that the limit of a standard convergent sequence in \overline{QF} is contained in ∂^{±}QF and that if a sequence ρ_{n} ∈ ∂QF converges standardly to ∂^{s}QF then ρ_{n} ∈ ∂^{s}QF for all large enough n, where s denotes + or −.

Theorem 6.9. Let ρ_{n} → ρ_{∞} be a standard convergent sequence in \overline{QF}. Then the sequence Ψ_{λ}(ρ_{n}) converges to Ψ_{λ}(ρ_{∞}) in P̂(S) for every λ ∈ ML_{ℕ}.

Proof. We only have to consider the case where ρ_{n} ∈ ∂QF for all n. Then ρ_{n} → ρ_{∞} in ∂^{±}QF. If Ψ_{λ}(ρ_{∞}) ∈ P (S) then the result follows from Theorem 6.3. Hence we suppose that Ψ_{λ}(ρ_{∞}) = ∞. (We remark that the argument below also works in the case where Ψ_{λ}(ρ_{∞}) ∈ P (S).) To obtain a contradiction, we assume that the sequence Ψ_{λ}(ρ_{n}) converges to some Σ′ ∈ P (S). Then for each n there exists a sequence ρ_{n,k} ∈ QF converging standardly to ρ_{n} ∈ ∂^{±}QF as k → ∞, which is taken by Ψ_{λ} to a convergent sequence Ψ_{λ}(ρ_{n,k}) → Ψ_{λ}(ρ_{n}) from Theorem 5.1. Then by a diagonal argument, we can choose a sequence ρ′_{n} in QF which converges standardly to ρ_{∞} and which satisfies Ψ_{λ}(ρ′_{n}) → Σ′. On the other hand, Ψ_{λ}(ρ′_{n}) → Ψ_{λ}(ρ_{∞}) = ∞ by Theorem 5.1, which is a contradiction. Now we have obtained the result.

Let \overline{Q_{0}} denote the closure of Q_{0} in P (S), not in P̂(S). From the observation in §3.3, we see that a sequence Σ_{n} ∈ \overline{Q_{0}} converges to Σ_{∞} ∈ ∂Q_{0} in P (S) if and only if the sequence ρ_{Σ_{n}} ∈ QF ⊔ ∂^{+}QF converges standardly to ρ_{Σ_{∞}} ∈ ∂^{+}QF. Hence, as a direct consequence of the above theorem, we obtain the following

Theorem 6.10. The map Gr_{λ} : Q_{0} → Q_{λ} is extended continuously to Gr_{λ} : \overline{Q_{0}} → P̂(S) for each λ ∈ ML_{ℕ}, where \overline{Q_{0}} is the closure of Q_{0} in P (S).

6.5 Discreteness of inverse images

Theorem 6.11. Let K be a given compact subset of T (S) and set QF_{K} = B_{K} ∪BK̄. Then the inverse image

hol^{−1}(QF_{K}) = ⨆_{λ∈ML_{ℕ}} Ψ_{λ}(QF_{K})

of QF_{K} in P (S) is discrete; that is, every connected component of hol^{−1}(QF_{K}) has open neighborhood which is disjoint from any other one.

Proof. Suppose for contradiction that there exist ρ ∈ \overline{QF_{K}} ∩ ∂^{±}QF and λ ∈ ML_{ℕ} such that Ψ_{λ}(ρ) is a limit of a sequence Σ_{n} in hol^{−1}(QF_{K})−Ψ_{λ}(QF_{K}). By Theorem 4.3, we may assume that the sequence Σ_{n} is contained in a component Q_{μ} of Q(S) with λ ≠ μ. Since the sequence ρ_{n} = hol(Σ_{n}) ∈ QF_{K} converges standardly to ρ, we have Σ_{n} = Ψ_{μ}(ρ_{n}) → Ψ_{μ}(ρ) by Theorem 5.1. But this contradicts the fact that Ψ_{λ}(ρ) ≠ Ψ_{μ}(ρ) (Corollary 3.7).

Compare the above theorem with the following

Theorem 6.12 ([It3]). Any pair of connected components of the inverse image

hol^{−1}(QF) = ⨆_{λ∈ML_{ℕ}} Ψ_{λ}(QF)

of QF have intersecting closures in P (S).

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Graduate School of Mathematics, Nagoya University, Nagoya 464-8602, Japan

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