EXOTIC PROJECTIVE STRUCTURES AND BOUNDARY OF QUASI-FUCHSIAN SPACE

KENTARO ITO

Abstract. Let P (S) denote the space of projective structures on a closed surface S. It is known that the subset Q(S) ⊂ P (S) of projective structures with quasi-Fuchsian holonomy has infinitely many connected components. In this paper, we investigate the configuration of these components. In particular, we show that the closure of any exotic component of Q(S) intersects the closure of the standard component of Q(S). As a consequence, Q(S) has connected closure in P (S). We also mention the complexity of the boundary of the quasi-Fuchsian space.

1. Introduction

Let S be an oriented closed surface of genus g > 1. A projective structure on S is a maximal system of local coordinates modeled on the Riemann sphere Ĉ, whose transition functions are Möbious transformations. For a given projective structure on S, we have a pair (f, ρ) of a local homeomorphism f from the universal cover S̃ of S to Ĉ, called a developing map, and a group homomorphism ρ of π_{1}(S) into PSL_{2}(C), called a holonomy representation. Let P (S) denote the space of all (marked) projective structures on S, and let V (S) denote the space of all conjugacy classes of representations of π_{1}(S) into PSL_{2}(C). Holonomy representations give a mapping hol : P (S) → V (S), which is called the holonomy mapping. It is known that the map hol is a local homeomorphism ([13]). The quasi-Fuchsian space QF (S) is the subspace of V (S) consisting of faithful representations whose holonomy images are quasi-Fuchsian groups.

In this paper, we investigate the subset Q(S) = hol^{−1}(QF (S)) of P (S). We say an element of Q(S) is standard if its developing map is injective; otherwise it is exotic. The set of standard projective structures with fixed underlying complex structure is well known as the image of the Teichmüller space under Bers embedding (see [5]). On the other hand, the existence of exotic projective structures was first shown by Maskit [21]. More investigations of exotic projective structures are found in [11], [12], [13], [25], [30] and [32]. As we shall see in Proposition 2.3, each connected component of Q(S) is biholomorphically isomorphic to QF (S). Moreover, as a consequence of the result of Goldman [12], the connected components of Q(S) are in one to one correspondence with the set ML_{Z}(S) of integral points of measured laminations (see 2.4 for precise definition). We denote by Q_{λ} the component of Q(S) corresponding to λ ∈ ML_{Z}(S), where Q_{0} is the component consisting of all standard projective structures.

Date: October 20, 1999.

1991 Mathematics Subject Classification. Primary 30F40; Secondary 30F30, 32G15, 57M50.

Recently McMullen [25, Appendix A] discovered the next phenomenon.

Theorem 1.1 (McMullen). There exists a sequence of exotic projective structures which converges to an element of ∂Q_{0}.

This phenomenon deeply depends on the following phenomenon in the theory of Kleinian groups: There is a sequence of quasi-Fuchsian groups whose algebraic limit is properly contained in the geometric limit. Such a sequence of quasi-Fuchsian groups used in the proof of Theorem 1.1 is essentially constructed in Anderson and Canary [2]. Related topics can be found in [7], [8] and [18].

Theorem 1.1 brings up naturally the following questions;

(1) Can we characterize the points on ∂Q_{0} which are limits of exotic projective structures?

(2) Can we characterize how a sequence of exotic projective structures can converge to a point of ∂Q_{0}?

As for the first question, we first remark that the holonomy image of the limit projective structure constructed in Theorem 1.1 is a regular b-group. Moreover, we will show that any element of ∂Q_{0} whose holonomy image is a degenerate group without accidental parabolics can not be an accumulation point of exotic projective structures (Corollary 3.5). This result has been announced by Matsuzaki already. We discuss this topic in Section 3.

In this paper, we are mainly concerned with the second question. Our first main result shows that there exists a sequence in any exotic component Q_{λ} which converges to a point of ∂Q_{0}.

Theorem A. For any λ ∈ ML_{Z}(S), we have \overline{Q_{0}} ∩ \overline{Q_{λ}} ≠ ∅. Especially, the closure of Q(S) in P (S) is connected.

The proof depends on the following observation: For a converging sequence of exotic projective structures, what component of Q(S) the sequence is contained is closedly related to how the algebraic limit is contained in the geometric limit of corresponding holonomy representations. In Section 5, we develop a technique to construct some sequences of representations with the same algebraic limit but with mutually distinct geometric limits. Using this technique, we can extend Theorem A to the following form.

Theorem B. For any finite set {λ_{i}}_{i=1}^{m} of ML_{Z}(S) satisfying i(λ_{j}, λ_{k}) = 0 for all j, k ∈ \{1, …, m\}, we have \overline{Q_{0}} ∩ \overline{Q_{λ_{1}}} ∩ ⋯ ∩ \overline{Q_{λ_{m}}} ≠ ∅. Where i(·, ·) denotes the geometric intersection number.

Since the holonomy mapping hol : P (S) → V (S) is a local homeomorphism, the complexity of Q(S) at ∂Q_{0} is inherited by the complexity of ∂QF (S). In fact, Theorem 1.1 implies that the closure of QF (S) in V (S) is not a manifold with boundary (Theorem A.1 in [25]). This shows the advantage of consideration of projective structures to investigate the quasi-Fuchsian space. As a consequence of Theorem B, we obtain the following Theorem C. For any positive integer n ∈ N, there exists a point [ρ] of ∂QF (S) such that U ∩ QF (S) consists of more than n components for any sufficiently small neighborhood U of [ρ].

Theorem A and B can be viewed as the projective structure anlogues of the works of Anderson and Canary [2] and Anderson, Canary and McCullough [3] in characterizing when components of the set of discrete faithful representations of finitely generated group G into PSL_{2}(C) have intersecting closures. Theorem C also can be viewed as the analogue of their work, while Theorem C describe how the closure of a unique component of the set of discrete faithful representations intersects itself.

This paper is organized as follows: In Section 2, we provide detailed definitions and basic properties of the spaces and maps with which we will be concerned. In Section 3, we investigate the relationship between sequences of exotic projective structures and algebraic and geometric limits of their holonomy representations. Section 4 and Section 5 are devoted to the proofs of Theorem A and Theorem B, respectively.

Acknowledgements. The author would like to express his gratitude to Hiroshige Shiga and Katsuhiko Matsuzaki for their encouragement and useful suggestions. He also appreciates the referee’s valuable comments on the previous manuscript of this paper.

2. Notation and basic facts

A Kleinian group G is a discrete subgroup of PSL_{2}(C), which acts on the hyperbolic space H^{3} as isometries, and on the sphere at infinity S_{∞}^{2} = Ĉ as conformal automorphisms. The region of discontinuity Ω(G) is the largest open subset of Ĉ on which G acts properly discontinuously, and the limit set Λ(G) of G is its compliment Ĉ − Ω(G). The quatient manifold N_{G} = H^{3} ∪ Ω(G)/G is called the Kleinian manifold of G. A quasi-Fuchsian group is a Kleinian group whose limit set is a Jordan curve and which contains no element interchanging the two components of its region of discontinuity. A quasi-Fuchsian group is obtained by a quasi-conformal deformation of a Fuchsian group.

2.1. Beltrami differentials. For a given Kleinian group G with Ω(G) ≠ ∅, a measurable function μ on Ĉ is called a Beltrami differential for G if

μ(g(z))\overline{g′(z)} = μ(z)g′(z)

holds for a.e. z ∈ Ĉ and for all g ∈ G. The space of all Beltrami differentials μ for G whose essential sup-norm satisfying ||μ||_{∞} < 1 is denoted by Belt(G)_{1}. For a given element μ ∈ Belt(G)_{1}, there exists a unique quasi-conformal map w : Ĉ → Ĉ satisfying the Beltrami equation wz̄ = μw_{z} and fixing 0, 1 and ∞. Throughout this paper, this normalized quasi-conformal map with the Beltrami coefficient μ will be denoted by w_{μ}. For more information about quasi-conformal map, see Lehto-Virtanen [20] for example. The quasi-conformal map w_{μ} induces a group isomorphism Θ_{μ} of G into PSL_{2}(C) satisfying w_{μ} ∘ g = Θ_{μ}(g) ∘ w_{μ} for all g ∈ G. For a G-invariant open set U ⊂ Ω(G), we denote by Belt(U, G)_{1} the subset of Belt(G)_{1} consisting of all elements with support in U.

2.2. Teichmüller space. Let S be an oriented closed surface of genus g > 1. The Teichmüller space T (S) consists of pairs (f, X), where X is a Riemann surface and f : S → X is an orientation preserving diffeomorphism. Two pairs (f_{1}, X_{1}) and (f_{2}, X_{2}) represent the same point in T (S), if there is a holomorphic isomorphism h : X_{1} → X_{2} such that h ∘ f_{1} is isotopic to f_{2}. It is known that the space T (S) is a 3g − 3 dimensional complex manifold, diffeomorphic to a cell.

There is another but equivalent definition of the Teichmüller space. We fix a Fuchsian group Γ acting on the upper half plane H = {z ∈ C : Imz > 0} such that S = H/Γ. Two elements μ, ν ∈ Belt(H, Γ)_{1} are called equivalent if w_{μ}|∂H = w_{ν}|∂H, or equivalently, Θ_{μ} = Θ_{ν}. The Teichmüller space T (Γ) (or T (S)) is the space of equivalence classes [μ] of elements μ in Belt(H, Γ)_{1}. For each t = [μ] ∈ T (S), let Γ_{t} denote the quasi-Fuchsian group Θ_{μ}(Γ), whose region of discontinuity is a union of H_{t} = w_{μ}(H) and H_{t}^{∗} = w_{μ}(H^{∗}), where H^{∗} = {z ∈ C : Imz < 0} is the lower half plane. The quasi-conformal map w_{μ}|H : H → H_{t} descends to a quasi-conformal map g_{t} : S = H/Γ → S_{t} = H_{t}/Γ_{t}, such that the pair (g_{t}, S_{t}) represents t ∈ T (S).

2.3. The space of projective structures. For a given projective structure on S, we obtain a local homeomorphism f : S̃ → Ĉ by lifting the structure to the universal cover S̃ of S and continuing the coordinates analytically. This map f is called a developing map of the projective structure. A developing map f induces a group homomorphism ρ : π_{1}(S) → PSL_{2}(C) satisfying f ∘ g = ρ(g) ∘ f for all g ∈ π_{1}(S), which is called a holonomy representation. This pair (f, ρ) is called a projective pair. Note that a projective structure determines a projective pair (f, ρ) uniquely up to the action of PSL_{2}(C); the action is defined by

(f, ρ) ↦ (A ∘ f, A ∘ ρ ∘ A^{−1})

for A ∈ PSL_{2}(C).

For each t ∈ T (S), let B_{2}(H_{t}, Γ_{t}) denote the space of holomorphic quadratic differentials for Γ_{t} on H_{t}, whose element is a holomorphic function φ on H_{t} satisfying

φ(γ(z))γ′(z)^{2} = φ(z)

for all γ ∈ Γ_{t}, z ∈ H_{t}. The space B_{2}(H_{t}, Γ_{t}) is a 3g − 3 dimensional complex vector space.

A projective structure determines naturally its underlying complex structure. The set of all projective structures with an underlying complex structure S_{t} = H_{t}/Γ_{t} is parametrized by B_{2}(H_{t}, Γ_{t}) as follows. For any projective structure on S_{t}, let f : H_{t} → Ĉ be its developing map. Then we assign an element S(f) ∈ B_{2}(H_{t}, Γ_{t}) to this projective structure, where S(f) is the Schwarzian derivative of f defined by S(f) = (f′′/f′)′ − 1/2(f′′/f′)^{2}. Conversely, for any element φ ∈ B_{2}(H_{t}, Γ_{t}), there is a holomorphic map f : H_{t} → Ĉ satisfying S(f) = φ, which descends to a projective structure on S_{t}. Here and hereafter, with this identification, we regard a pair (t, φ) as a marked projective structure, where t ∈ T (S) and φ ∈ B_{2}(H_{t}, Γ_{t}).

Let P (S) denote the holomorphic cotangent bundle over T (S) with projection π : P (S) → T (S). Then, each fiber π^{−1}(t) over t ∈ T (S) is identified with B_{2}(H_{t}, Γ_{t}). We regard P (S) as the space of marked projective structures. For any projective structure (t, φ) ∈ P (S), let f_{t,φ} : H_{t} → Ĉ denote its developing map and ρ̄_{t,φ} : Γ_{t} → PSL_{2}(C) its holonomy representation satisfying f_{t,φ} ∘ γ = ρ̄_{t,φ}(γ) ∘ f_{t,φ} for any γ ∈ Γ_{t}. A representation ρ_{t,φ} = ρ̄_{t,φ} ∘ Θ_{μ} of π_{1}(S) = Γ into PSL_{2}(C) is also called a holonomy representation, where μ ∈ Belt(H, Γ)_{1} is a representative of t ∈ T (S). It is known that, for any projective structure (t, φ), the holonomy image ρ_{t,φ}(π_{1}(S)) is a non-abelian subgroup of PSL_{2}(C).

A sequence {ρ_{n}} of representations of π_{1}(S) into PSL_{2}(C) is said to converge algebraically to ρ if ρ_{n}(g) converges to ρ(g) in PSL_{2}(C) for any g ∈ π_{1}(S). Let V (S) denote the space of all conjugacy classes [ρ] of representations ρ : π_{1}(S) → PSL_{2}(C) such that ρ(π_{1}(S)) is non-abelian. A sequence {[ρ_{n}]} in V (S) converges to [ρ] ∈ V (S) if there is a sequence of representatives of {[ρ_{n}]} which converges algebraically to a representative of [ρ]. It is known that V (S) is 6g − 6 dimensional complex manifold (see, for example, [23, Theorem 4.21]). The holonomy map

hol : P (S) → V (S)

is defined by hol(t, φ) = [ρ_{t,φ}]. The basic fact is that the holonomy map is a holomorphic local homeomorphism ([13], see also [9] and [14]).

2.4. Grafting. Let S denote the set of homotopy classes of non-trivial simple closed curves on S. By abuse of notation, we also denote a representative of C ∈ S by C. Let ML_{Z}(S) denote the set of integral points of measured laminations on S. Namely, each element λ ∈ ML_{Z}(S) is written as a formal summation ∑ n_{j}C_{j}, where {n_{j}} are positive integers and {C_{j}} are mutually distinct disjoint elements in S. We shall contain the “zero” measured lamination in ML_{Z}(S).

Let X be a Riemann surface marked by S. A canonical projective structure on S is provideded by the Fuchsian uniformization X = H/G, where H is the upper half plane and G is a Fuchsian group acting on H. For any element λ = ∑ n_{j}C_{j} of ML_{Z}(S), we can construct a new projective structure on S by cutting X along each C_{j} and “grafting” some projective annulus at each cut locus. More precisely, let l_{X}(C_{j}) denote the hyperbolic length of geodesic representative of C_{j} on X and let

A_{j} = (C − iR^{+})/ ⟨z ↦ e^{l_{X}(C_{j})}z⟩

be the annulus equipped with natural projective structure. Then new projective structure on S is obtained by cutting X along geodesic representatives of each C_{j} and inserting A_{j} for n_{j}-times. This new projective structure is said to be obtained by grafting along λ. (See [12], [17], [25] and [32] for more information.)

We explain this grafting operation in the contex of complex analysis, for our later use. The following construction is due to Maskit [21] and our explanation is based on that of Gallo [11]. An element (t, φ) ∈ P (S) is called a Fuchsian projective structure if f_{t,φ} is injective and ρ_{t,φ}(π_{1}(S)) is a Fuchsian group.

For simplicity, we first explain for the case that λ ∈ ML_{Z}(S) is a simple closed curve with weight 1. Take a Fuchsian projective structure (t, φ) and an element C ∈ S. We denote the holonomy image ρ_{t,φ}(π_{1}(S)) by G. We may assume that the image of developing map f_{t,φ}(H_{t}) coincides with the upper half plane H, and that the imaginary axis iR^{+} projects onto C via the covering map H → S_{t} = H/G. Choose α ∈ (0, π/2) so that B_{α} = {z ∈ H : π/2 − α < arg z < π/2 + α} projects onto a collar about C in S_{t}. Take a C^{1} homeomorphism

v : [π/2 − α, π/2 + α] → [π/2 − α, 5π/2 + α]

satisfying v(π/2 − α) = π/2 − α, v(π/2 + α) = 5π/2 + α, and v′(π/2 − α) = v′(π/2+α) = 1. We now define a local homeomorphism W : H → Ĉ as follows. Let g ∈ G be a generator for the stabilizer of iR^{+} in G and set D_{α} = ⋃_{h∈G/⟨g⟩} h(B_{α}). Let W (z) = z for z ∈ H − D_{α}, and W (z) = re^{iv(θ)} for z = re^{iθ} ∈ B_{α}. For z ∈ h(B_{α}) with some h ∈ G, let W (z) = h∘W ∘h^{−1}(z). One can easily verify that W ∘h = h∘W for all h ∈ G and hence that μ = Wz̄/W_{z} ∈ Belt(H, G)_{1}.

Now let μ̂ ∈ Belt(H_{t}, Γ_{t})_{1} be the pull back f_{t,φ}^{∗}(μ) of μ via developing map f_{t,φ}, which is defined by

f_{t,φ}^{∗}(μ) = (μ ∘ f_{t,φ})\overline{f′_{t,φ}}/f′_{t,φ}.

Let ν ∈ Belt(H, Γ)_{1} be a representative of t ∈ T (S). Let t′ ∈ T (S) be the equivalent class of the Beltrami coefficient ν′ of the quasi-conformal map wμ̂∘w_{ν} : Ĉ → Ĉ. Since W ∘ f_{t,φ} ∘ (wμ̂)^{−1} is locally conformal on H_{t′} = wμ̂(H_{t}), we can take its Schwarzian derivative φ′ ∈ B_{2}(H_{t′}, Γ_{t′}). Now we obtain a new projective structure (t′, φ′) with surjective developing map f_{t′,φ′} which commutes the following diadram;

H ⟶^{w_{ν}} H_{t} ⟶^{f_{t,φ}} Ĉ
id↓  ↓^{wμ̂}  ↓^{W}
H ⟶^{w_{ν′}} H_{t′} ⟶^{f_{t′,φ′}} Ĉ.

Moreover one can see that ρ_{t′,φ′} = ρ_{t,φ} from the fact that W ∘ h = h ∘ W for all h ∈ G. It is shown in [11, Lemma 3.1] that the element (t′, φ′) ∈ P (S) does not depend on the choice of α, v, and ν. The projective structure (t′, φ′) is said to be obtained from (t, φ) by grafting along C and is denoted by Gr_{C}(t, φ).

An important remark is that, for (t, φ) = Gr_{C}(t, φ), the subset f_{t′,φ′}^{−1}(Λ(G))/Γ_{t′} in S_{t′} consists of two simple closed curves each of which is homotopic to C (see [12]). This can be seen as follows: We first note that the limit set Λ(G) of G = ρ_{t′,φ′}(π_{1}(S)) coincides with R ∪ {∞}. Since f_{t,φ}^{−1}(B_{α}) projects onto a collar about C in S_{t} via the covering map H_{t} → S_{t}, wμ̂(f_{t,φ}^{−1}(B_{α})) is also projected onto a collar about C in S_{t′} via the covering map H_{t′} → S_{t′}. Since the developing map f_{t′,φ′} maps wμ̂(f_{t,φ}^{−1}(B_{α})) onto the multi-sheeted domain {z ∈ Ĉ : π/2 − α < arg z < 5π/2 + α}, wμ̂(f_{t,φ}^{−1}(B_{α})) ∩ f_{t′,φ′}^{−1}(Λ(G)) consists of two connected components each of which is projected onto a simple closed curve homotopic to C.

The grafting operation can be naturally extended to ML_{Z}(S). For example, if λ = nC ∈ ML_{Z}(S), we only have to change v into v : [π/2 − α, π/2 + α] → [π/2 − α, 2nπ + π/2 + α]. Let Gr_{λ}(t, φ) denote the projective structure obtaind from (t, φ) by grafting along λ ∈ ML_{Z}(S).

As we have observed, the grafting operator does not change holonomy representations. Conversely, Goldman [12] showed that all projective structures with Fuchsian holonomy are obtained by grafting.

Theorem 2.1 (Goldman). For any Fuchsian projective structure (t, φ),

hol^{−1}(hol(t, φ)) = {Gr_{λ}(t, φ)}_{λ∈ML_{Z}(S)}.

2.5. Quasi-conformal deformations of projective structures. Let AH(S) denote the subset of V (S) consisting of discrete faithful representations. The quasi-Fuchsian space QF (S) is the subset of AH(S) consisting of faithful representations whose images are quasi-Fuchsian groups. It is known that AH(S) is a closed subset in V (S)(see [15, Theorem 1]) and that the interior intAH(S) of AH(S) coincides with QF (S)(see [31, Theorem A]). (It is conjectured that \overline{QF (S)} = AH(S), which is so called Bers-Thurston conjecture.) We denote the subset hol^{−1}(AH(S)) of P (S) by K(S) and hol^{−1}(QF (S)) by Q(S). Then, since the holonomy map hol : P (S) → V (S) is a local homeomorphism, one obtains intK(S) = Q(S).

We now introduce the notion of a quasi-conformal deformation of a projective structure with quasi-Fuchsian holonomy, which was developed by Shiga and Tanigawa in [30]. Fix an element (t, φ) ∈ Q(S) and denote its holonomy image ρ_{t,φ}(π_{1}(S)) by G. For each μ ∈ Belt(G)_{1}, we take the pull back μ̂ = f_{t,φ}^{∗}(μ) ∈ Belt(H_{t}, Γ_{t})_{1} of μ, which determines a new point t′ ∈ T (S) in the same manner described in 2.4. Since w_{μ} ∘ f_{t,φ} ∘ (wμ̂)^{−1} is locally conformal on H_{t′}, we can take its Schwarzian derivative φ′ ∈ B_{2}(H_{t′}, Γ_{t′}). Now we obtain a new projective structure (t′, φ′), a quasi-conformal deformation of (t, φ), satisfying

\begin{cases}f_{t′,φ′} = w_{μ} ∘ f_{t,φ} ∘ (wμ̂|H_{t})^{−1} : H_{t′} → Ĉ, \\ ρ_{t′,φ′} = Θ_{μ} ∘ ρ_{t,φ} : π_{1}(S) → PSL_{2}(C), \end{cases}

where Θ_{μ} is the group isomorphism of G into PSL_{2}(C) induced by w_{μ}. We define a map

Ψ̃_{t,φ} : Belt(G)_{1} → P (S)

by Ψ̃_{t,φ}(μ) = (t′, φ′). Two elements μ, ν ∈ Belt(G)_{1} are said to be equivalent if Θ_{μ} is PSL_{2}(C)-conjugate to Θ_{ν}. The quotient space of Belt(G)_{1} by this equivalent relation can be naturally identified with the quasi-Fuchsian space QF (S).

Lemma 2.2 (cf. [30]). The map Ψ̃_{t,φ} descends to a map

Ψ_{t,φ} : QF (S) → P (S).

Proof. For any equivalent two elements μ, ν ∈ Belt(G)_{1}, we will show that Ψ̃_{t,φ}(μ) = Ψ̃_{t,φ}(ν). The same argument in [10] reveals that there is a path c_{τ}, τ ∈ [0, 1] in Belt(G)_{1} jointing μ and ν and contained in the equivalence class [μ] of μ. Note that hol ∘ Ψ̃_{t,φ}(c_{τ}) is constant on τ ∈ [0, 1]. Since the map hol is a local homeomorphism, it implies that Ψ̃_{t,φ}(c_{τ}) is constant on τ ∈ [0, 1] and that Ψ̃_{t,φ}(μ) = Ψ̃_{t,φ}(ν).

Using the map in Lemma 2.2, we can show the following Proposition 2.3. For any connected component Q of Q(S),

hol|Q : Q → QF (S)

is a biholomorphic map. Moreover, Ψ_{t,φ} = (hol|Q)^{−1} holds for any (t, φ) ∈ Q. Therefore the map Ψ_{t,φ} does not depend on the choice of (t, φ) ∈ Q.

Proof. To show that hol|Q is biholomorphic, it suffices to show that hol|Q is bijective, since the map hol is a local biholomorphism. Fix an element (t, φ) ∈ Q. Note that Ψ_{t,φ}(QF (S)) ⊂ Q, since QF (S) is connected and Ψ_{t,φ} is continuous. It can be easily seen by definition that (hol|Q) ∘ Ψ_{t,φ} is the identity map of QF (S). Therefore we only have to show that Ψ_{t,φ} ∘ (hol|Q) is the identity map of Q. To this end, we will show that a subset

Q′ = {(s, ψ) ∈ Q : (s, ψ) = Ψ_{t,φ} ∘ hol(s, ψ)}

of Q is non-empty, open and closed. Since Ψ_{t,φ} ∘ (hol|Q) is cotinuous, Q′ is closed. Moreover, Ψ_{t,φ}([ρ_{t,φ}]) = (t, φ) implies (t, φ) ∈ Q′, and hence Q′ ≠ ∅. Take (s, ψ) ∈ Q′ and its neighborhood U in Q such that hol|U is injective. Let V be a neighborhood of (s, ψ) contained in U and satisfying Ψ_{t,φ} ∘ hol(V) ⊂ U. Note that, for any (s′, ψ′) ∈ V, hol(Ψ_{t,φ} ∘ hol(s′, ψ′)) = (hol ∘ Ψ_{t,φ}) ∘ hol(s′, ψ′) = hol(s′, ψ′). Since both Ψ_{t,φ} ∘ hol(s′, ψ′) and (s′, ψ′) are contained in U and hol|U is injective, we have Ψ_{t,φ} ∘ hol(s′, ψ′) = (s′, ψ′) for all (s′, ψ′) ∈ V. Therefore Q′ is open.

2.6. Components of Q(S). Take a Fuchsian projective structure (t, φ). By Proposition 2.3, each connected component of Q(S) contains a unique projective structure whose holonomy representation coincides with [ρ_{t,φ}]. But, from Theorem 2.1, these projective structures are written in the form {Gr_{λ}(t, φ)}_{λ∈ML_{Z}(S)}. Therefore we obtain the decomposition of Q(S) into its connected components;

Q(S) = ∐_{λ∈ML_{Z}(S)} Q_{λ},

where Q_{λ} is the component containing Gr_{λ}(t, φ). Note that this suffix does not depend on the choice of Fuchsian projective structure (t, φ). Recall that an element of Q(S) is called standard if its developing map is injective; otherwise exotic. Since any element of Q_{λ} is obtained from Gr_{λ}(t, φ) by a quasi-conformal deformation, one can easily see that Q_{0} is the component consisting of all standard projective structures, and that any element of Q_{λ}(λ ≠ 0) is an exotic projective structure. Note that, since Gr_{λ}(t, φ)(λ ≠ 0) has surjective developing map, any exotic projective structure has surjective developing map. Moreover, we can characterize the component of Q(S) in which an element (t, φ) of Q(S) is contained, as follows.

Lemma 2.4. Take an element (t, φ) ∈ Q(S) and denote ρ_{t,φ}(π_{1}(S)) by G. Then (t, φ) is contained in a component Q_{λ} corresponding to λ = ∑ n_{j}C_{j} ∈ ML_{Z}(S) if and only if the subset f_{t,φ}^{−1}(Λ(G))/Γ_{t} of S_{t} consists of disjoint unions of 2n_{j} simple closed curves each of which is homotopic to C_{j} for all j.

3. Exotic projective structures and limits of representations

In this section, we investigate the relationship between sequences of exotic projective structures and algebraic and geometric limits of their holonomy representations. We begin with the definition of geometric convergence of Kleinian groups.

Definition 3.1. Let X be a locally compact Hausdorff space. We denote by C(X) the set of all closed subset of X. A sequence {A_{n}} of closed subsets of X converges to a closed subset A ⊂ X in the Hausdorff topology on C(X) if every element x ∈ A is the limit of a sequence {x_{n} ∈ A_{n}} and if every accumulation point of every sequence {x_{n} ∈ A_{n}} lies in A. A sequence of Kleinian groups {G_{n}} is said to converge geometrically to a group Ĝ if {G_{n}} converges to Ĝ in the Hausdorff topology on C(PSL_{2}(C)).

We recall some basic facts on the convergence of representations. Let {ρ_{n}} be a sequence of discrete faithful representations of π_{1}(S) into PSL_{2}(C) which converges algebraically to ρ_{∞}. Then ρ_{∞} is also a discrete faithful representation (see [15, Theorem 1]). Moreover, there is a subsequence of {G_{n} = ρ_{n}(π_{1}(S))} converging geometrically to a kleinian group Ĝ which contains G_{∞} = ρ_{∞}(π_{1}(S)) (see [16, proposition 3.8]). The following theorem is due to Kerckhoff and Thurston [18, Corollary 2.2].

Theorem 3.2 (Kerckhoff-Thurston). Let {ρ_{n} : π_{1}(S) → PSL_{2}(C)} be an algebraically convergent sequence of faithful representations whose images {G_{n} = ρ_{n}(π_{1}(S))} are quasi-Fuchsian groups. Assume that {G_{n}} converges geometrically to Ĝ. Then, {Λ(G_{n})} converges to Λ(Ĝ) in the Hausdorff topology on C(Ĉ).

The following Lemma 3.3 plays an important role in this paper, especially in the proof of Theorem A. Since the situation under which we consider Lemma 3.3 is somewhat complicated, we first describe it:

Let {(t_{n}, φ_{n})} be a sequence in Q(S) converging to an element (t, φ) in K(S). Take a sequence of projective pairs {(f_{t_{n},φ_{n}}, ρ_{t_{n},φ_{n}})} and a projective pair (f_{t,φ}, ρ_{t,φ}) such that {ρ_{t_{n},φ_{n}}} converges algebraically to ρ_{t,φ}. Put G_{n} = ρ_{t_{n},φ_{n}}(π_{1}(S)) and G_{∞} = ρ_{t,φ}(π_{1}(S)). Moreover, we assume that {G_{n}} converges geometrically to a Kleinian group Ĝ. Since {t_{n}} converges to t in T (S), one can take a smooth quasi-conformal map ω_{n} : S_{t} → S_{t_{n}} such that ω_{n} ∘ g_{t} is homotopic to g_{t_{n}}, where g_{t} : S → S_{t} and g_{t_{n}} : S → S_{t_{n}} are markings for t and t_{n} respectively, and that the maximal dilatation

\frac{1 + ||(ω_{n})z̄/(ω_{n})_{z}||_{∞}}{1 − ||(ω_{n})z̄/(ω_{n})_{z}||_{∞}}

of ω_{n} tends to 1 as n → ∞.

In this situation, we have the following

Lemma 3.3. The sequence {ω_{n}^{−1}(f_{t_{n},φ_{n}}^{−1}(Λ(G_{n}))/Γ_{t_{n}})} converges to f_{t,φ}^{−1}(Λ(Ĝ))/Γ_{t} in the Hausdorff topology on C(S_{t}).

Proof. We first observe that {f_{t_{n},φ_{n}}} converges to f_{t,φ} locally uniformly in H_{t}. Since {(t_{n}, φ_{n})} converges to (t, φ), {t_{n}} converges to t in T (S) and {φ_{n}} converges to φ locally uniformly in H_{t}. Therefore, one can take a sequence of projective pairs {(f̌_{t_{n},φ_{n}}, ρ̌_{t_{n},φ_{n}})} and a projective pair {(f̌_{t,φ}, ρ̌_{t,φ})} such that {f̌_{t_{n},φ_{n}}} converges to f̌_{t,φ} locally uniformly in H_{t}. Chose an element A_{n} ∈ PSL_{2}(C) so that f̌_{t_{n},φ_{n}} = A_{n} ∘ f_{t_{n},φ_{n}} holds. Then, since both {ρ_{t_{n},φ_{n}}} and {ρ̌_{t_{n},φ_{n}} = A_{n} ∘ ρ_{t_{n},φ_{n}} ∘ A_{n}^{−1}} are algebraically convergent sequences, {A_{n}} converges to some element A ∈ PSL_{2}(C). Therefore, {f_{t_{n},φ_{n}}} also converges to f_{t,φ} locally uniformly in H_{t}.

Take a sequence of lifts ω̃_{n} : H_{t} → H_{t_{n}} of ω_{n} which converges to the identity locally uniformly in H_{t}. Then {f_{t_{n},φ_{n}} ∘ ω̃_{n}} also converges to f_{t,φ} locally uniformly in H_{t}. Since, from Theorem 3.2, {Λ(G_{n})} converges to Λ(Ĝ) in the Hausdorff topology on C(Ĉ), one can easily check that {ω̃_{n}^{−1}(f_{t_{n},φ_{n}}^{−1}(Λ(G_{n})))} converges to f_{t,φ}^{−1}(Λ(Ĝ)) in the Hausdorff topology on C(H_{t}). This implies that the sequence {ω_{n}^{−1}(f_{t_{n},φ_{n}}^{−1}(Λ(G_{n}))/Γ_{t_{n}})} converges to f_{t,φ}^{−1}(Λ(Ĝ))/Γ_{t} in the Hausdorff topology on C(S_{t}).

Remark. The above lemma implies that the shape of f_{t,φ}^{−1}(Λ(Ĝ))/Γ_{t} restricts the shape of f_{t_{n},φ_{n}}^{−1}(Λ(G_{n}))/Γ_{t_{n}}, and hence the component of Q(S) in which (t_{n}, φ_{n}) is contained by Lemma 2.4. This is the fundamental idea of the proof of Theorem A.

A Kleinian group G is called geometrically finite if it has a finite-sided fundamental domain in H^{3}. A Kleinian group G is said to be a b-group if it has the only one simply connected invariant component of Ω(G), which is denoted by Ω_{0}(G). A geometrically finite b-group is said to be regular. A degenerate group is a b-group with Ω_{0}(G) = Ω(G). For a b-group G, take a Riemann mapping f : Ω_{0}(G) → H, which induce a group isomorphism χ_{f} : G → fGf^{−1}. An accidental parabolic element g in G is a parabolic element such that χ_{f}(g) is a loxodromic element in fGf^{−1}.

Let U(S) denote the subset of P (S) consisting of all projective structures whose developing maps are injective. Then U(S) is closed in P (S), containing Q_{0}, and contained in K(S). Since intAH(S) = QF (S), one can easily see that intU(S) = Q_{0}, and that ∂Q_{0} ⊂ ∂U(S). Note that, for an element (t, φ) in ∂U(S), its holonomy image G = ρ_{t,φ}(π_{1}(S)) is a b-group with an invariant component f_{t,φ}(H_{t}) = Ω_{0}(G) (see [19]).

Using Lemma 3.3, we can characterize a sequence of exotic projective structures converging to an element of ∂U(S) by algebraic and geometric limits of their holonomy representations. Proposition 3.4. In the same situation in Lemma 3.3, with additional assumption that (t, φ) ∈ ∂U(S), the followings are equivalent;

(1) (t_{n}, φ_{n}) are exotic projective structures for large enough n,

(2) Ω_{0}(G_{∞}) ∩ Λ(Ĝ) ≠ ∅. Remark. The “(2) ⇒ (1)” part of Proposition 3.4 is due to McMullen [25]. In fact, he constructs a sequence of representations satisfying (2), and apply the “(2) ⇒ (1)” part to show Theorem 1.1. Later, we will explain his arguments more precisely. Proof of Proposition 3.4. Recall that any exotic projective structure has surjective developing map. Hence, a projective structure (t_{n}, φ_{n}) ∈ Q(S) is exotic if and only if its developing map f_{t_{n},φ_{n}} is surjective. Therefore the condition (1) is equivalent to the condition that f_{t_{n},φ_{n}}^{−1}(Λ(G_{n}))/Γ_{t_{n}} ≠ ∅ for large enough n. Using Lemma 3.3, it turns out to be equivalent to the condition f_{t,φ}^{−1}(Λ(Ĝ))/Γ_{t} ≠ ∅. But this condition is equivalent to the condition f_{t,φ}(H_{t}) ∩ Λ(Ĝ) ≠ ∅. Since f_{t,φ}(H_{t}) = Ω_{0}(G_{∞}) holds for (t, φ) ∈ ∂U(S), we have completed the proof.

As a consequence of the “(1) ⇒ (2)” part of Proposition 3.4, we have the following assertion due to Matsuzaki (oral communication, see also [23, Section 7.4]).

Corollary 3.5 (Matsuzaki). If a projective structure (t, φ) in ∂U(S) is an accumulation point of exotic projective structures, its holonomy image ρ_{t,φ}(π_{1}(S)) contains accidental parabolics.

Proof. Assume that there is a sequence of exotic projective structures {(t_{n}, φ_{n})} converging to a projective structure (t, φ) ∈ ∂U(S) whose holonomy image contains no accidental parabolics. Then the following theorem due to Thurston (see Ohshika [27, Corollary 6.1]) implies that the geometric limit Ĝ of {G_{n} = ρ_{t_{n},φ_{n}}(π_{1}(S))} coincides with the algebraic limit G_{∞} = ρ_{t,φ}(π_{1}(S)). This contradicts Proposition 3.4.

Theorem 3.6 (Thurston). Let {ρ_{n}} be a sequence of discrete faithful representations of π_{1}(S) into PSL_{2}(C) converging algebraically to ρ_{∞}. Assume that G_{n} = ρ_{n}(π_{1}(S)) and G_{∞} = ρ_{∞}(π_{1}(S)) contain no accidental parabolics. Then {G_{n}} converges geometrically to G_{∞}.

We remark that a b-group with no accidental parabolics is a degenerate group (see [5]). Therefore Corollary 3.5 implies that an element of ∂U(S) whose holonomy image is a degenerate group without accidental parabolics can not be an accumulation point of exotic projective structures. Moreover, we obtain the next corollaries.

Corollary 3.7. The subset of ∂Q_{0} of projective structures which can not be accumulation points of exotic projective structures is dense in ∂Q_{0}.

Proof. A similar argument in [5, p.598] and [24, p.221] reveals that the subset of ∂Q_{0} of projective structures whose holonomy images contain no accidental parabolics is dense in ∂Q_{0}. Then the assertion follows immediately from Corollary 3.5.

Corollary 3.8. For any t_{0} ∈ T (S), there exists a projective structure (t_{0}, φ) in ∂Q_{0} whose holonomy image ρ_{t_{0},φ}(π_{1}(S)) is a regular b-group, such that there is no sequence of exotic projective structures converging to (t_{0}, φ).

Proof. Fix t_{0} ∈ T (S) and consider a subspace T = Q_{0} ∩ B_{2}(H_{t_{0}}, Γ_{t_{0}}) of B_{2}(H_{t_{0}}, Γ_{t_{0}}). The space T is coincident with the image of, so called, the Bers embedding of the Teichmüller space into B_{2}(H_{t_{0}}, Γ_{t_{0}}) (see [29]). Note that the boundary ∂T of T in B_{2}(H_{t_{0}}, Γ_{t_{0}}) is contained in ∂Q_{0} ∩ B_{2}(H_{t_{0}}, Γ_{t_{0}}). It is known by McMullen [24] that the subset ∂′T ⊂ ∂T of projective structures whose holonomy images are regular b-groups is dense in ∂T. Let (t_{0}, ψ) be a point in ∂T such that ρ_{t_{0},ψ}(π_{1}(S)) is a degenerate group and take a sequence {(t_{0}, φ_{n})} in ∂′T converging to (t_{0}, ψ). Now suppose that the assertion were false. Then, by the diagonal argument, one can take a sequence of exotic projective structures converging to (t_{0}, ψ). This contradicts Corollary 3.5.

4. The proof of Theorem A

We first recall some basic facts of the quasi-Fuchsian space. Take a faithful representation ρ : π_{1}(S) → PSL_{2}(C) whose image G = ρ(π_{1}(S)) is a quasi-Fuchsian group. Then the Kleinian manifold N_{G} = (H^{3} ∪ Ω(G))/G is homeomorphic to S × [0, 1] and ∂N_{G} = Ω(G)/G consists of two Riemann surfaces X_{1} and X_{2}. We assume that ∂N_{G} is equipped with the orientation induced from that of Ĉ. Then ∂N_{G} = X_{1} ∪ X_{2}, combined with markings induced from ρ, determines a point in T (S) × T (\overline{S}), where \overline{S} denotes S with its orientation reversed. Moreover, this assignment induces a holomorphic bijection (see [4]),

qf : T (S) × T (\overline{S}) → QF (S).

A subset B_{t} = qf(\{t\} × T (\overline{S})) of QF (S) for some t ∈ T (S) is called a vertical Bers slice. On the other hand, B^{t} = qf(T (S) × \{t\}) for some t ∈ T (\overline{S}) is called a horizontal Bers slice. Note that the bounbary of any vertical Bers slice is contained in hol(∂Q_{0}), while the boundary of any horizontal Bers slice is not.

The mapping class group Mod(S) is the group consisting of isotopy classes of orientation preserving homeomorphism of S. Recall that Mod(S) acts naturally on T (S) and on T (\overline{S}).

We devote the rest of this section to the proof of the following theorem.

Theorem A. For any λ ∈ ML_{Z}(S), we have \overline{Q_{0}} ∩ \overline{Q_{λ}} ≠ ∅. Especially, the closure of Q(S) in P (S) is connected.

The second statement can be easily seen from the first statement. Hence we consentrare our attention to the first statement.

4.1. Proof for special case. Here, we prove Theorem A for the case that λ = C ∈ S. Our aim is to show that there exists a sequence in Q_{C} which converges to an element of ∂Q_{0}. This proceeds as follows.

1. First we review the proof of Theorem 1.1:

Let (u, v) be any pair of Riemann surfaces in T (S)×T (\overline{S}) and τ ∈ Mod(S) be the Dehn twist around C. We will see that the sequence {[ρ_{n}] = qf(τ^{n}u, τ^{2n}v)} converges algebarically to a point [ρ_{∞}] on the boundary of some vertical Bers slice B_{t}. Let (t, φ) be a point of ∂Q_{0} such that hol(t, φ) = [ρ_{∞}]. Since the holonomy map hol : P (S) → V (S) is a local homeomorphism, one can take a sequence {(t_{n}, φ_{n})} in Q(S) converging to (t, φ) and satisfying hol(t_{n}, φ_{n}) = [ρ_{n}]. For large enough n, (t_{n}, φ_{n}) turns out to be exotic and the proof of Theorem 1.1 is completed.

We will show that (t_{n}, φ_{n}) is contained in Q_{C} for large enough n in the following steps.

2. Let {(f_{t_{n},φ_{n}}, ρ_{t_{n},φ_{n}})} and (f_{t,φ}, ρ_{t,φ}) be projective pairs corresponding {(t_{n}, φ_{n})} and (t, φ), respectively, such that {ρ_{t_{n},φ_{n}}} converges algebraically to ρ_{t,φ}. Let Ĝ be the geometric rimit of the sequence {G_{n} = ρ_{t_{n},φ_{n}}(π_{1}(S))}. In Lemma 4.1, we will see that the subset f_{t,φ}^{−1}(Λ(Ĝ))/Γ_{t} in S_{t} consists of two components and is contained in an annulus whose core is homotopic to C. Ricall that {ω_{n}^{−1}(f_{t_{n},φ_{n}}^{−1}(Λ(G_{n}))/Γ_{t_{n}})} converges to f_{t,φ}^{−1}(Λ(Ĝ))/Γ_{t} in the Hausdorff topology on C(S_{t}) by Lemma 3.3. Therefore, one may expect that f_{t_{n},φ_{n}}^{−1}(Λ(G_{n}))/Γ_{t_{n}} consists of two simple closed curves each of which is homotopic to C for large enough n. If it were true, (t_{n}, φ_{n}) is contained in Q_{C} for large enough n by Lemma 2.4, and the proof is completed. We justify the above expectation in the following steps.

3. It is easy to see that all components of f_{t_{n},φ_{n}}^{−1}(Λ(G_{n}))/Γ_{t_{n}} are simple close curves homotopic to C for large enough n (Lemma 4.2). Hence, all what we have to show is that any component of f_{t_{n},φ_{n}}^{−1}(Λ(G_{n}))/Γ_{t_{n}} does not join with another components as n tends to ∞.

4. In Lemma 4.3, we will see that any two components of f_{t_{n},φ_{n}}^{−1}(Λ(G_{n}))/Γ_{t_{n}} are separeted by some annulus whose core is homotopic to C and whose modulus is larger than m_{0}, where m_{0} does not depend on n.

5. We will see that the hyperbolic distance between two boundary components of an annulus in a Riemann surface can be estimated below by using the modulus of the annulus (Lemma 4.4). Hence the hyperbolic distance between any two components of ω_{n}^{−1}(f_{t_{n},φ_{n}}^{−1}(Λ(G_{n}))/Γ_{t_{n}}) is bounded below by some positive constant L > 0 which does not depends on n. Therefore, one can see that f_{t_{n},φ_{n}}^{−1}(Λ(G_{n}))/Γ_{t_{n}} consists of two simple closed curves each of which is homotopic to C for large enough n, and can complete the proof of Theorem A for the case that λ is a simple closed curve.

We now fill in the details.

Step 1. Theorem 1.1 is due to McMullen [25, Appendix A]. In our sketch of this proof, we make use of a variation of Thurston’s hyperbolic Dehn surgery theorem, which is due to Comar [8] (see also [2, Theorem 2.2] and [6]). Related arguments can be found in [7] and [18]. The “adding twist” technique, discovered by Anderson and Canary in [2], also plays an important role in this proof.

Sketch of proof of Theorem 1.1. Let Ĝ be a geometrically finite Kleinian group whose Kleinian manifold NĜ = (H^{3} ∪ Ω(Ĝ))/Ĝ is homeomorphic to S × [0, 1] − C × {1/2}. The existence of such a Kleinian group Ĝ is guaranteed by Thurston’s geometrization theorem (see [26]). Here, the tubular neighborhood of C × {1/2} corresponds to the rank two cusp end of NĜ. We fix a basis ⟨γ, δ⟩ for the fundamental group of the rank two cusp so that γ is homotopic to C × {0} and δ is trivial in S × [0, 1]. By performing (n, 1) Dehn filling on the cusp (n ∈ N), we obtain a sequence of representations {β_{n} : Ĝ → PSL_{2}(C)} which satisfies the following conditions (see [8]);

• G_{n} = β_{n}(Ĝ) is a quasi-Fuchsian group,

• The kernel of β_{n} is normally generated by γ^{n}δ,

• {G_{n}} converges geometrically to Ĝ, and

• {β_{n}} converges algebraically to the identity representation of Ĝ.

S  f_{C}  C × {1/2} C  S × {0} S × [0, 1] S × {1}

Figure 1. The wrapping map f_{C}.

Let f_{0} be the inclusion map S → S × {1/4} ⊂ NĜ and denote by (f_{0})_{∗} the induced group homomorphism of π_{1}(S) into Ĝ. Then we obtain a sequence of faithful representations ρ′_{n} = β_{n} ∘ (f_{0})_{∗} of π_{1}(S) onto quasi-Fuchsian groups G_{n}. By modifying β_{n} slightly, if necessary, we may assume that ∂N_{G_{n}} is conformally isomorphic to ∂NĜ for all n. Then the above representations are expressed as

[ρ′_{n}] = qf(u, τ^{n}v),

where (u, v) ∈ T (S) × T (\overline{S}) is the complex structure on ∂NĜ combined with trivial markings, and τ ∈ Mod(S) is the Dehn twist around C.

Now we add a twist to f_{0}. More precisely, we construct an immersion f_{C} : S → NĜ which is homotopic to f_{0} in S ×[0, 1] but not in S ×[0, 1]−C ×{1/2} in the following way. Let A be a tubular neighborhood of C in S. Then the map f_{C}|(S−A) is defined by f_{C}(x) = (x, 1/4), and the map f_{C}|A is defined so that f_{C}(A) wraps once around the tubular neighborhood of C × {1/2}, see Figure 1. This immersion f_{C} is called the wrapping map associated to C.

Again, we obtain a sequence of faithful representations ρ_{n} = β_{n} ∘ (f_{C})_{∗} of π_{1}(S) onto quasi-Fuchsian groups G_{n}, which can be expressed as

[ρ_{n}] = qf(τ^{n}u, τ^{2n}v).

The sequence {ρ_{n}} converges algebraically to ρ_{∞} = (f_{C})_{∗}. We denote ρ_{∞}(π_{1}(S)) by G_{∞}. We can show that G_{∞} is a regular b-group and, moreover, that [ρ_{∞}] lies on the boundary of some vertical Bers slice B_{t} (see Lemma 4.1(1)). Therefore, there exists an element (t, φ) ∈ ∂Q_{0} with hol(t, φ) = [ρ_{∞}]. Since the holonomy map hol : P (S) → V (S) is a local homeomorphism, one can take a sequence {(t_{n}, φ_{n})} in Q(S) converging to (t, φ) and satisfying hol(t_{n}, φ_{n}) = [ρ_{n}]. Since f_{C} : S → NĜ is

Figure 2. The subset f_{t,φ}^{−1}(Λ(Ĝ))/Γ_{t} in S_{t}.

not homotopic to a map into ∂NĜ, G_{∞} does not represent the fundamental group of either component of ∂NĜ and hence Ω_{0}(G_{∞}) ∩ Λ(Ĝ) ≠ ∅ (see also Lemma 4.1(2)). Therefore, Proposition 3.4 implies that (t_{n}, φ_{n}) are exotic. This completes the proof of Theorem 1.1.

Step 2. Our aim is to show that the sequence {(t_{n}, φ_{n})} constructed above is contained in Q_{C}. To this end, we examine the shape of f_{t,φ}^{−1}(Λ(Ĝ))/Γ_{t} ⊂ S_{t} in the following lemma (see Figure 2).

Lemma 4.1. (1) G_{∞} is a regular b-group. Moreover, [ρ_{∞}] lies on the boundary of

some vertical Bers slice B_{t}.

(2) The subset f_{t,φ}^{−1}(Λ(Ĝ))/Γ_{t} in S_{t} consists of two connected components and is contained in an annulus whose core is homotopic to C.

Proof. During this proof, the reader is adviced to refer Figure 3.

(1) Since G_{∞} is a finitely generated subgroup of the geometrically finite Kleinian group Ĝ with Ω(Ĝ) ≠ ∅, it is also geometrically finite (see, for example, [23, Theorem 3.11]). Moreover, since G_{∞} has a parabolic element corresponding to C, it is not a quasi-Fuchsian group. Therefore, to show that G_{∞} is a regular b-group, we only have to show that Ω(G_{∞}) has a simply connected invariant component.

Let A be a closed annular neighborhood of C in S. By deforming the wrapping map f_{C} : S → NĜ in its homotopy class, we may assume that f_{C} maps S − A onto (S − C) × {0} and that f_{C}(intA) ⊂ intNĜ. We take a lift f̃_{C} : S̃ → H^{3} ∪ Ω(Ĝ) of f_{C} satisfying

f̃_{C} ∘ g = ρ_{∞}(g) ∘ f̃_{C} for all g ∈ π_{1}(S),

where π_{1}(S) is regarded as the covering transformation group of the universal covering map p : S̃ → S. Since the map f_{C} is π_{1}-injective, we may assume that the map f̃_{C} is an embedding, and hence the image f̃_{C}(S̃) of S̃ is simply connected.

Fix a component Ã_{0} of Ã = p^{−1}(A). Let g_{0} ∈ π_{1}(S) be a generator for the stabilizer of Ã_{0} in π_{1}(S). Let ⟨γ′, δ′⟩ be a rank two parabolic subgroup of Ĝ which is conjugate to ⟨γ, δ⟩ in Ĝ and satisfying ρ_{∞}(g_{0}) = γ′. Since ⟨γ′⟩ stabilizes f̃_{C}(Ã_{0}),

R C̃_{1}  C̃_{3}  C̃_{2}

Figure 3. Λ(Ĝ) and Ω_{0}(G_{∞}) (the shaded part).

each of two components C̃_{1}, C̃_{2} of f̃_{C}(∂Ã_{0}) forms a simple closed curve together with the common fixed point of ⟨γ′, δ′⟩. Note that C̃_{1} and C̃_{2} projects onto C ⊂ S ×{0} ⊂ ∂NĜ via the covering map Ω(Ĝ) → ∂NĜ. Since f_{C}(A) wraps once around a tubular neighborhood of C × {1/2} in NĜ, we may assume that C̃_{2} = δ′C̃_{1} holds. Let R be the crescent-like domain in Ĉ lying between C̃_{1} and C̃_{2}, and let D be the domain which is cut out of H^{3} by f̃_{C}(Ã_{0}) and is facing R. Note that R ∩ f̃_{C}(S̃− Ã) = ∅ and that D is precisely invariant under the subgroup ⟨γ′⟩ of G_{∞}, that is, (γ′)^{l}(D) = D for any l ∈ Z and D ∩ g(D) = ∅ for any g ∈ G_{∞} − ⟨γ′⟩. Therefore, f̃_{C} is homotopic to an embedding F : S̃ → Ĉ satisfying

F ∘ g = ρ_{∞}(g) ∘ F for all g ∈ π_{1}(S),

and the homotopy is constant on S̃ − Ã. Since G_{∞} acts on F (S̃) properly discontinuously, there is a component Δ of Ω(G_{∞}) containing F (S̃). One can easily see that Δ coincides with F (S̃) because F : S̃ → Ĉ descends to an homeomorphism S → F (S̃)/G_{∞} ⊂ Δ/G_{∞}. Since F (S̃) is simply connected, G_{∞} is a regular b-group with F (S̃) = Ω_{0}(G_{∞}).

A result of Abikoff [1] implies that any representation whose image is a regular bgroup lies on the boundary of some (vertical or horizontal) Bers slice. Since F : S̃ → Ĉ descends to an orientation preserving homeomorphism S → Ω_{0}(G_{∞})/G_{∞} ⊂ N_{G_{∞}} which induces the representation [ρ_{∞}], one can see that [ρ_{∞}] lies on the boundary of a vertical Bers slice B_{t} for some t ∈ T (S).

(2) We first remark that the limit set Λ(Ĝ) is connected since each component of Ω(Ĝ) is simply connected, the latter can be seen from the fact that each component of ∂N_{G} is incompressible in NĜ.

From the above argument, we have F (Ã) = ⋃_{g∈G_{∞}/⟨γ′⟩} g(R). Since

Ω_{0}(G_{∞}) − ⋃_{g∈G_{∞}/⟨γ′⟩} g(R) = F (S̃) − F (Ã) ⊂ Ω(Ĝ),

we have

Ω_{0}(G_{∞}) ∩ Λ(Ĝ) = ( ⋃_{g∈G_{∞}/⟨γ′⟩} g(R)) ∩ Λ(Ĝ) = ⋃_{g∈G_{∞}/⟨γ′⟩} g(R ∩ Λ(Ĝ)).

Therefore, we concentrate our attention to R ∩ Λ(Ĝ). Let G′ be a subgroup of Ĝ representing the fundamental group of S × {0} ⊂ ∂NĜ. By conjugating G′ in Ĝ, if necessary, we may assume that C̃_{1} ⊂ Ω_{0}(G′) and that C̃_{2} ⊂ Ω_{0}(δ′G′δ′^{−1}). Note that Ω_{0}(G′)∩Ω_{0}(δ′G′δ′^{−1}) = ∅, since δ′ ∉ G′. One can easily see that there exists a unique lift C̃_{3} of C ⊂ S × {1} ⊂ ∂NĜ contained in R and terminating in the common fixed point of ⟨γ′, δ′⟩ at both ends. The curve C̃_{3} divides R into two crescent-like domains R_{1} and R_{2}. Since C̃_{1}, C̃_{2} and C̃_{3} are contained in distinct components of Ω(Ĝ), R_{j} ∩ Λ(Ĝ) ≠ ∅ for j = 1, 2. Moreover, since Λ(Ĝ) is connected, R_{j} ∩ Λ(Ĝ), j = 1, 2, is also connected. Therefore, the subset F^{−1}(Λ(Ĝ))/π_{1}(S) in S consists of two connected components and is contained in the annulus A whose core is homotopic to C. Since F : S̃ → Ĉ can be regarded as the developing map of the projective structure on S corresponding to (t, φ) ∈ P (S), we obtain the assertion.

Step 3. Let {λ_{n}} be the sequence of ML_{Z}(S) satisfying (t_{n}, φ_{n}) ∈ Q_{λ_{n}}.

Lemma 4.2. λ_{n} ∈ {kC : k ∈ N} ⊂ ML_{Z}(S) for large enough n.

Proof. For simplicity, we denote f_{t,φ}^{−1}(Λ(Ĝ))/Γ_{t} by Λ̂ and ω_{n}^{−1}(f_{t_{n},φ_{n}}^{−1}(Λ(G_{n}))/Γ_{t_{n}}) by Λ_{n}. Take an open annulus A in S_{t} hwose core is homotopic to C and which containing Λ̂. Then K = S_{t} − A is a compact set. Now suppose that there exists a subsequence {n_{j}} such that λ_{n_{j}} ∉ {kC : k ∈ N}. Then, from the characterization of Λ_{n_{j}} in Lemma 2.4, one can easily see that Λ_{n_{j}} ∩ K ≠ ∅ for all j. Since K is compact, any sequence {z_{n_{j}} ∈ Λ_{n_{j}} ∩ K} has an accumulation point z_{∞} ∈ K. But the Hausdorff convergence Λ_{n} → Λ̂ (Lemma 3.3) implies that z_{∞} ∈ Λ̂, which contradicts to Λ̂ ∩ K = ∅.

Step 4. For an annulus A with a conformal structure, the modulus m(A) of A is defined by m(A) = (2π)^{−1} log c when A is conformally equivalent to a round annulus {z ∈ C : 1 < |z| < c}. We will make use of the monotonicity of moduli; if an annulus A contains disjoint essential annuli A_{1} and A_{2}, then m(A_{1}) + m(A_{2}) ≤ m(A) (see [20, Lemma 6.3]).

Lemma 4.3. There exists a positive constant m_{0} > 0, independent of n, satisfying the following: Any curve on S_{t_{n}} joining any two connected components of f_{t_{n},φ_{n}}^{−1}(Λ(G_{n}))/Γ_{t_{n}} traverses some annulus whose core is homotopic to C and whose modulus is larger than m_{0}.

Proof. Fix a positive integer k ∈ N. We consider a sequence {(s_{n}, ψ_{n})} = {Ψ_{kC}([ρ_{n}])} in Q_{kC}, where

Ψ_{kC} = (hol|Q_{kC})^{−1} : QF (S) → Q_{kC}.

We will show that there exists a positive constant m_{0}, independent of k and n, satisfying the following: Any curve in S_{s_{n}} joining any two connected components of f_{s_{n},ψ_{n}}^{−1}(Λ(G_{n}))/Γ_{s_{n}} traverses some annulus whose core is homotopic to C and whose modulus is larger than m_{0}. Once it has shown, since (t_{n}, φ_{n}) = Ψ_{k_{n}C}([ρ_{n}]) for λ_{n} = k_{n}C, we obtain the assertion in Lemma 4.3.

The proof depends on the particular form of [ρ_{n}] = qf(τ^{n}u, τ^{2n}v). For simplicity, we assume that [ρ_{0}] = qf(u, v) is a Fuchsian representation. (The following argument, with a slight modification, works out without this assumption.) Let (p, ϕ) denote the projective structure in Q_{0} such that [ρ_{p,ϕ}] = [ρ_{0}]. Then (s_{0}, ψ_{0}) = Gr_{kC}(p, ϕ). To obtain (s_{0}, ψ_{0}) from (p, ϕ), we perform the same construction as described in 2.4. We use the same notation and normalization as in 2.4; for example, G_{0} = ρ_{p,ϕ}(π_{1}(S)) is a Fuchsian group acting on H, g ∈ G_{0} is a generator of cyclic subgroup which stabilizes B_{α}, etc. In addition, we prepare some notations; let B_{α}^{∗} = {z ∈ H^{∗} : 3π/2 − α < arg z < 3π/2 + α} be the complex conjugation of B_{α}, put B̂_{α} = B_{α} ∪ B_{α}^{∗} and set D̂_{α} = ⋃_{h∈G_{0}/⟨g⟩} h(B̂_{α}).

Recall that f_{s_{0},ψ_{0}}^{−1}(Λ(G_{0}))/Γ_{s_{0}} ⊂ S_{s_{0}} consists of 2k simple closed curves each of which is homotopic to C. Moreover, observe that f_{s_{0},ψ_{0}}^{−1}(D̂_{α})/Γ_{s_{0}} contains 2k + 1 annular domains A_{1}, … A_{2k+1} each of whose core is homotopic to C. There exists exactly one connected component of f_{s_{0},ψ_{0}}^{−1}(Λ(G))/Γ_{s_{0}} lying between A_{j} and A_{j+1} for every j ∈ {1, …, 2k}. Therefore, any curve on S_{s_{0}} joining any two components of f_{s_{0},ψ_{0}}^{−1}(Λ(G_{0}))/Γ_{s_{0}} traverses some A_{j}. Note that the developing map f_{s_{0},ψ_{0}} induce a natural conformal isomorphism ξ_{j} : A_{j} → B_{α}^{(∗)}/⟨g⟩ for all j, where B_{α}^{(∗)} is B_{α} or B_{α}^{∗}.

One can take an element μ_{n} ∈ Belt(G_{0})_{1} with supp(μ_{n}) ⊂ D̂_{α/2} such that the quasi-conformal map w_{μ_{n}} : Ĉ → Ĉ induces the quasi-conformal deformation [ρ_{n}] = qf(τ^{n}u, τ^{2n}v) of [ρ_{0}] = qf(u, v). (See [33] for an explicit description of μ_{n}.) This element μ_{n} ∈ Belt(G_{0})_{1} also induces the quasi-conformal deformation (s_{n}, ψ_{n}) of (s_{0}, ψ_{0}), as described in 2.5. Recall that the quasi-conformal map wμ̂_{n}|H_{s_{0}} : H_{s_{0}} → H_{s_{n}} with the Beltrami coefficient μ̂_{n} = f_{s_{0},ψ_{0}}^{∗}(μ_{n}) descends to a quasi-conformal map gμ̂_{n} : S_{s_{0}} → S_{s_{n}}.

The two components of ξ_{j}^{−1}((B_{α}^{(∗)} − B_{α/2}^{(∗)})/⟨g⟩) ⊂ A_{j} are denoted by A′_{j} and A′′_{j}, each of which is essential sub-annulus in A_{j}. An easy calculation reveals that m(A′_{j}) = m(A′′_{j}) = α/2l for all j, where l is the exponent of g, that is, g(z) = e^{l}z. Since gμ̂_{n} is conformal on A′_{j} ∪ A′′_{j}, combining with the monotonicity of moduli, we have

m(gμ̂_{n}(A_{j})) ≥ m(gμ̂_{n}(A′_{j})) + m(gμ̂_{n}(A′′_{j}))
= m(A′_{j}) + m(A′′_{j})
= α/l.

Put m_{0} = α/l. Then any curve on S_{s_{n}} = gμ̂_{n}(S_{s_{0}}) joining any two connected components of f_{s_{n},ψ_{n}}^{−1}(Λ(G_{n}))/Γ_{s_{n}} = gμ̂_{n}(f_{s_{0},ψ_{0}}^{−1}(Λ(G_{0}))/Γ_{s_{0}}) traverses some gμ̂_{n}(A_{j}) whose core is homotopic to C and whose modulus is larger than m_{0}. We have completed the proof of Lemma 4.3.

Step 5. The following lemma implies that the hyperbolic distance between two boundary components of an annulus in a Riemann surface can be estimated below by using the modulus of the annulus. The essencial tool in this proof is Grötzsch’s module theorem (see [20]). For t ∈ T (S), we denote the hyperbolic distance of z_{1}, z_{2} ∈ S_{t} by d_{t}(z_{1}, z_{2}) and the hyperbolic length of the closed geodesic representing C ∈ S by l_{t}(C).

Lemma 4.4. Let t ∈ T (S) and C ∈ S. Let A ⊂ S_{t} be an annular domain such that ∂A consists of two simple closed curves C_{1}, C_{2} each of which is homotopic to C. Then there is a positive constant I = I(l_{t}(C), m(A)) > 0 which depends only on l_{t}(C) and m(A) such that

d_{t}(C_{1}, C_{2}) > I(l_{t}(C), m(A)),

where d_{t}(C_{1}, C_{2}) = inf{d_{t}(z_{1}, z_{2}) : z_{j} ∈ C_{j}(j = 1, 2)}.

Proof. Let H = {z ∈ C : Imz > 0} denote the upper half plane. Take a holomorphic covering map p_{1} : H → S_{t} so that the imaginary axis iR^{+} projects onto a simple closed curve homotopic to C. Let Ã denote the connected component of p_{1}^{−1}(A) which connect 0 and ∞. Then Ã is stabilized by a cyclic group ⟨g⟩ which is generated by g(z) = z exp(l_{t}(C)). Since m(H/⟨g⟩) = π/l_{t}(C), there is a holomorphic covering map p_{2} : H → R = {w ∈ C : 1 < |w| < exp(2π^{2}/l_{t}(C))} whose covering transformation group is ⟨g⟩. Let λ_{R}(w)|dw| denote the complete hyperbolic metric on R, and d_{R}(·, ·) the distance with respect to λ_{R}(w)|dw|. Note that A is conformally equivalent to A′ = p_{2}(Ã) and that d_{t}(C_{1}, C_{2}) = d_{R}(C′_{1}, C′_{2}), where C′_{1} and C′_{2} are components of ∂A′. One can easily see that there is a positive constant I_{1} = I_{1}(l_{t}(C)) which depends only on l_{t}(C) such that λ_{R}(w)|dw| > I_{1}(l_{t}(C))|dw|. Therefore we obtain

d_{R}(C′_{1}, C′_{2}) > I_{1}(l_{t}(C))d_{e}(C′_{1}, C′_{2}), (4.1)

where d_{e}(·, ·) is the distance with respect to the Euclidean metric |dw|.

We can assume that C′_{1} (resp. C′_{2}) is the inner (resp. outer) component of ∂A′. Take w_{1} ∈ C′_{1} and w_{2} ∈ C′_{2} satisfying d_{e}(w_{1}, w_{2}) = d_{e}(C′_{1}, C′_{2}). Let D ⊂ C be a connected component of C − C′_{2} which contains 0. Take a Riemann mapping h : Δ → D such that h(0) = 0 and h^{−1}(w_{1}) = r > 0. Now, Koebe’s one-quarter theorem and Koebe’s distorsion theorem (see, for example, [28, p.9]) implies that there is a positive constant I_{2} = I_{2}(1 − r) such that

d_{e}(C′_{1}, C′_{2}) > I_{2}(1 − r). (4.2)

Let μ(r) denote the modulus of the domain {z ∈ C : |z| < 1, z ∉ [0, r]} for 0 < r < 1. Then Grötzsch’s module theorem asserts that m(A) ≤ μ(r) for any annular domain A ⊂ Δ = {z ∈ C : |z| < 1} which separates 0 and r from ∂Δ (see [20, Chapter II] for more informations). Since h^{−1}(A′) separates 0 and r from ∂Δ, Grötzsch’s theorem implies that m(A′) ≤ μ(r). Since μ is a monotone decreasing function, there is a positive constant I_{3} = I_{3}(m(A′)) such that

1 − r > I_{3}(m(A′)). (4.3)

From the inequalities (4.1)-(4.3), we obtain

d_{R}(C′_{1}, C′_{2}) > I_{1}(l_{t}(C))I_{2}(I_{3}(m(A′)).

Since d_{R}(C′_{1}, C′_{2}) = d_{t}(C_{1}, C_{2}) and m(A′) = m(A), we obtain the assertion.

Using Lemma 4.4, we can finally prove the next lemma, and can complete the proof of Theorem A for the special case.

Lemma 4.5. (t_{n}, φ_{n}) ∈ Q_{C} for large enough n.

Proof. Since maximal dilatations of quasi-conformal maps ω_{n} : S_{t} → S_{t_{n}} tend to 1, there exists a positive constant m_{1} > 0 such that moduli of ω_{n}^{−1} ∘ gμ̂_{n}(A_{j}) exseed m_{1} for sufficiently large n. Since any curve in S_{t} joining any two components of Λ_{n} = ω_{n}^{−1}(f_{t_{n},φ_{n}}^{−1}(Λ(G_{n}))/Γ_{t_{n}}) traverses some ω_{n}^{−1} ∘ gμ̂_{n}(A_{j}), Lemma 4.4 implies that there exists a positive constant L > 0 such that the hyperbolic distance of any two components of Λ_{n} is bounded below by L for sufficiently large n. Then, from Lemma 3.3 and Lemma 4.1(2), one can easily see that Λ_{n} consists of two connected components each of which is homotopic to C. Therefore, from Lemma 2.4, we obtain the assertion.

4.2. Proof for general case. Take an arbitrary element λ = ∑_{j=1}^{l} n_{j}C_{j} ∈ ML_{Z}(S) and let Ĝ be a geometrically finite Kleinian group whose Kleinian manifold NĜ is homeomorphic to S × [0, 1] − ∪_{j=1}^{l}C_{j} × {1/2}. Let f_{λ} : S → NĜ be an immersion such that S minus every annular neighborhoods of C_{j} is mapped into S × {0} by inclusion and the image of each annular neighborhood of C_{j} is wrapping n_{j} times around C_{j} × {1/2}. This immersion is called the wrapping map associated to λ.

Now we perform simultaneous (n, 1) Dehn filling on all the cusps to obtain a sequence of quasi-Fuchsian representations {β_{n} : Ĝ → G_{n}} as before. Moreover, we obtain a sequence {ρ_{n} = β_{n} ∘ (f_{λ})_{∗}} of representations converging algebraically to ρ_{∞} = (f_{λ})_{∗}, and the corresponding sequence {(t_{n}, φ_{n})} of projective structures converging to (t, φ) ∈ ∂Q_{0}. Then we can show, in the same manner in the proof of Lemma 4.1, that the subset f_{t,φ}^{−1}(Λ(Ĝ))/Γ_{t} in S_{t} consists of disjoint unions of 2n_{j} connected components contained in annuli whose core are homotopic to C_{j} for all j. Again, we can show that (t_{n}, φ_{n}) ∈ Q_{λ} for large enough n. Finally, we have completed the proof of Theorem A.

4.3. A remark. Although it is difficult to understand exactly the shape of the closed set \overline{Q_{0}} ∩ \overline{Q_{λ}}, one can see that it is not a compact subset of P (S).

Proposition 4.6. For any λ ∈ ML_{Z}(S), \overline{Q_{0}} ∩ \overline{Q_{λ}} is not compact in P (S).

Proof. We proof this, again, for the case that λ = C ∈ S. Let Ĝ be the same as in the proof of Theorem A for special case. Let A be an annular domain in a component S × {1} of conformal boundary ∂NĜ, whose core is homotopic to C. Observe that A can be conformally embedded in S_{t}, where t ∈ T (S) is the underlying complex structure of the limit point (t, φ) ∈ ∂Q_{0} constructed in the proof of Theorem A. If we deform the conformal structure on S × {1} so that m(A) → ∞, then the hyperbolic length of C on S × {1} tends to 0. Then t diverges in T (S), and hence (t, φ) ∈ ∂Q_{0} diverges in P (S).

5. The proofs of Theorems B and C

For C_{1}, C_{2} ∈ S, the geometric intersection number i(C_{1}, C_{2}) is the minimum number of points in which the representations of C_{1} and C_{2} must intersects. This can be naturally extended to ML_{Z}(S). The main aim of this section is to prove the next theorem.

Theorem B. For any finite set {λ_{i}}_{i=1}^{m} ⊂ ML_{Z}(S) satisfying i(λ_{j}, λ_{k}) = 0 for all j, k ∈ \{1, …, m\}, we have \overline{Q_{0}} ∩ \overline{Q_{λ_{1}}} ∩ ⋯ ∩ \overline{Q_{λ_{m}}} ≠ ∅.

To prove this theorem, we first prepair the following Lemma 5.1, which provides us a method to construct some sequences of representations with the same algebraic limit but with mutually distinct geometric limits.

For a Kleinian group G, let M_{G} = H^{3}/G denote the interior of the Kleinian manifold N_{G} = (H^{3} ∪ Ω(G))/G.

Take an element λ = ∑_{j=1}^{l} n_{j}C_{j} ∈ ML_{Z}(S) and an ordered set of positive integers k = {k_{1}, …, k_{l}}. For this pair (λ, k), we define a new element λ/k of ML_{Z}(S) by

λ/k = ∑_{j=1}^{l}[n_{j}/k_{j}]C_{j},

where [n_{j}/k_{j}] is the largest integer which does not exceed n_{j}/k_{j}.

Lemma 5.1. Let λ = ∑_{j=1}^{l} n_{j}C_{j} be an element of ML_{Z}(S), and let Ĝ be a geometrically finite Kleinian group such that NĜ is homeomorphic to S × [0, 1] − ⋃_{j=1}^{l} C_{j} × {1/2}. Then, for any ordered set of positive integers k = {k_{1}, …, k_{l}}, there exists a geometrically finite Kleinian group Ĝ′ such that

• NĜ′ is also homeomorphic to S × [0, 1] − ⋃_{j=1}^{l} C_{j} × {1/2}, and

• the wrapping map f_{λ} : S → MĜ associated to λ and the wrapping map f_{λ′} : S → MĜ′ associated to λ′ = λ/k induce the same representation up to conjugation in PSL_{2}(C), that is, [(f_{λ})_{∗}] = [(f_{λ′})_{∗}].

Proof. Take an element λ̄ = ∑_{j=1}^{l} m_{j}C_{j} ∈ ML_{Z}(S) satisfying m_{j} = n_{j} − [n_{j}/k_{j}]k_{j} for all j. Note that 0 ≤ m_{j} < k_{j} hold for all j. We consider the wrapping map fλ̄ : S → MĜ and denote (fλ̄)_{∗}(π_{1}(S)) by G. Recall that G is a regular b-group, and that M_{G} is homeomorphic to the interior of S × [0, 1]. Via the canonical projection p_{1} : M_{G} → MĜ, the map fλ̄ is lifted to an embedding f̃λ̄ : S → M_{G} which is a homotopy equivalence.

We now choose a basis ⟨γ_{j}, δ_{j}⟩ ⊂ Ĝ for the fundamental group of each rank two cusp in MĜ corresponding to C_{j}×{1/2} as before, but with an additional assumption that γ_{j} is contained in G. We take a subgroup

Ĝ′ = ⟨G, δ_{1}^{k_{1}}, …, δ_{l}^{k_{l}}⟩

of Ĝ. Since m_{j} < k_{j} hold for all j, one can take Jordan domains {D_{j}, D′_{j}}_{j=1}^{l} in Ω(G) such that

• both D_{j} and D′_{j} are stabilized by ⟨γ_{j}⟩,

• δ_{j}^{k_{j}} maps ∂D_{j} onto ∂D′_{j} and the interior of D_{j} onto the exterior of D′_{j}, and • {D_{j}, D′_{j}}_{j=1}^{l} are projected onto 2l disjoint cusp neighborhoods in ∂N_{G} via the

covering map Ω(G) → ∂N_{G}.

Now one can apply the Klein-Maskit combination theorem II [22, Theorem E.5 (xi)] inductively to show that NĜ′ is homeomorphic to S × [0, 1] − ⋃_{j=1}^{l} C_{j} × {1/2}. Moreover, note that the canonical projection p_{2} : M_{G} → MĜ′ maps the end of M_{G} corresponding to S × {0} homeomorphically onto the end of MĜ′ corresponding to S × {0}. Therefore p_{2} ∘ f̃λ̄ is homotopic to the inclusion map f_{0} : S → MĜ′ onto S × {1/4}.

We now consider the canonical projection p_{3} : MĜ′ → MĜ. Note that p_{3} ∘ f_{0} is homotopic to fλ̄. Observe that the restriction of p_{3} on the tubular neighborhood of C_{j} × {1/2} in MĜ′ is a k_{j}-fold covering map onto the tubular neighborhood of C_{j} × {1/2} in MĜ. We now perform a surgery to obtain the wrapping map f_{λ′} : S → MĜ′ from f_{0} : S → MĜ′ as before. Then p_{3} ∘ f_{λ′} is homotopic to the wrapping map f_{λ} : S → MĜ associated to λ. Therefore [(p_{3} ∘ f_{λ′})_{∗}] = [(f_{λ})_{∗}], and since (p_{3})_{∗} : Ĝ′ → Ĝ is the inclusion map, we have [(f_{λ′})_{∗}] = [(f_{λ})_{∗}].

Now Theorem B follows immediately from Lemma 5.1.

Proof of Theorem B. For any finite set {λ_{i}}_{i=1}^{m} ⊂ ML_{Z}(S) satisfying i(λ_{j}, λ_{k}) = 0 for all j, k ∈ {1, …, m}, one can easily find an element λ = ∑_{j=1}^{l} n_{j}C_{j} ∈ ML_{Z}(S) and ordered sets of positive integers k_{i} = {k_{1}^{(i)}, …, k_{l}^{(i)}} satisfying λ_{i} = λ/k_{i} for all i. By Lemma 5.1, there exist Kleinian groups Ĝ and Ĝ_{i}(i = 1, …, m) such that the wrapping maps f_{λ} : S → MĜ and f_{λ_{i}} : S → MĜ_{i} induce the same representations [(f_{λ})_{∗}] = [(f_{λ_{i}})_{∗}] for all i. Take the element (t, φ) ∈ ∂Q_{0} such that hol(t, φ) = [(f_{λ})_{∗}]. By performing simultaneous (n, 1) Dehn filling on all the cusps in MĜ_{i}, we obtain a sequence {(t_{n}^{(i)}, φ_{n}^{(i)})} in Q_{λ_{i}} converging to (t, φ). We have completed the the proof.

As a consequence of Theorem B, we obtain the following Theorem C. For any positive integer n ∈ N, there exists a point [ρ] ∈ ∂QF (S) such that U ∩ QF (S) consists of more than n components for any sufficiently small neighborhood U of [ρ].

Proof. For any positive integer n ∈ N, a finite set {kC}_{k=1}^{n} in ML_{Z}(S) satisfies the condition in Theorem B. Since the holonomy map hol : P (S) → V (S) is local homeomorphism, we obtain the assertion.

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Department of Mathematics, Tokyo Institute of Technology, Tokyo 152-8551 Japan

  E-mail address: itoken@@math.titech.ac.jp