SCHOTTKY GROUPS AND BERS BOUNDARY OF TEICHMÜLLER SPACE

KENTARO ITO

Abstract. We will show that every Kleinian groups on a Bers boundary of the Teichmüller space is an algebraic limit of a sequence of Schottky groups. To show this, we extend the action of the mapping class group on a Bers slice to that on a class of function groups whose invariant components are covering some fixed Riemann surface. An important observation is that the orbit of every maximal cusp is dense in a Bers boundary.

1. Introduction

In this paper, we extend the action of the mapping class group on a Bers slice to that on a wider class (which will be called an extended Bers slice) of Kleinian groups. Here, we explain the fundamental idea how to extend the action of the mapping class group.

Let S be an oriented compact surface possibly with boundary ∂S, and let T (S) be the Teichmüller space of comprete hyperbolic structures on the interior of S with finite area. Let V (S) be the space of conjugacy classes of representations of π_{1}(S) into PSL_{2}(C). The subspace QF (S) of V (S) consisting of discrete faithful representations whose images are quasi-Fuchsian groups is naturally identified with a product of Teichmüller spaces T (S) × T (S); that is, there exists a holomorphic isomorphism

Q : T (S) × T (S) → QF (S).

The mapping class group Mod(S) of S naturally acts on T (S), and hence on the Bers slice B_{X} = Q({X} × T (S)) for every X ∈ T (S);

Q(X, Y) ↦ Q(X, σY),

where σ ∈ Mod(S). The representation Q(X, σY) has another description as follows;

Q(X, σY) = Q(σ^{−1}X, Y) ∘ σ_{∗}^{−1},

where σ_{∗} is the group isomorphism of π_{1}(S) induced by σ. The right side of the above equation suggests us a possibility to define the action of Mod(S) even when Y is pinched or degenerated. In this view point, Bers [3] extended the action of Mod(S) to that on the closure of B_{X} and, in this paper, we extend the action to that on the subset C_{X} of V (S) consisting of representations whose images are function groups with invariant components covering X ∈ T (S). The set C_{X} is called an extended Bers slice for X.

Date: June22, 1999.

1991 Mathematics Subject Classification. 30F40, 57M50.

Our main theorem is the following (see Lemma 6.1 and Theorem 6.4):

Main Theorem . The subset S_{X} of C_{X} of Schottky groups consists of exactly one orbit under the action of Mod(S) and the set of accumulation points of S_{X} contains the Bers boundary ∂B_{X}.

We remark that, in the case of genus 2, it was shown by Gallo [8] that the set of accumulation points of S_{X} contains the Bers boundary ∂B_{X}. But our method is different from that of Gallo.

This paper is organized as follows: In section 2, we give a definition of an extended Bers slice C_{X} on which we define the action of the mapping class group. In section 3, we obtain a sufficient condition so that the action of Mod(S) is continuous at an element of C_{X} (Corollary 3.2). In section 5, we show that the orbit of every maximal cusp is dense in the Bers boundary ∂B_{X} (Theorem 5.6). In section 6, we prove our main theorem as a corollary of preceding sections. In section 5 and 6, one of the crucial tool is Thurston’s compactness theorem [23], which will be introduced in section 4.

Acknowledgements . The author would like to express his gratitude to Hiroshige Shiga for his encouragement and useful suggestions.

2. Preliminaries

Let S be a compact oriented surface of negative Euler characteristic possibly with boundary ∂S. The Teichmüller space T (S) is the set of equivalence classes of pairs (f, X); where X is a hyperbolic Riemann surface of finite area and f : int(S) → X is a homeomorphism from the interior of S. Two pairs (f, X) and (g, Y) are equivalent if there is a conformal map ψ : X → Y such that ψ ∘f is isotopic to g. The mapping class group Mod(S) is the group consisting of isotopy classes of orientation preserving homeomorphisms of S. There is a natural action of σ ∈ Mod(S) on T (S) by

σ(f, X) = (f ∘ σ^{−1}, X).

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 by conformal automorphisms. The limit set of G in Ĉ is denoted by Λ(G) and its compliment Ĉ − Λ(G), which is called the region of discontinuity of G, is denoted by Ω(G). A Kleinian group G is called a function group if its region of discontinuity Ω(G) has an invariant component Ω_{0}(G). If a function group has exactly two invariant components, it is called a quasi-Fuchsian group; otherwise, it has a unique invariant component.

For a given X ∈ T (S), let Γ be a Fuchsian group acting on the unit disc Δ = {z ∈ Ĉ : |z| < 1} such that X = Δ/Γ. We define the space of bounded holomorphic quadratic differentials on Δ for Γ by

B_{2}(Γ) = {φ|φ is holomorphic on Δ, φ ∘ γ(γ′)^{2} = φ for ∀γ ∈ Γ and ||φ||_{∞} < ∞},

where ||φ||_{∞} is the hyperbolic sup-norm of φ defined by sup_{z∈Δ}(1 − |z|^{2})^{2}|φ(z)|. With this norm ||φ||_{∞}, B_{2}(Γ) is a finite dimensional complex Banach space.

For φ ∈ B_{2}(Γ), we associate a pair (f_{φ}, ρ_{φ}), called a normalized projective structure on X; where,

f_{φ} : Δ → Ĉ

is a meromorphic local homeomorphism whose Schwarzian derivative S(f_{φ}) is equal to φ, and

ρ_{φ} : Γ → PSL_{2}(C)

is a group homomorphism satisfying f_{φ} ∘ γ = ρ_{φ}(γ) ∘ f_{φ} for all γ ∈ Γ. Moreover, f_{φ} is normalized by the conditions f_{φ}(0) = 0, f′_{φ}(0) = 1 and f′′_{φ}(0) = 0. Then there is a bijective correspondence between normalized projective structures and B_{2}(Γ).

We denote by C(Γ) the set of φ ∈ B_{2}(Γ) such that the map f_{φ} is a covering map. For φ ∈ C(Γ), G = ρ_{φ}(Γ) is a function group (possibly with torsion) and f_{φ}(Δ) is an invariant component of G (see [12] for more information). Furthermore, we denote C_{0}(Γ) by the set of φ ∈ C(Γ) such that the map f_{φ} : Δ → f_{φ}(Δ) ⊂ Ĉ induces a conformal isomorphism X = Δ/Γ → f_{φ}(Δ)/ρ_{φ}(Γ);

C_{0}(Γ) = {φ ∈ C(Γ)|X = Δ/Γ ≅ f_{φ}(Δ)/ρ_{φ}(Γ)}.

For φ ∈ C_{0}(Γ), G = ρ_{φ}(Γ) may have an elliptic element whose fixed points are not contained in the invariant component Ω_{0}(G) = f_{φ}(Δ). In [14], one can find examples of elements of C(Γ) but not of C_{0}(Γ).

Let V (S) denote the space of conjugacy classes [ρ] of irreducible representations ρ : π_{1}(S) → PSL_{2}(C) such that ρ(g) is parabolic for every g ∈ π_{1}(∂S). We also use the notation (ρ, G) to represent an element [ρ] ∈ V (S) whose image is G = ρ(π_{1}(S)). The space V (S) is a manifold endowed with the algebraic topology; a sequence of representations ρ_{n} : π_{1}(S) → PSL_{2}(C) converges algebraically to a representation ρ : π_{1}(S) → PSL_{2}(C) if ρ_{n}(g) → ρ(g) in PSL_{2}(C) for all g ∈ π_{1}(S).

It is known by Kra [11] that the map

hol : B_{2}(Γ) → V (S)

assigning the conjugacy class [ρ_{φ}] of the representation ρ_{φ} : π_{1}(S) ≅ Γ → PSL_{2}(C) to φ ∈ B_{2}(Γ) is a holomorphic embedding. (Here and hereafter, we frequently identify a representation of π_{1}(S) with a representation of Γ.) For any X ∈ T (S), we define subsets Ĉ_{X} and C_{X} of V (S) by

Ĉ_{X} = hol(C(Γ)), and C_{X} = hol(C_{0}(Γ)).

We call C_{X} an extended Bers slice, on which we will define an action of the mapping class group. We can define C_{X} directly as a subset of V (S) consisting of function groups whose invariant components are covering X ∈ T (S); more precisely, a representation (ρ, G) ∈ V (S) is an element of C_{X} if G is a function group with an invariant component Ω_{0}(G) and ρ is induced by the composition of the inclusion map ι : Ω_{0}(G)/G ↪ N_{G} into the Kleinian manifold N_{G} = H^{3} ∪ Ω(G)/G with a conformal isomorphism g : X → Ω_{0}(G)/G.

A Bers slice B_{X} is the subset of C_{X} consisting of faithful representations whose images are quasi-Fuchsian groups. It is known by Bers [2] that a Bers slice B_{X} is (anti-holomorphically) isomorphic to the Teichmüller space T (S), and that it is relatively compact in V (S). A set ∂B_{X} = B̄_{X} −B_{X} is called a Bers boundary, where B̄_{X} is the closure of B_{X} in V (S). Moreover, we denote by B̂_{X} the subset of C_{X} consisting of faithful representations. It is conjectured that B̄_{X} = B̂_{X} in Bers [2].

The followings are the sets which we want to consider in this paper.

B_{X} ⊂ B̄_{X} ⊆ B̂_{X} ⊂ C_{X} ⊂ Ĉ_{X}.

Example . In the case that S is a closed surface, a typical example of an element of C_{X} − B̂_{X} is a Schottky group. A Kleinian group G is a Schottky group if its Kleinian manifold N_{G} = H^{3} ∪ Ω(G)/G is homeomorphic to a handle body H_{g}. Let G be a Schottky group which uniformizes X, that is X = Ω(G)/G = ∂N_{G}, then a representation ρ : π_{1}(S) ≅ π_{1}(∂N_{G}) → G ≅ π_{1}(N_{G}) induced by the inclusion map ∂N_{G} ↪ N_{G} is an element of C_{X} but not of B̂_{X}.

Lemma 2.1. For any X ∈ T (S), C_{X} is a compact subset of V (S).

Proof. To show that C_{X} = hol(C_{0}(Γ)) is compact, we will show that C_{0}(Γ) is closed and bounded subset of B_{2}(Γ). Since it is known by Kra and Maskit [14] that C(Γ) is a closed and bounded subset of B_{2}(Γ) (i.e. Ĉ_{X} is compact), we only have to show that C_{0}(Γ) is closed. Let φ_{n} ∈ C_{0}(Γ) be a sequence converging to φ ∈ C(Γ). Let (f_{n}, ρ_{n}) and (f, ρ) be normalized projective structures corresponding to φ_{n} and φ, respectivery. Then f_{n} converges to f locally uniformly on Δ. Suppose that the map g : X → f(Δ)/ρ(Γ) induced from f : Δ → f(Δ) is not injective. Then there are two points x, y ∈ X(x ≠ y) such that g(x) = g(y), and hence there are lifts x̃, ỹ ∈ Δ of x and y such that f(x̃) = f(ỹ). Since f_{n}(x̃) → f(x̃) and f_{n}(ỹ) → f(ỹ), the hyperbolic distances d_{n} between f_{n}(x̃) and f_{n}(ỹ) on f_{n}(Δ) tend to 0 as n → ∞. On the other hand, since φ_{n} ∈ C_{0}(Γ), f_{n}(Δ)/ρ_{n}(Γ) are conformally equivalent to X. It implies that d_{n} is equal to the hyperbolic distance between x and y on X for all large n. This is a contradiction.

For a given Kleinian group G with Ω(G) ≠ ∅, we denote by B(U, G)_{1} the space of measurable Beltrami differentials μ for G satisfying ||μ||_{∞} = ess.sup|μ| < 1 with support in an open set U ⊂ Ĉ. For μ ∈ B(U, G)_{1}, there is a unique quasiconformal homeomorphism

w_{μ} : Ĉ → Ĉ

satisfying (w_{μ})z̄ = μ × (w_{μ})_{z} (a.e.) and fixing 0, 1 and ∞. Two elements μ, ν ∈ B(U, G)_{1} are equivalent (denoted by μ ∼ ν) if w_{μ} and w_{ν} induce the same group isomorphism; that is, w_{μ} ∘ g ∘ (w_{μ})^{−1} = w_{ν} ∘ g ∘ (w_{ν})^{−1} for all g ∈ G.

Let Γ be a Fuchsian group acting on the unit disc Δ such that X = Δ/Γ. Let (ρ, G) ∈ C_{X} and let (f : Δ → Ω_{0}(G), ρ : π_{1}(S) → G) be the corresponding projective structure. For μ ∈ B(Δ, Γ)_{1}, the push-forward f_{∗}μ is an element of B(Ω_{0}(G), G)_{1} which is defined locally by the pull-back

(f^{−1})^{∗}μ = μ ∘ f^{−1} × \overline{(f^{−1})′}/(f^{−1})′

of μ via some branch of f^{−1}. Then, the map

f_{∗} : B(Δ, Γ)_{1} → B(Ω_{0}(G), G)_{1},

assigning f_{∗}μ to μ is an isomorphism. The quotient space B(Δ, Γ)_{1}/ ∼ is naturally identified with the Teichmüller space T (S). We define a subspace QC_{0}(ρ) of V (S) by

QC_{0}(ρ) = {[ρ′] ∈ V (S)|ρ′(γ) = w_{μ} ∘ ρ(γ) ∘ (w_{μ})^{−1}, γ ∈ π_{1}(S), μ ∈ B(Ω_{0}(G), G)_{1}}.

This space QC_{0}(ρ) is also identified with the quotient space B(Ω_{0}(G), G)_{1}/ ∼. Then, it was shown by Maskit [17] (see also Kra [13]) that the map f_{∗} : B(Δ, Γ)_{1} → B(Ω_{0}(G), G)_{1} descends to an unbranched covering map

Ψ_{ρ} : T (S) → QC_{0}(ρ)

with Ψ_{ρ}(X) = [ρ]. One can easily see that Ψ_{ρ}(Y) ∈ C_{Y} for all Y ∈ T (S) and that if [ρ] ∈ C_{X} then [ρ] ∘ σ_{∗}^{−1} ∈ C_{σX} for all σ ∈ Mod(S), here σ_{∗} is the group isomorphism of π_{1}(S) induced by σ. Now we define the action of σ ∈ Mod(S) on C_{X} by

[ρ] ↦ [ρ]^{σ} = Ψ_{ρ}(σ^{−1}X) ∘ σ_{∗}^{−1}.

One can easily check that [ρ]^{σ_{1}∘σ_{2}} = ([ρ]^{σ_{2}})^{σ_{1}} holds for any σ_{1}, σ_{2} ∈ Mod(S). This action of Mod(S) coincides with the natural action on B_{X} ≅ T (S) described in Introduction. We also remark that the action restricted to B̂_{X} is the same with that defined by Bers in [3].

3. Continuity of the action of the mapping class group

In this section, we obtain a sufficient condition for [ρ] ∈ C_{X} so that the action of Mod(S) at [ρ] is continuous. The same result, for the case that C_{X} is replaced by B̂_{X}, was obtained by Bers [3].

We first show the continuity under base change.

Proposition 3.1. Let [ρ] = (ρ, G) be an element of C_{X} such that all components of Ω(G)/G except for X = Ω_{0}(G)/G have (if there exist) no moduli of deformation. Then the following holds:

If [ρ_{n}] → [ρ] in C_{X} then Ψ_{ρ_{n}}(Y) → Ψ_{ρ}(Y) in C_{Y} for all Y ∈ T (S).

Proof. Let φ_{n} and φ be elements in C_{0}(Γ) such that hol(φ_{n}) = [ρ_{n}] and hol(φ) = [ρ], respectively. Since C_{0}(Γ) is compact and the map hol : B_{2}(Γ) → V (S) is injective, φ_{n} → φ in C_{0}(Γ). Let (f_{n}, ρ_{n}) and (f, ρ) be normarized projective structures for φ_{n} and φ, respectively. Then f_{n} converges to f locally uniformly on Δ. Let μ ∈ B(Δ, Γ)_{1} be a representative of Y ∈ T (S). We may assume that μ is continuous. Put μ̂_{n} = (f_{n})_{∗}μ ∈ B(Ω_{0}(G_{n}), G_{n})_{1} and μ̂ = f_{∗}μ ∈ B(Ω_{0}(G), G)_{1}, where G_{n} = ρ_{n}(π_{1}(S)) and G = ρ(π_{1}(S)). Since {wμ̂_{n}} fix 0, 1 and ∞ and their dilatations are uniformly bounded, it has a subsequence (which we denote by the same symbol) {wμ̂_{n}} converging uniformly to some quasiconformal homeomorphism w_{∞} of Ĉ. Since the representatives of Ψ_{ρ_{n}}(Y) are induced by wμ̂_{n} ∘ f_{n}, {Ψ_{ρ_{n}}(Y)} converges algebraically to the conjugacy class of a representation induced by w_{∞} ∘ f. Therefore, we only have to show that w_{∞} and wμ̂ induce the same group isomorphism from G into PSL_{2}(C).

Since injectivity radii (with respect to the Poincaré metric on Δ) of f_{n} are uniformly bounded below (see [14], Lemma 5.1), for any z ∈ Ω_{0}(G), there is an open neighborhood U of z and suitable branches of inverse maps f_{n}^{−1} and f^{−1} on U such that f_{n}^{−1} converges to f^{−1} uniformly on U. Hence one can see that μ̂_{n} converges to μ̂ locally uniformly on Ω_{0}(G). Therefore, wμ̂_{n} ∘(wμ̂)^{−1} converges to a conformal map w_{∞} ∘ (wμ̂)^{−1} locally uniformly on wμ̂(Ω_{0}(G)), and hence, the Beltrami coefficient of w_{∞} is equal to μ̂ almost everywhere on Ω_{0}(G). Since there is no essential deformation on Ω(G) − Ω_{0}(G) by assumption and on Λ(G) by Sullivan’s rigidity theorem [22], w_{∞} and wμ̂ induce the same group isomorphism.

Corollary 3.2. Let [ρ] be an element of C_{X} satisfying the same condition as in Proposition 3.1. Then the action of Mod(S) is continuous at [ρ]; that is, if [ρ_{n}] → [ρ] in C_{X} then [ρ_{n}]^{σ} → [ρ]^{σ} for all σ ∈ Mod(S).

Proof. By Proposition, Ψ_{ρ_{n}}(σ^{−1}X) → Ψ_{ρ}(σ^{−1}X) for all σ ∈ Mod(S). Therefore, [ρ_{n}]^{σ} = Ψ_{ρ_{n}}(σ^{−1}X) ∘ σ_{∗}^{−1} converges algebraically to [ρ]^{σ} = Ψ_{ρ}(σ^{−1}X) ∘ σ_{∗}^{−1}.

Remark . In [10], Kerckhoff and Thurston showed that there is a Bers slice B_{X} and a point [ρ] ∈ ∂B_{X} at which the action of Mod(S) is not continuous.

4. Thurston’s compactness theorem

In this section, we introduce Thurston’s compactness theorem [23], which will play an important roll in the following sections.

Let M be a compact 3-manifold with boundary ∂M. A closed curve γ on ∂M is said to be compressible if it is null homotopic in M but not in ∂M; otherwise it is incompressible. A proper map f : (A, ∂A) → (M, ∂M) of an annulus A into M is said to be essential if f_{∗} : π_{1}(A) → π_{1}(M) is an injection and f is not homotopic (as a map of pairs) to a map into ∂M.

Definition 4.1. Let M be a compact 3-manifold with boundary ∂M. Let λ be a system of non-trivial homotopically distinct simple closed curves on ∂M. Then a pair (M, λ) is doubly incompressible if

(1) every compressible simple closed curve on ∂M intersects λ at least three times,

(2) there are no essential annuli with boundary in ∂M − λ, and

(3) every maximal abelian subgroup of π_{1}(∂M −λ) is mapped to a maximal abelian subgroup of π_{1}(M).

Let M be a compact 3-manifold with boundary ∂M. We denote by AH(M) the space of conjugacy classes [ρ] = (ρ, G) of discrete faithful representations ρ : π_{1}(M) → PSL_{2}(C), where G = ρ(π_{1}(M)). The space AH(M) is equipped with the algebraic topology. Let γ be an incompressible closed curve on ∂M. For (ρ, G) ∈ AH(M), l_{ρ}(γ) denotes the length of the geodesic representative of γ in the hyperbolic manifold H^{3}/G, or l_{ρ}(γ) = 0 if ρ(γ) is parabolic. We define AH(M, λ, K) to be the set of (ρ, G) ∈ AH(M) such that l_{ρ}(λ) ≤ K, where l_{ρ}(λ) is the total sum of the lengths of every component of λ.

Theorem 4.2 (Thurston [23]). If (M, λ) is doubly incompressible, then AH(M, λ, K) is compact for all K.

A curve system λ = {α_{j}}_{j=1}^{N} on S is called homotopically independent if it has the following properties: (1) each α_{j} is a simple closed curve on S and for i ≠ j, α_{i} ∩ α_{j} = ∅, (2) each α_{j} is non-trivial and not freely homotopic to a component of ∂S, and (3) for i ≠ j, α_{i} is not freely homotopic to α_{j}. A homotopically independent curve system λ = {α_{j}}_{j=1}^{N} on S is maximal if it divides S into a union of pairs of pants. (If S is a surface of type (g, n); that is, S is a closed surface of genus g with n open disc removed, then N = 3g − 3 + n.) A pair (λ, λ′) of maximal curve systems on S is called binding S if they have no curves in common and if (after suitable deformation of λ and λ′ by homotopy) each component of S − (λ ∪ λ′) is a simply connected domain or an annulus containing a component of ∂S in its boundary.

The following lemma is discussed in more general setting in Ohshika [21].

Lemma 4.3. Let (λ′, λ′′) be a pair of maximal curve systems which binds S. For this pair, we define a maximal curve system λ on ∂(S × I), where I = [0, 1] is a closed interval, as

λ = (λ′ × {0}) ∪ (λ′′ × {1}) ∪ (∂S × {1/2}).

Then (S × I, λ) is doubly incompressible.

Proof. We only consider the case of ∂S ≠ ∅, since the case of ∂S = ∅ is easier. If ∂S ≠ ∅, then S × I is homeomorphic to a handle body H_{g} of some genus g. We identify S × I and H_{g} via this homeomorphism. We first check the condition (1) in the Definition 4.1. Let γ be a compressible simple closed curve on ∂H_{g}. Since S × {0} and S × {1} contain no compressible curves, γ must intersect a component of ∂S ×{1/2}. If i(γ, λ) ≤ 2 (here i(·, ·) denotes the geometric intersection number), one can easily see that the only possible case is the following one: there exists a component δ of ∂S × {1/2} and a component W of ∂H_{g} − ((λ′ × {0}) ∪ (λ′′ × {1})) homeomorphic to a four-time-punctured sphere such that γ ∪δ ⊂ W and i(γ, δ) = 2. Let α and β be components of ∂W such that α, β and γ bound a pair of pants T. Take a base point x in T. By abuse of notation, α, β and γ also denote the elements of π_{1}(∂H_{g}, x) contained in T. Since (λ′, λ′′) is binding, the curves α and β form a rank 2 free subgroup ⟨α, β⟩ of π_{1}(∂H_{g}, x) which is mapped into π_{1}(H_{g}, x) injectively. Hence γ = α · β ∈ π_{1}(∂H_{g}, x) is incompressible, which is a contradiction.

Now we check the condition (2). Suppose that there exists an essential annulus f : (A, ∂A) → (H_{g}, ∂H_{g}) with boundary in ∂H_{g} − λ. let γ and γ′ be images of ∂A on ∂H_{g}. Then, since i(γ ∪ γ′, ∂S × {1/2}) = 0, γ and γ′ may be assumed to be contained in S × {0, 1}. Let p : S × I → S be the canonical retraction. Then, p(γ) is homotopic to p(γ′) in S. Since (λ′, λ′′) is binding S, γ and γ′ must be contained in the same component of S × {0, 1}. Now the retraction above gives a homotopy between f and a map into ∂H_{g}. This is a contradiction.

Finally, we check the condition (3). Since all abelian subgroups of π_{1}(∂H_{g} − λ) or π_{1}(H_{g}) are isomorphic to Z, we only have to show that all primitive element of π_{1}(∂H_{g} − λ) is also primitive in π_{1}(H_{g}). But this is trivial since H_{g} is homotopic to S.

5. Orbit density for maximal cusps

Let [ρ] ∈ B̂_{X}. The accidental parabolic locus of [ρ] is a homotopically independent curve system λ = {α_{j}} on S such that ρ(α_{j}) is (a conjugacy class of) a parabolic element of G = ρ(π_{1}(S)) for every j, and no simple closed curve which is not homotopic to a component of λ have this property. For [ρ] ∈ B̂_{X}, its accidental parabolic locus is uniquely determined up to homotopy. An element [ρ] ∈ B̂_{X} is called maximal cusp if its accidental parabolic locus is maximal. It is a well known facts that every maximal cusp is contained in ∂B_{X} and that, for any maximal curve system λ on S, there exists a unique maximal cusp whose accidental parabolic locus is λ (see Abikoff [1] and Maskit [16]).

Lemma 5.1. Let [ρ] ∈ C_{X} and let λ = {α_{j}} be a maximal curve system on S. If ρ(α_{j}) are parabolic for all j, [ρ] is the maximal cusp in ∂B_{X} whose accidental parabolic locus is λ.

Proof. We only have to show that [ρ] is a faithful representation. Suppose that ρ : π_{1}(S) → G is not faithful. Then the covering map p : Ω_{0}(G) → X = Ω_{0}(G)/G is not universal, where Ω_{0}(G) is a unique invariant component of G. Then, by the planarity theorem (see [18], X.A.4), there exists a non-trivial simple closed curve δ on X and δ̃ on Ω_{0}(G) such that p|δ̃: δ̃ → δ is a finite-sheeted covering map; say ksheeted. Let g ∈ G be a generator of the subgroup of G stabilizing δ̃. Since λ ⊂ X is maximal and δ is not parallel to a component of λ, δ must intersect some component of λ, say α_{1}. We may assume that the number of intersection of δ and α_{1} is equal to i(δ, α_{1}). Let α̃_{1} be a lift of α_{1} on Ω_{0}(G) which intersects δ̃. Let h be a parabolic element which conjugates to ρ(α_{1}) in G and stabilizing α̃_{1}. By adjoining the fixed point of h to α̃_{1}, we obtain a simple closed curve, which divides Ĉ into two domains; let D be one of the two domains. If k > 1, we require that D satisfies D ∩ g(D) = ∅. Let η_{1} be a component of D ∩ δ̃ and β be an arc in α̃_{1} which connects end points of η_{1}. Let δ̃_{1} = η_{1} ∪ β and let δ̃_{2} be the closed curve δ̃ with η_{1}, g(η_{1}), …, g^{k−1}(η_{1}) replaced by β, g(β), …, g^{k−1}(β). Then, for j = 1, 2, δ̃_{j} projects to a simple closed curve δ_{j} on X such that p|δ̃_{j} : δ̃_{j} → δ_{j} is a finite-sheeted covering map. Moreover, note that i(δ_{j}, λ) is strictly less than i(δ, λ) for j = 1, 2. Since δ is non-trivial and δ = δ_{1} · δ_{2}, either δ_{1} or δ_{2} are non-trivial. After a finite number of steps as above, we obtain a non-trivial simple closed curve δ′ such that i(δ′, λ) = 0 and that, for a lift δ̃′ of δ′, p|δ̃′ : δ̃′ → δ′ is a finite-sheeted covering map. This is a contradiction.

For a simple closed curve α on S, let D_{α} ∈ Mod(S) denote the Dehn twist (once) around α.

Proposition 5.2. Let (λ′, λ′′) be a binding pair of maximal curve systems on S. Let [ρ] ∈ ∂B_{X} be a maximal cusp whose accidental parabolic locus is λ′′. Put σ = D_{α_{1}} ∘ ⋯ ∘ D_{α_{N}} ∈ Mod(S), where λ′ = {α_{j}}_{j=1}^{N}. Then the sequence {[ρ]^{σ^{n}}}_{n∈Z} converges to the maximal cusp [ρ_{∞}] whose accidental parabolic locus is λ′ as |n| → ∞.

In the proof of Proposition 5.2, we will make use of the following two lemmas; the first is due to Canary [5] and the second is a well known technical lemma.

Lemma 5.3 (Canary [5]). Given A > 0, there exists a constant R such that if G is a non-elementary, torsion-free Kleinian group such that every incompressible closed geodesic on Σ = Ω(G)/G has hyperbolic length at least A, then for any closed curve γ on Σ,

l_{N}(γ) ≤ Rl_{Σ}(γ),

where l_{N}(γ) and l_{Σ}(γ) are hyperbolic length of geodesic representatives of γ in N = H^{3}/G and in Σ, respectively.

Lemma 5.4. Let F_{2} be a rank 2 free group and let {χ_{n} : F_{2} → PSL_{2}(C)} be a sequence of discrete faithful representations which converges algebraically to χ_{∞}. If {χ′_{n} = ψ_{n} · χ_{n} · ψ_{n}^{−1}} also converges algebraically to χ′_{∞} for a sequence {ψ_{n}} of PSL_{2}(C), then ψ_{n} converges to some element ψ_{∞} in PSL_{2}(C).

Proof of 5.2. Our argument is almost parallel to that of Kerckhoff and Thurston [10] (see also [4]). We denote by AH_{∂S}(S × I) the set of [χ] ∈ AH(S × I) such that χ(γ) are parabolic for all γ ∈ π_{1}(∂S × I). Then, for a given maximal cusp [ρ] ∈ ∂B_{X}, AH_{∂S}(S × I) is properly embedded into V (S) so that [ρ] ∈ AH_{∂S}(S × I) and hence QC_{0}(ρ) ⊂ AH_{∂S}(S × I). Let λ be a maximal curve system on ∂(S × I) as in Lemma 4.3, so that (S × I, λ) is doubly incompressible. Then the sequence {[ρ̄_{n}] = Ψ_{ρ}(σ^{−n}X)}_{n∈Z} in QC_{0}(ρ) is contained in

AH_{∂S}(S × I, λ, K) = AH_{∂S}(S × I) ∩ AH(S × I, λ, K)

for some K, since

lρ̄_{n}(λ′′ × {1}) = lρ̄_{n}(∂S × {1/2}) = 0

and

lρ̄_{n}(λ′ × {0}) ≤ Rl_{σ^{−n}X}(λ′ × {0}) = Rl_{X}(λ′ × {0}),

where R is a constant in Lemma 5.3. Since AH(S × I, λ, K) is compact by Theorem 4.2, and AH_{∂S}(S × I) is closed in AH(S × I), AH_{∂S}(S × I, λ, K) is compact. Therefore {[ρ̄_{n}]}_{n∈Z} has a convergent subsequence. On the other hand, since C_{X} is compact (Lemma 2.1), {[ρ]^{σ^{n}}}_{n∈Z} also has a convergent subsequence. We also denote this subsequence by the same symbol; in fact, in the following argument, we will show that this subsequence converges to a unique maximal cusp, therefore {[ρ]^{σ^{n}}}_{n∈Z} converges without passing to a subsequence. Take representatives ρ_{n} of [ρ]^{σ^{n}} converging to a representation ρ_{∞}. Then ρ̄_{n} = ρ_{n} ∘ σ^{n} are representatives of [ρ̄_{n}]. Therefore, there are elements ψ_{n} ∈ PSL_{2}(C) such that ψ_{n} · ρ̄_{n} · ψ_{n}^{−1} converges to a representation ρ̄_{∞}.

Take a component α of λ′ and let T be a component of S − λ′ containing α in its boundary. Let α′(≠ α) be a component of λ′ or a component of ∂S contained in the boundary of T. Choose a base point x in T and regard π_{1}(S) = π_{1}(S, x). By abuse of notation, α and α′ also denote the elements of π_{1}(S, x) contained in T. Since ρ_{n}(α) = ρ̄_{n}(α) and ρ_{n}(α′) = ρ̄_{n}(α′) and since ρ_{n} converges on α and α′, by Lemma 5.4, the elements ψ_{n} ∈ PSL_{2}(C) may be taken to be the identity.

One can find non-trivial elements γ_{1}, γ_{2} ∈ π_{1}(S, x) each of which intersects α twice in the opposite direction and does not intersect any other components of λ′, and that ⟨γ_{1}, γ_{2}⟩ is a rank 2 free subgroup of π_{1}(S, x). Then

ρ̄_{n}(γ_{1}) = ρ_{n}(α^{n}) · ρ_{n}(γ_{1}) · ρ_{n}(α^{−n})

and

ρ̄_{n}(γ_{2}) = ρ_{n}(α^{n}) · ρ_{n}(γ_{2}) · ρ_{n}(α^{−n})

holds. Since both ρ_{n} and ρ̄_{n} converge on γ_{1} and γ_{2}, Lemma 5.4 again implies that ρ_{n}(α^{n}) converges to an element α̂ in PSL_{2}(C). Since ρ_{n}(α) commutes with ρ_{n}(α^{n}) for all n, ρ_{∞}(α) commutes with α̂. If ⟨ρ_{∞}(α), α̂⟩ were isomorphic to Z, then ρ_{∞}(α^{k}) = α̂^{l} for some integers k and l, and thus ρ_{n}(α^{nl−k}) → id. This contradicts the fact that [ρ_{n}] are discrete faithful representations. Therefore we conclude that ⟨ρ_{∞}(α), α̂⟩ is a rank 2 parabolic subgroup in PSL_{2}(C). Hence, ρ_{∞}(α) is parabolic. The same argument works well for all components of λ′. Therefore, by Lemma 5.1, we can conclude that [ρ_{∞}] is a maximal cusp whose accidental parabolic locus is λ′.

Lemma 5.5. For any two maximal curve systems λ = {α_{j}}_{j=1}^{N} and λ′ = {β_{j}}_{j=1}^{N} on S, there exists a maximal curve system ν = {γ_{j}}_{j=1}^{N} such that the pairs (λ, ν) and (ν, λ′) are binding S.

Proof. There exists a simple closed curve δ on S such that i(δ, α_{j}) > 0 for all j (see [6]). Put σ = D_{α_{1}} ∘ ⋯ ∘ D_{α_{N}} and put δ_{n} = σ^{n}(δ). If i(β_{j}, λ) = 0 then β_{j} = α_{i} for some i and hence i(β_{j}, δ_{n}) > 0 for all n. If i(β_{j}, λ) > 0 then i(β_{j}, α_{i}) > 0 for some i. In this case, i(β_{j}, δ_{n}) > 0 for all but finitely many n. Therefore, for sufficiently large n, i(β_{j}, δ_{n}) > 0 holds for all j. Fix such n and let γ_{1} = δ_{n}. Choose simple closed curves γ_{2}, …, γ_{N} so that ν = {γ_{j}}_{j=1}^{N} is a maximal curve system. This ν satisfies the desired condition.

It was shown by McMullen [20] that the set of maximal cusps is dense in ∂B_{X}. Since the manner of decomposition of S into pairs of pants up to Mod(S) is finite, the set of maximal cusps in ∂B_{X} decomposes into finitely many orbits under the action of Mod(S). The next theorem shows that each orbit is dense in ∂B_{X}.

Theorem 5.6. For any maximal cusp [ρ] ∈ ∂B_{X}, its orbit {[ρ]^{σ}}_{σ∈Mod(S)} under the action of Mod(S) is dense in ∂B_{X}.

Proof. Since the set of maximal cusps is dense in ∂B_{X}, we only have to show that, for arbitrary fixed two maximal cusps [ρ] and [ρ′] in ∂B_{X}, the orbit {[ρ]^{σ}}_{σ∈Mod(S)} of [ρ] contains a sequence converging to [ρ′]. Let λ and λ′ are accidental parabolic loci for [ρ] and [ρ′], respectively. Then we can find a maximal curve system ν = {γ_{j}}_{j=1}^{N} such that the pairs (λ, ν) and (ν, λ′) are binding (Lemma 5.5). Put σ = D_{γ_{1}} ∘ ⋯ ∘ D_{γ_{N}} and τ = D_{β_{1}} ∘ ⋯ ∘ D_{β_{N}}, where λ′ = {β_{j}}_{j=1}^{N}. Then [ρ]^{σ^{n}} converges to an maximal cusp [ρ′′] ∈ ∂B_{X} whose accidental parabolic locus is ν by Proposition 5.2. Similarly [ρ′′]^{τ^{n}} converges to [ρ′]. Since the action of Mod(S) is continuous at maximal cusps (Corollary 3.2), we can find a desired sequence by a diagonal method.

6. Orbits of Schottky groups and Bers boundary

In this section, we assume that S is a closed surface of genus g. We denote by S_{X} the set of [ρ] = (ρ, G) ∈ C_{X} such that G is a Schottky group.

Lemma 6.1. The set S_{X} consists of one orbit under the action of Mod(S); that is, S_{X} = {[ρ]^{σ}}_{σ∈Mod(S)} for any [ρ] ∈ S_{X}.

Proof. Let (ρ_{1}, G_{1}) and (ρ_{2}, G_{2}) be arbitrary two elements of S_{X}. Then there exists a homeomorphism N_{G_{1}} → N_{G_{2}} such that the restriction of this map to the boundaries is a quasiconformal map Ω_{0}(G_{1})/G_{1} → Ω_{0}(G_{2})/G_{2}. Now one can see that [ρ_{2}] = [ρ_{1}]^{σ}, where σ ∈ Mod(S) is an isotopy class of a homeomorphism of S induced by the quasiconformal map.

A Kleinian group is called geometrically finite if it has a finite sided convex fundamental polyhedron in H^{3}.

Lemma 6.2 (Hejhal [9], Matsuzaki [19]). Each element [ρ] ∈ S_{X} is an isolated point in C_{X}. On the other hand, if torsion-free, geometrically finite element [ρ] ∈ C_{X} is isolated in C_{X}, then [ρ] ∈ S_{X}.

Proof. The first statement is due to Hejhal [9], who showed that each Schottky group [ρ] ∈ Ĉ_{X} is isolated in Ĉ_{X}. Conversely, let [ρ] ∈ C_{X} be an isolated point of C_{X}. Since the same argument of Lemma 2.1 reveals that Ĉ_{X} − C_{X} is closed, [ρ] is also isolated in Ĉ_{X}. It was shown by Matsuzaki ([19], Theorem 3) that, if a torsion-free, geometrically finite element [ρ] ∈ Ĉ_{X} is isolated in Ĉ_{X}, then [ρ] is a Schottky group. Thus, the second statement is proved.

Remark . In Matsuzaki [19], it is obtained the necessary and sufficient condition for a (not necessarily torsion-free) geometrically finite element of Ĉ_{X} to be isolated in Ĉ_{X}.

For [ρ] ∈ S_{X}, the following lemma gives a characterization of the elements of Mod(S) which stabilize [ρ].

Lemma 6.3. Let [ρ] = (ρ, G) ∈ S_{X} and σ ∈ Mod(S). Then the followings are equivalent:

(1) [ρ]^{σ} = [ρ],

(2) σ_{∗}(ker ρ) = ker ρ, and

(3) σ can be extended to a homeomorphism of the Kleinian manifold N_{G}, where σ is regarded as a homeomorphism of X = ∂N_{G}.

Proof. (1) ⇒ (2) and (3) ⇒ (2) are trivial. (2) ⇒ (1) can be seen from [19, Theorem 2]. We will show that (2) ⇒ (3). Let (f, ρ) be the projective structure corresponding to [ρ]. We may assume that σ : X → X is a quasiconformal map. Let σ̃ : Δ → Δ be a lift of σ : X → X. If σ_{∗}(ker ρ) = ker ρ, then σ̃ descends to a quasiconformal map σ̂ : f(Δ) → f(Δ), because the covering group f : Δ → f(Δ) is ker ρ. Since G = ρ(π_{1}(S)) is geometrically finite and Ω(G) = f(Δ), Marden’s isomorphism theorem [15] implies that σ̂ can be extended to a G-compatible quasiconformal automorphism of Ĉ. This quasiconformal map can be extended to a G-compatible homeomorphism of H^{3} ∪ Ĉ, which descends to a homeomorphism of N_{G}.

Theorem 6.4. The set of accumulation points of S_{X} contains ∂B_{X}.

Remark . It is known by Gallo [7] that there is an accumulation point of S_{X} which is not contained in ∂B_{X}. This can be seen also from a slight modification of the following argument.

Proof of 6.4. Let [ρ] = (ρ, G) ∈ S_{X}. We claim that, for some σ ∈ Mod(S), the sequence {[ρ]^{σ^{n}}}_{n∈Z} converges to some maximal cusp [ρ_{∞}] ∈ ∂B_{X} as |n| → ∞. If it has shown, the similar argument in Theorem 5.6 reveals that the claim of the theorem holds. In fact, for any element [ρ′] ∈ ∂B_{X}, there exists a sequence {τ_{n}} in Mod(S) such that [ρ_{∞}]^{τ_{n}} converges to [ρ′] by Theorem 5.6. Since the action of Mod(S) is continuous at maximal cusps (Corollary 3.2), we can find a sequence in S_{X} which converges to [ρ′] by a diagonal method.

Now we will show that, for some σ ∈ Mod(S), the sequence {[ρ]^{σ^{n}}}_{n∈Z} converges to some maximal cusp [ρ_{∞}] ∈ ∂B_{X} as |n| → ∞. Note that the Kleinian manifold N_{G} is homeomorphic to a handle body H_{g}. With the identification G = π_{1}(H_{g}), AH(H_{g}) is properly embedded in V (S) so that [ρ] ∈ AH(H_{g}) and hence QC_{0}(ρ) ⊂ AH(H_{g}). Let Σ be a surface with boundary ∂Σ such that Σ × I is homeomorphic to H_{g}. (For example, let Σ be a surface of type (1, g − 1).) We can find a pair (λ′, λ′′) of maximal curve systems on Σ which binds Σ (cf. Lemma 5.3). For this pair, we define a maximal curve system λ on S = ∂(Σ × I) = ∂H_{g}, as

λ = (λ′ × {0}) ∪ (λ′′ × {1}) ∪ (∂Σ × {1/2}).

Then, by Lemma 4.3, (H_{g}, λ) is doubly incompressible and hence, by Theorem 4.2, AH(H_{g}, λ, K) is compact. Put σ = D_{α_{1}} ∘ ⋯ ∘ D_{α_{N}} ∈ Mod(S), where λ = {α_{j}}_{j=1}^{N}. Then {[ρ̄_{n}] = Ψ_{ρ}(σ^{−n}X)}_{n∈Z} is contained in a compact set AH(H_{g}, λ, K) of V (S) for some K, since

lρ̄_{n}(λ) ≤ Rl_{σ^{−n}X}(λ) = Rl_{X}(λ),

where R is a constant in Lemma 5.3. Hence, {[ρ̄_{n}]}_{n∈Z} has a convergent subsequence. On the other hand, since C_{X} is compact, {[ρ]^{σ^{n}}}_{n∈Z} also has a convergent subsequence. Take representatives ρ̄_{n} of [ρ̄_{n}] converging to a representation ρ̄_{∞}. Then ρ_{n} = ρ̄_{n} ∘ σ^{−n} are representatives of [ρ]^{σ^{n}}. Therefore, there are elements ψ_{n} ∈ PSL_{2}(C) such that ψ_{n} · ρ_{n} · ψ_{n}^{−1} converges to a representation ρ_{∞}.

Take a component α of λ and let T be a component of S − λ containing α in its boundary. Let α′(≠ α) be a component of λ contained in the boundary of T. Choose a base point x in T and regard π_{1}(S) = π_{1}(S, x). By abuse of notation, α and α′ also denotes the elements of π_{1}(S, x) contained in T. Note that ⟨α, α′⟩ is a rank 2 free subgroup of π_{1}(S, x) which is mapped into π(H_{g}, x) injectively, and that ρ̄_{n}|⟨α, α′⟩ are discrete faithful representations. Moreover since ρ_{n}|⟨α, α′⟩ = ρ̄_{n}|⟨α, α′⟩, by Lemma 5.4, the elements ψ_{n} ∈ PSL_{2}(C) may be taken to be the identity.

One can find non-trivial elements γ_{1}, γ_{2} ∈ π_{1}(S, x) each of which intersects α twice in the opposite direction and does not intersect any other components of λ, and that ⟨γ_{1}, γ_{2}⟩ is a rank 2 free subgroup of π_{1}(S, x) which is mapped into π_{1}(H_{g}, x) injectively. Then

ρ_{n}(γ_{1}) = ρ̄_{n}(α^{−n}) · ρ̄_{n}(γ_{1}) · ρ̄_{n}(α^{n})

and

ρ_{n}(γ_{2}) = ρ̄_{n}(α^{−n}) · ρ̄_{n}(γ_{2}) · ρ̄_{n}(α^{n})

holds. Since both ρ_{n} and ρ̄_{n} converge on γ_{1} and γ_{2}, Lemma 5.4 again implies that ρ̄_{n}(α^{n}) converges to an element α̂ in PSL_{2}(C). The same argument in the proof of Proposition 5.2 reveals that ⟨ρ̄_{∞}(α), α̂⟩ is a rank 2 parabolic subgroup in PSL_{2}(C). Hence, ρ̄_{∞}(α) is parabolic. Recall that ρ_{n}(α) = ρ̄_{n}(α) for all n. Therefore, ρ_{∞}(α) also would be a parabolic element. The same argument works well for all components of λ. Hence, by Lemma 5.1, we can conclude that [ρ_{∞}] is a maximal cusp whose accidental parabolic locus is λ.

References

[1] W. Abikoff, On boundaries of Teichmüller spaces and on Kleinian groups:III, Acta Math. 134 (1975), 211-234.

[2] L. Bers, On boundaries of Teichmüller spaces and on Kleinian groups:I, Ann. of Math. 91 (1970), 570–600.

[3] , The action of the modular group on the complex boundary, Ann. of Math. Stud. 97 (1981), 33–52.

[4] J. F. Brock, Iteration of mapping classes on a Bers slice: examples of algebraic and geometric limits of hyperbolic 3-manifolds, Contemporary Math. 211 (1997), 81–106.

[5] R. D. Canary, The Poincaré metric and a conformal version of a theorem of Thurston, Duke Math. J. 64 (1991), 349–359.

[6] A. Fathi, F. Raudenbach, and V. Poenaru, Travaux de Thurston sur le surfaces, Astérisque 66-67 (1979).

[7] D. M. Gallo, Some special limits of Schottky groups, Proc. Amer. Math. Soc. 118 (1993), 877–883.

[8] , Schottky groups and the boundary of Teichmüller space: genus 2, Contemporary Math. 169 (1994), 283–305.

[9] D. A. Hejhal, On Schottky and Koebe-like uniformizations, Duke Math. J. 55 (1987), 267–286.

[10] S. P. Kerckhoff and W. P. Thurston, Non-continuity of the action of the modular group at Bers’ boundary of Teichmuller space, Invent. Math. 100 (1990), 25–47.

[11] I. Kra, A generalization of a theorem of Poincaré, Proc. Amer. Math. Soc. 27 (1971), 299–302.

[12] , Deformations of Fuchsian groups, II, Duke Math. J. 38 (1971), 499–508.

[13] , On spaces of Kleinian groups, Comm. Math. Helv. 47 (1972), 53–69.

[14] I. Kra and B. Maskit, Remarks on projective structures, Ann. of Math. Stud. 97 (1981), 343–359.

[15] A. Marden, The geometry of finitely generated Kleinian groups, Ann. of Math. 99 (1974), 384–462.

[16] B. Maskit, On boundaries of Teichmüller spaces and on kleinian groups:II, Ann. of Math. 91 (1970), 607–639.

[17] , Self-maps of Kleinian groups, Amer. J. Math. 93 (1971), 840–856.

[18] , Kleinian groups, Springer-Verlag, 1988.

[19] K. Matsuzaki, Projective structures inducing covering maps, Duke Math. J. 78 (1995), 413– 425.

[20] C. T. McMullen, Cusps are dense, Ann. of Math. 133 (1991), 217–247.

[21] K. Ohshika, Geometrically finite Kleinian groups and parabolic elements, Proc. Edinburgh Math. Soc. 41 (1998), 141–159.

[22] D. P. Sullivan, On the ergodic theory at infinity of an arbitrary discrete group of hyperbolic motions, Ann. of Math. Stud. 97 (1981), 465–496.

[23] W. P. Thurston, Hyperbolic structures on 3-manifolds, III: Deformations of 3-manifolds with incompressible boundary, preprint.

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

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