Annales Academiæ Scientiarum Fennicæ Mathematica Volumen 24, 1999, 305–342

THE OUTSIDE OF THE TEICHMÜLLER SPACE OF PUNCTURED TORI IN MASKIT’S EMBEDDING

Jouni Parkkonen

University of Jyväskylä, Department of Mathematics P.O. Box 35, FIN-40351 Jyväskylä, Finland; parkkone@math.jyu.fi

Abstract. We show that for each cusp on the boundary of Maskit’s embedding ℳ ⊂ H of the Teichmüller space of punctured tori there is a sequence of parameters in the complement of \overline{ℳ} converging to the cusp such that the parameters correspond to discrete groups with elliptic elements. Using Tukia’s version of Marden’s isomorphism theorem we identify them as cusps on the boundary of certain deformation spaces of Koebe groups with a non-simply connected invariant component.

1. Introduction

A Kleinian group G is a discrete group of Möbius transformations, that is, of conformal bijections of the extended complex plane Ĉ = C ∪ {∞}. We use matrix notation for Möbius transformations, identifying the transformation

(1.1)  z ↦ \frac{az + b}{cz + d}

with the matrix

(1.2)  ( \begin{matrix}a & b \\ c & d \end{matrix} ) ∈ PSL(2, C).

The extended complex plane is divided into two disjoint sets according to the type of the action of G on these sets: the set of discontinuity, or the ordinary set Ω(G) where the group G acts discontinuously, and the limit set Λ(G) = Ĉ\ Ω(G). For material on Kleinian groups, we refer to Maskit [21].

Let G be a Kleinian group. If X ⊂ Ĉ, then

(1.3)  Stab(X) = Stab_{G}(X) = {g ∈ G : g(X) = X}.

1991 Mathematics Subject Classification: Primary 30F40; Secondary 32G15.

Research supported by the Academy of Finland (Grant 34082) and Magnus Ehrnroothin säätiö of the Finnish Society of Sciences and Letters.

is the stabilizer of X in G. A component of the ordinary set such that Stab_{G}(Ω_{0}) = G is called an invariant component of G. G is a Fuchsian group of the first kind if Ω(G) consists of two G-invariant round disks. G is called a terminal b-group of type (g, n) if it has a simply connected invariant component Ω_{0}(G) ⊂ Ω(G) such that Ω_{0}/G is a Riemann surface of genus g with n punctures, and if (Ω \ Ω_{0})/G is a collection of 2g − 2 + n thrice punctured spheres.

The (quasiconformal) deformation space or Teichmüller space 𝒯 (G) of a Kleinian group G is (1.4) 𝒯 (G) = {w: Ĉ →Ĉ quasiconformal : w ∘ g ∘ w^{−1} ∈ PSL(2, C) for all g ∈ G}/ ∼,

where w_{1} ∼ w_{2} if there is a Möbius transformation A ∈ PSL(2, C) such that

(1.5)  w_{1} ∘ g ∘ w_{1}^{−1} = A ∘ w_{2} ∘ g ∘ w_{2}^{−1} ∘ A^{−1}  for all g ∈ G.

If G is finitely generated, then 𝒯 (G) is a finite dimensional complex manifold. If G is a torsion-free terminal b-group such that the quotient Riemann surface Ω_{0}(G)/G has genus g and n punctures, then 𝒯 (G) is naturally isomorphic to 𝒯 (g, n), the Teichmüller space of Riemann surfaces with finite area, genus g, and n punctures. The complex dimension of this space is 3g − 3 + n. See [2], [14], [17] and the references given there for more details on these spaces and the identifications between the various related spaces.

The Maskit embedding [19], [15] of the Teichmüller space 𝒯 (g, n) is defined by first identifying 𝒯 (g, n) with 𝒯 (G), where G is a torsion-free terminal bgroup as above. 𝒯 (G) is then analytically embedded in C^{3g−3+n} using traces and/or fixed points of a suitable collection of group elements.

In this paper we study the complex one-dimensional case of once-punctured tori, (g, n) = (1, 1). This is essentially the only one-dimensional case; 𝒯 (0, 4) is canonically biholomorphic to 𝒯 (1, 1). In fact, the conformal map identifying 𝒯 (1, 1) with 𝒯 (0, 4) has a particularly simple form if one uses Maskit’s embedding for both spaces. For more details on this we refer to Kra [15, Section 6].

Maskit’s embedding of 𝒯 (1, 1) can be represented as the set ℳ of parameters μ ∈ H for which the group G[μ] generated by two transformations

(1.6)  S = ( \begin{matrix}1 & 2 \\ 0 & 1 \end{matrix} )  or  S(z) = z + 2,

and

(1.7)  T [μ] = ( \begin{matrix}−iμ & −i \\ −i & 0 \end{matrix} )  or  T [μ](z) = \frac{1}{z} + μ,

is a terminal b-group representing a punctured torus on its invariant component Ω_{0}(G). Let

(1.8)  S̃ = T [μ]^{−1}S^{−1}T [μ] = ( \begin{matrix}1 & 0 \\ −2 & 1 \end{matrix} ) .

The outside of T (1, 1) in Maskit’s embedding 307

From (1.6)–(1.8) we see that for any μ ∈ C, G[μ] contains a discrete Fuchsian subgroup

(1.9)  Γ_{0} = ⟨S,S̃ ⟩,

the level 2 principal congruence subgroup of PSL(2, Z), that keeps both H and H^{∗} fixed. Moreover, Λ(Γ_{0}) = R ∪ {∞}. For parameters μ ∈ ℳ it is useful to think of G = G[μ] as the HNN extension of ⟨S,S̃ ⟩ by T [μ]. It follows from Maskit’s second combination theorem, [21, Theorem VII.E.5], that this is the case for all parameters μ such that Im μ > 2, implying H + 2i ⊂ ℳ .

Let Ω_{0}(G[μ]) = Ω_{0}[μ] be the invariant component of G[μ]. The quotient surface Ω(G[μ])/G[μ] consists of the disjoint union of a punctured torus Ω_{0}(G[μ])/G[μ] and a thrice punctured sphere. The punctures of the thrice punctured sphere H^{∗}/ Stab(H^{∗}) correspond to the three parabolic conjugacy classes of S, S̃ and

(1.10)  K =S̃S = ( \begin{matrix}1 & 2 \\ −2 & −3 \end{matrix} )

in ⟨S,S̃ ⟩. Note that for μ ∈ ℳ there are only two parabolic conjugacy classes in G[μ]: S and S̃ are conjugate in G[μ] by (1.8), and all parabolic elements of G[μ] are conjugate to parabolics of ⟨S,S̃ ⟩ by Maskit’s second combination theorem, [21, Theorem VII.E.5]. The parabolic transformation K =S̃S corresponds to the puncture on the torus component of the quotient Riemann surface.

The set ℳ and its boundary have been studied in detail by Keen and Series in [10] and by Wright in the unpublished manuscript [31]. In this note we consider discrete groups generated by S and T [μ] for parameters μ ∈ C \ \overline{ℳ} . We can restrict our attention to the case μ ∈ \overline{H} \ ℳ , as a simple calculation shows that for all μ ∈ C

(1.11)  G[μ] = EG[−μ]E,

for E(z) = −z, E^{2} = 1. In other words, G[−μ] is discrete if and only if G[μ] is discrete.

We consider the following questions:

(1) For which parameters μ ∈ C \\overline{ℳ} is the group G[μ] discrete?

(2) If G[μ] is discrete, what is the geometry of the quotient spaces Ω(G[μ])/G[μ] and H^{3}/G[μ]? If μ ∈ ℳ , the geometry of the quotient space H^{3}/G[μ] is well understood.

The boundary of the convex core (See Section 5 for the terminology used here) is a pleated surface in the sense of Thurston [29] with two components, a totally geodesic sphere with three punctures, and a punctured torus that is planar outside a measured geodesic lamination. Keen and Series [10] studied the associated pleating structure on ℳ in detail. In particular, they showed that the pleating ray, the locus of points in ℳ with a fixed pleating lamination (fixed as a projective measured lamination) is homeomorphic to a line, with exactly one endpoint on ∂ ℳ . If the pleating lamination is a simple closed geodesic, the associated pleating ray is said to be rational. Each rational pleating ray 𝒫_{p/q} ends at a cusp μ_{p/q} ∈ ℳ , such that G[μ_{p/q}] is a geometrically finite Kleinian group that represents the disjoint union of two thrice punctured spheres.

Keen and Series showed that each rational pleating ray is contained in the real locus of a polynomial in μ, namely the trace of an element W_{p/q} of G[μ] representing the pleating locus on Ω(G[μ])/G[μ]. Using a slight modification the circle chain method of Keen and Series [10] and Wright [31] (see Sections 6 and 8), a result of McShane, Parker and Redfern [23] on cusp groups, and Maskit’s second combination theorem [22] we prove that if we only consider parameters in the real locus associated with a pleating ray, we can determine which parameters close to the cusp correspond to a discrete group.

9.4. Theorem. On the extension 𝒫p/q^{+} of each rational pleating ray there is an open neighborhood 𝒰 of the cusp μ_{p/q} on 𝒫p/q^{+} such that if μ ∈ 𝒫p/q^{+} ∩ 𝒰 and | tr W_{p/q}| = 2 cos(π/n) for some n ∈ N, n ≥ 3, then

(1.12)  G[μ] = F ∗_{W_{r/s}[S,T [μ]]} .

For these values of μ, G[μ] is a Kleinian group representing a thrice punctured sphere and a sphere with a puncture and two branch points of order n on its ordinary set.

If μ ∈ (𝒫p/q^{+} ∩ 𝒰 ) \\overline{ℳ} is not of this form, then G[μ] is not discrete.

A similar observation for the Bers embedding of Teichmüller space is made by Kra and Maskit in [16], and for the so-called Riley slice of Schottky space by Riley in the introduction to [27].

We do not have an estimate for the size of the neighborhood 𝒰 or how big n has to be for a fixed rational number p/q to guarantee that the group is discrete. Computer experiments suggest that the groups should be discrete for all n ≥ 2. See Remark 9.7 for a special case.

We also study groups that are HNN extensions of triangle groups that uniformize a pair of spheres with a puncture and two cone points of equal orders. In Sections 2–5 we introduce a pleating structure on the deformation spaces ℳ_{n} of groups of this form, analogous to [10], concentrating on rational pleating rays. The motivation for studying these groups is that we can show that the discrete groups found in Theorem 9.4 are conjugate to groups that correspond to cusps on the boundaries of the spaces ℳ_{n}. The deformation spaces ℳ_{n} are treated in [1] and [26]. For details on the pleating structure on the Riley slice see [11] and [13].

9.6. Corollary. Let p/q ∈ Q, n ∈ N, n ≥ 3, and p′ ∈ Z such that pp′ = 1 mod q. Let μ_{p′/q,n} be the boundary point of ℳ_{n} on the ray 𝒫_{p′/q}^{n}. If the group G[μ_{p/q}(n)] is discrete, then it is conjugate to G_{n}[μ_{p′/q,n}] by a Möbius transformation.

Acknowledgments. I am grateful to Linda Keen for suggesting the problem “What happens to pleating rays on the other side of the boundary of ℳ ?” at the special session in honor of Bernard Maskit’s 60th birthday at the AMS meeting at Hartford, Connecticut, in March 1995. The work presented in this paper is the result of my attempt to understand this question.

Part of the research for this paper was done during my stay at SUNY at Stony Brook in 1994–1995 and at Institut Henri Poincaré, Centre Emile Borel in 1996. I would like to thank these institutes and the organizers of the Semester on Low Dimensional Geometry and Topology (CEB/IHP) for their hospitality.

I would also like to thank Francis Bonahon for explaining to me the topologist’s point of view, and Curtis T. McMullen and David J. Wright for the computer programs used to produce the limit sets of Kleinian groups and the picture of the deformation space. Figures 1 and 11 were computed using McMullen’s program. Figures 2, 9 and part of Figure 5 were produced with Wright’s program.

Finally, I would like to express my gratitude to the referee for their valuable comments.

2. Deformation spaces of Koebe groups of type (1, 1, ∞)

In this section we consider Kleinian groups that can be obtained from hyperbolic triangle groups by an HNN construction conjugating two maximal cyclic subgroups. These groups include torsion-free terminal b-groups (see [15]) and a class of Koebe groups (see [26]) where one simple closed geodesic on the corresponding punctured torus is represented, respectively, by a parabolic or an elliptic transformation.

2.1. Definition. A Kleinian group G that has an invariant component Ω_{0} ⊂ Ω(G) is called a Koebe group if any other component Δ′ ⊂ Ω(G) \ Ω_{0} is a round disk. If the stabilizers of the disk components are hyperbolic triangle groups, the Koebe group is terminal. If the invariant component Ω_{0} is simply connected, the group is called a b-group. In this paper Koebe group usually means a terminal Koebe group that is not a b-group.

In this paper a presentation of a Kleinian group contains, in addition to the usual presentation of a group, a list of conjugacy classes that are assumed to consist of parabolic transformations. Sometimes the group may contain additional parabolic elements not included in this list.

A Fuchsian group F acting on the hyperbolic plane H is called a (hyperbolic) triangle group if H/F is a sphere with p punctures and and e projections of fixed points of elliptic elements of F such that p + e = 3. We refer to the projections of elliptic fixed points on Ω(G)/G as cone points. The signature of a triangle group is a triple (n_{1}, n_{2}, n_{3}), n_{i} ∈ {2, 3, …, ∞} that records the orders of the transformations of F corresponding to the cone points and punctures. In particular, a puncture on H/F is denoted by ∞ in the signature.

Let G be a Kleinian group. An elliptic Möbius transformation g ∈ G is a primitive element of G, if it is conjugate to a rotation by 2π/n, n ∈ Z, and if any elliptic element in G with the same fixed points as g is a power of g. Let x ∈Ĉ be the fixed point of a parabolic transformation of G. A parabolic transformation g ∈ G is primitive if it is not a positive power of any other transformation of G.

2.2. Definition. Let F_{n}, 3 ≤ n ≤ ∞, be a triangle group of signature (n, n, ∞). If 3 ≤ n < ∞, two primitive elliptic transformations A and B are canonical generators for F_{n}, if F_{n} has the presentation

(2.1)  F_{n} = ⟨A_{n}, B_{n} : A_{n}^{n} = B_{n}^{n} = id, K_{n} = B_{n}A_{n} parabolic⟩.

If n = ∞, two primitive parabolic transformations A_{∞} and B_{∞} in F_{∞} are canonical generators if F_{∞} has the presentation

(2.2)  F_{∞} = ⟨A_{∞}, B_{∞} : A_{∞}, B_{∞}, and K_{∞} = B_{∞}A_{∞} parabolic⟩.

Let F_{n} be as above. We wish to construct a new Kleinian group G = ⟨F_{n}, C⟩ that represents a punctured torus on an invariant component of its set of discontinuity. This amounts to the following gluing construction. Let X be a component of the quotient Ω(F_{n})/F_{n}. We cut out disks around the elliptic special points (for n finite) or punctures (for n = ∞) on X corresponding to the canonical generators A_{n} and B_{n}, and glue the boundaries of the remaining surface together, thus producing a punctured torus. This can be achieved by adding a new generator C_{n}[τ] satisfying

(2.3)  C_{n}^{−1}A_{n}C_{n} = B_{n}^{−1}

such that the group G_{n} = ⟨F_{n}, C⟩ is discrete and G_{n} has the presentation

(2.4) G_{n} = ⟨A_{n}, B_{n}, C_{n} : A_{n}^{n} = B_{n}^{n} = id, C_{n}^{−1}A_{n}C_{n} = B_{n}^{−1}, K_{n} = B_{n}A_{n} parabolic⟩,

and similarly for n = ∞; see [15] or [26] for details. If G_{n} has the presentation (2.4), then it is the HNN extension of F_{n} by C_{n}, denoted

(2.5)  G_{n} = F_{n} ∗_{C_{n}} .

There is a one complex parameter family of transformations C_{n}[μ] for each n = 3, 4, …, ∞ satisfying condition (2.3). We describe the two cases briefly, and refer to [15] and [26] for details.

n = ∞. It is convenient to normalize the torsion-free triangle group F_{∞} so that it acts discontinuously on the union of the upper and lower half planes. We use a normalization where the generators are

(2.6)  S = A_{∞} = ( \begin{matrix}1 & 2 \\ 0 & 1 \end{matrix} )  and S̃ = B_{∞} = ( \begin{matrix}1 & 0 \\ −2 & 1 \end{matrix} ) .

The HNN extensions of F_{∞} are groups of the form

(2.7)  G[μ] = G_{∞}[μ] = ⟨S,S̃ ⟩∗_{T [μ]} = ⟨S, T [μ]⟩,

where

(2.8)  T [μ] = ( \begin{matrix}−iμ & −i \\ −i & 0 \end{matrix} ) ,

satisfying

(2.9)  S̃ = T [μ]^{−1}S^{−1}T [μ].

It was shown in [15] that for Im μ > 2, G[μ] is a terminal b-group uniformizing a punctured torus. The Maskit embedding of the Teichmüller space of once-punctured tori, ℳ is the open set of parameters μ ∈ H, containing {μ ∈ H | Im μ > 2} for which G[μ] is a terminal b-group. For these parameters

(2.10)  G[μ] = F_{∞} ∗_{T [μ]} .

3 ≤ n < ∞. After normalization we can assume that F_{n} is generated by two elliptic transformations

(2.11)  A_{n} = ( \begin{matrix}e^{−iπ/n} & 0 \\ 0 & e^{iπ/n} \end{matrix} ) ,

and

(2.12)

B_{n} = ( \begin{matrix}i sin(π/n) cosh d_{n} − cos(π/n) & −i sin(π/n) sinh d_{n} \\ i sin(π/n) sinh d_{n} & −i sin(π/n) cosh d_{n} − cos(π/n) \end{matrix} )

= ( \begin{matrix}2i/ sin(π/n) − e^{iπ/n} & −2i cot(π/n) \\ 2i cot(π/n) & −2i/ sin(π/n) − e^{−iπ/n} \end{matrix} ) ,

where

(2.13)  d_{n} = arcosh \frac{cos^{2} π/n + 1}{sin^{2} π/n} = 2 arcosh \frac{1}{sin(π/n)}

is the hyperbolic distance between the fixed points of A and B in D. The general form of the transformation C_{n} satisfying (2.3) is

(2.14)

C_{n}[μ] = ( \begin{matrix}\sqrt{μ} sinh(d_{n}/2) & −\sqrt{μ} cosh(d_{n}/2) \\ \sqrt{μ}^{−1}cosh(d_{n}/2) & −\sqrt{μ}^{−1}sinh(d_{n}/2) \end{matrix} )

= ( \begin{matrix}\sqrt{μ} cot(π/n) & −\sqrt{μ} / sin(π/n) \\ 1/(\sqrt{μ} sin(π/n)) & − cot(π/n)/\sqrt{μ} \end{matrix} ) .

Clearly the choice of the branch of the square root does not change C[μ] as a Möbius transformation. It was shown in [26] that for

(2.15)  |μ| > coth^{2}(d_{n}/4) = (\frac{1 + sin(π/n)}{cos(π/n)})^{2},

the group

(2.16)  G_{n}[μ] = ⟨F_{n}, C_{n}[μ]⟩

is a Koebe group that represents a punctured torus on its invariant component, and that the map G_{n}[μ] ↦ μ is a global coordinate for the deformation space of any fixed Koebe group G_{n}[μ_{0}]. We denote the image of the deformation space of G_{n}[μ_{0}] under this map by ℳ_{n}.

Figure 1. The limit set of a Kleinian group from ℳ_{4} corresponding to the parameter μ = 4.32 + 1.26i ≈ 4.5 e^{0.28} .

If μ ∈ ℳ_{n}, n finite, the invariant component Ω_{0} of G_{n}[μ] is infinitely connected, see Figure 1. There are homotopically nontrivial simple closed curves in Ω_{0} that are stabilized by conjugates of the finite cyclic group generated by A_{n}. These loops project n-to-1 to a non-dividing simple closed geodesic α on the punctured torus Ω_{0}/G_{n}[μ].

The mapping class group of a Riemann surface Y is the group of orientation preserving homeomorphisms of Y modulo those homotopic to the identity map. The deformation space 𝒯 (G_{n}[μ]) is 𝒯 (1, 1) factored by the action of the subgroup of the mapping class group generated by the nth power of the Dehn twist about this special curve. Complex analytically 𝒯 (G_{n}[μ]), 3 ≤ n < ∞, is a punctured disk, in ℳ_{n} the puncture is ∞. More detailed treatments of these deformation spaces can be found in [1] and [26].

The quotient space M = H^{3}/G_{n} is a three-orbifold whose topology can be described as follows: Let X = Ω_{0}(G_{n})/G_{n} be a punctured torus, and let α ⊂ X be the simple closed curve on X corresponding to the elliptic element A_{n}. The orbifold M is obtained from X ×[0, 1] by adding a singular two-handle D_{n}×[0, 1]. Here D_{n} is a disk with a metric cone singularity of order n in the origin, that is, D_{n} is the quotient of the unit disk D by the action of a finite cyclic group generated by a rotation z ↦ e^{2πi/n}z. The curve ∂ D_{n}× \frac{1}{2} is identified with α×\{0\}, and ∂ D_{n} × [0, 1] is identified with an annular neighborhood of α in X × {0}. The line {0} × [0, 1] ⊂ M is the singular locus of M, a line that connects the two cone points of order n in Ω(G_{n})/G_{n}.

Figure 2. This picture shows a 2 to 1 covering of ℳ_{21} by “the trace plane” ℳ̃_{n} . The plotted parameter is i tr C_{n}[τ] = i cot(π/21)(τ − 1/τ). The lifts of integral pleating rays 𝒫_{m}^{21} , m = 0, 1, …, 20 are shown. See Lemma 2.3 for more details on the covering, and Section 5 for the definition of pleating rays.

Although a parameter μ ∈ ℳ_{n} defines the transformation C_{n}[μ] uniquely as a Möbius transformation, the trace of C_{n} is not a single-valued analytical function on ℳ_{n}. The expression of C_{n} involves τ = \sqrt{μ} in an essential way, and the choice of the branch of the square root causes an ambiguity. Therefore, it is sometimes convenient to work in a 2 − 1 covering space. This is the parameterization used in Figure 2.

2.3. Lemma. Let

(2.17)  ℳ̃_{n} = {μ̃ ∈ C : ∃μ ∈ ℳ_{n} such that μ̃ = tr C[μ]}.

The map μ̃ ↦ μ is a 2 − 1 analytical covering map ℳ̃_{n} → ℳ .

Proof. Let μ = τ^{2} . By C_{n}[τ] we mean the expression of C_{n}[μ] with a fixed choice of the square root. With this notation,

(2.18)  tr C[τ] = cot(π/n)(τ − 1/τ).

ℳ_{n} is connected, and clearly no parameter from the unit circle can be contained in ℳ_{n}. Thus, ℳ_{n} ⊂ {μ = τ^{2} : |τ| > 1}. The map τ ↦ τ − (1/τ) maps the outside of the unit disk injectively onto the complement of the interval [−2i, 2i] ⊂ C.

(2.19)

[figure: \{τ ∈ C : |τ| > 1\}; z↦z^{2}; 2−1; τ↦τ−(1/τ); 1−1; C \ [−2i cot \frac{π}{n}, 2i cot \frac{π}{n}] ⊃ \{tr C[τ] : τ^{2} ∈ ℳ_{n}\}; \{τ : |τ| > 1\}⊃ ℳ_{n}.]

Combining these observations, it is clear that the map tr C[τ] ↦ μ is 2 to 1 from ℳ̃_{n} = {tr C[τ] : μ ∈ ℳ_{n}}.

3. Spaces of measured laminations

Keen and Series studied the Maskit embedding ℳ of 𝒯 (1, 1) in [10]. They related the parameters in ℳ to the geometry of the quotient of H^{3} by the action of the group G_{∞}[μ]. In order to do the same kind of investigation for ℳ_{n} we study the space of measured geodesic laminations on the punctured torus, and define the appropriate space of measured laminations associated with ℳ_{n}, n finite. The main difference to ℳ comes from the fact that there is a simple closed geodesic γ on Ω(G_{n})/G_{n} that is represented by an elliptic element of G_{n}. On the orbifold M = H/G_{n}, γ is a loop that winds once around the singular locus of M.

3.1. Definition. Let Y be a hyperbolic Riemann surface. A geodesic lamination on Y is a closed subset of Y that is the disjoint union of a collection of simple (not necessarily closed) geodesics on Y . A transverse measure σ on a geodesic lamination λ assigns to any embedded image I of [0, 1] transversal to λ, a Borel measure with finite total mass such that the measure is supported on I ∩λ, and the measure is invariant under isotopies that preserve the lamination λ. The pair (λ, σ) is a measured geodesic lamination. The lamination λ is the support of σ.

We denote the space of measured geodesic laminations on a Riemann surface Y by ℳℒ (Y). We use the weak topology for measures on the space ℳℒ (Y): A sequence (λ_{n}, σ_{n}) ∈ ℳℒ (Y) converges (weakly) to (λ, σ) ∈ ℳℒ (Y) if

(3.1)  lim_{n→∞} ∫_{I} φ dσ_{n} = ∫_{I} φ dσ

for any interval I transversal to λ and λ_{n} for all n, and for all compactly supported continuous functions φ. The space of projective measured laminations on Y is

(3.2)  𝒫ℳℒ (Y) = ℳℒ (Y) \ {0}/(γ, σ) ∼ (γ, r σ),

where r > 0. Using the topology induced from ℳℒ (Y) by the above projection, 𝒫ℳℒ (Y) is homeomorphic to S^{6g−7+2p}, where g is the genus of Y and p is the number of punctures on Y .

The structure of those geodesic laminations that support a transverse measure is more restricted than that of a general geodesic lamination.

3.2. Proposition. Let X be a punctured torus. Let λ ⊂ X be the support of a nonzero measured geodesic lamination. Then either

(1) λ is a simple closed geodesic and X \ λ is a sphere with a puncture and two geodesic boundary components of equal length , or

(2) λ is the closure of an infinite simple geodesic, and X \λ is a punctured bigon.

Proof. [29, Proposition 9.5.2].

For the general case we refer the reader to Thurston [29, Section 9] and to Otal [25, Section A.3]. For the rest of the paper we will only be concerned with the case of a punctured torus. If Y and Y′ are two homeomorphic Riemann surfaces of finite area, the associated spaces of measured laminations are canonically isomorphic. We denote the space of measured laminations on a punctured torus by ℳℒ (1, 1), and the corresponding projectivized space by 𝒫ℳℒ (1, 1).

The space 𝒫ℳℒ (1, 1) is naturally identified with R̂ = R ∪ {∞}: a geodesic lamination on a punctured torus can be identified with a geodesic on the square Euclidean torus, and the geodesics on the Euclidean torus are in 1 to 1 correspondence with their slopes r ∈ R ∪ {∞} in the standard uniformization of the square torus as C/Z^{2} . This identifies simple closed geodesics on the punctured torus with Q̂ = Q ∪ {∞}. We will call a simple closed geodesic on the punctured torus that is identified with r ∈ Q the r-curve. For more details on this identification see e.g. Birman and Series [3] Appendix, or Series [28].

4. G_{n} and simple closed curves on punctured tori

The set of simple closed curves on a Riemann surface Y can be embedded into 𝒫ℳℒ (Y), by associating a Dirac mass to each intersection of the simple closed curve with a transversal. With this identification, simple closed curves form a dense countable subset in 𝒫ℳℒ (Y). In this section we describe methods from combinatorial group theory that will be used for bookkeeping of group elements that correspond to simple closed curves in the groups G_{n}. In the following definition the “relation” a_{2}^{∞} = 1 means that the element a_{2} ∈ G has infinite order.

4.1. Definition. Let

(4.1)  G = ⟨a_{1}, a_{2} : a_{2}^{n} = 1⟩ =: ⟨a_{1}⟩ ∗ ⟨a_{2} : a_{2}^{n} = 1⟩,

n = 2, 3, …, ∞. A word

(4.2)  W = a_{ν_{1}}^{α_{1}}a_{ν_{2}}^{α_{2}} ⋯ a_{ν_{p}}^{α_{p}},

where p ∈ N, α_{i} ∈ Z, ν_{1} = ±1, is cyclically reduced in G (with generators a_{1} , a_{2} ) if

(1) ν_{i} ≠ ν_{i+1} for 1 ≤ i ≤ p − 1, ν_{1} ≠ ν_{p}, and

(2) 1 ≤ α_{i} ≤ n if ν_{i} = 2.

It is easy to see from the presentation (2.4) that for parameters μ ∈ ℳ_{n},

(4.3)  G_{n}[μ] = ⟨A_{n}⟩ ∗ ⟨C_{n}[μ]⟩.

We will make use of the following fact:

4.2. Theorem. Two cyclically reduced words in the group G_{n}[μ] for μ ∈ ℳ_{n} are conjugate if and only if they are cyclic permutations of each other.

Proof. [18, Theorem 1.4].

Let us first consider the case of terminal b-groups. A simple closed curve on the punctured torus corresponds to a conjugacy class of transformations of G[μ]. As described in Section 3, the free homotopy classes of simple closed curves on a punctured torus that are not parallel to the puncture, can be enumerated by Q ∪ {∞}. We will sketch a method of producing a representative

(4.4)  W_{p/q} = W_{p/q}[μ] = W_{p/q}[S, T [μ]]

called the p/q -word that is cyclically reduced in G[μ] with the generators S^{−1} and T [μ] such that W_{p/q} represents the p/q -curve on Ω_{0}[μ]/G[μ]. See e.g. [10], [24], or [4] for proofs.

The special words can be defined inductively using the Farey sequence as follows. Two rational numbers p/q and p′/q′ (where q, q′ > 0 and (p, q), (p′, q′) are relatively prime) are called (Farey) neighbors if

(4.5)  pq′ − p′q = ±1.

In the construction of the p/q -words, we first set

(4.6)  W_{1/0} = W_{∞} = S^{−1}  and  W_{0/1} = W_{0} = T.

If a/b < c/d (with the convention 1/0 = ∞ > r for all r ∈ Q) are neighbors, we require

(4.7)  W_{(a+c)/(b+d)} = W_{c/d}W_{a/b}.

Note that we get the words for negative integers n = −1, −2, … by solving the equations

(4.8)  W_{n+1} = W_{∞}W_{n}.

This process assigns a unique word W_{p/q} to each rational number p/q ∈ Q. It is easy to see from the construction of the words that for p ≥ 0, p is the number of S^{−1} ’s and q is the number of T ’s in W_{p/q}. For p < 0, −p is the number of S ’s in the word W_{p/q}.

4.3. Proposition. Let p/q and r/s be Farey neighbors. Then

(1) G[μ] = ⟨W_{p/q}, W_{r/s}⟩ , and

(2) If p/q < r/s, then

(4.9)  W_{r/s}^{−1}W_{p/q}^{−1}W_{r/s}W_{p/q} = T^{−1}S^{−1}T S = K,

where

(4.10)  K =S̃S = ( \begin{matrix}1 & 2 \\ −2 & −3 \end{matrix} ) .

Proof. See [10].

Let us now consider the case of finite n. There is a canonical surjective homomorphism

(4.11)  Φ_{n}: ⟨a⟩ ∗ ⟨b⟩ → ⟨a⟩ ∗ ⟨b : b^{n} = 1⟩.

Clearly some of the different words project to the same group element under Φ_{n}. In particular, this means that we do not get a 1-1 correspondence between the elements of G_{n} represented by the words W_{p/q} and free homotopy classes of simple closed curves on the punctured torus. However, the identifications take place in a controlled manner.

4.4. Proposition. Let n ∈ {3, 4, …, ∞}, and μ ∈ ℳ_{n}. The transformations W_{p/q}[A_{n}, C[μ]] and W_{p′/q′}[A_{n}, C[μ]] are conjugate in G_{n}[μ] if and only if p′/q′ = p/q + kn for some k ∈ Z. W_{p/q}[A_{n}, C[μ]] is not conjugate with any W_{p′/q′}^{−1}[A_{n}, C[μ]], p′/q′ ∈ Q ∪ {∞}.

Proof. Let [p/q] be the integral part of p/q. If we write W_{p/q}[A_{n}, C[μ]] in terms of W_{[p/q]}[A_{n}, C[μ]] and W_{[p/q]+1}[A_{n}, C[μ]] as a word V_{p/q}[W_{[p/q]}, W_{[p/q]+1}], then clearly V_{p/q}[W_{0/1}, W_{1/1}] = W_{p/q−[p/q]}[A_{n}, C[μ]]. As W_{m+kn} = W_{m} for all k ∈ Z as elements of G_{n}, the words W_{p/q}[A_{n}, C[μ]] and W_{kn+p/q}[A_{n}, C[μ]] represent the same element in G_{n}. We claim there are no further conjugacies among the special words.

The group G_{n} is isomorphic to the free product ⟨A_{n} : A_{n}^{n} = 1⟩ ∗ ⟨C⟩. Clearly we can restrict our attention to the case 0 ≤ p/q < n. The words W_{p/q}, p/q ∈ Q∩[0, n) are cyclically reduced in G_{n}. By Theorem 4.2, two elements of the group G_{n} = ⟨A_{n} : A_{n}^{n} = 1⟩ ∗ ⟨C⟩ are conjugate if their cyclically reduced presentations are cyclic permutations of one another. For a word W_{p/q}, p is the number of C^{−1} ’s and q is the number of A_{n} ’s. Both of these quantities are invariant under cyclic permutation, implying that W_{p/q} is conjugate with W_{p′/q′}, with 0 ≤ p/q, p′/q′ < n, only if p/q = p′/q′.

To prove the second statement, we observe that we get a cyclically reduced word W̃ _{p′/q′}^{−1} from W_{p′/q′}^{−1} , without changing the group element that the words represent in G_{n}, by formally inverting W_{p′/q′}, and then replacing each A^{−j}, 1 ≤ j ≤ n − 1, by A^{n−j}. W_{p/q} has only negative powers of C, and W̃ _{p′/q′}^{−1} has only positive powers of C, so by the same reasoning as above, no W_{p/q} can be conjugate to any W_{p′/q′}^{−1} .

The mapping class group has a natural action on ℳℒ (Y) and on 𝒫ℳℒ (Y). On 𝒫ℳℒ (1, 1) this action is the same as the action of PSL(2, Z) on R̂ if we use the identification of 𝒫ℳℒ (1, 1) with R̂ indicated in Section 3. In particular, the Dehn twist Dehn_{α} about the simple closed curve on Ω_{0}/G[μ] that corresponds to the conjugacy class of the parabolic transformation S, acts on R̂ as a translation by 1. Proposition 4.4 means that if two simple closed curves on the punctured torus Ω_{0}/G_{n} differ by a power of Dehn_{α}^{n}, they correspond to the same group element in G_{n}. The action of ⟨Dehn_{α}^{n}⟩ fixes ∞, which corresponds to the projective measured lamination with support α, and it is discontinuous on R = 𝒫ℳℒ (1, 1) \ {α}. The quotient space

(4.12)  𝒫ℳℒ _{n}(1, 1) = 𝒫ℳℒ (1, 1)/⟨Dehn_{α}^{n}⟩.

is the natural space of laminations associated with the deformation spaces ℳ_{n}.

5. Pleating structure for ℳ_{n}

Let G_{n}[μ], n ∈ {3, 4, …, ∞}, μ ∈ ℳ_{n}, be a Koebe group uniformizing a punctured torus as in Section 2. Let 𝒞 (Λ(G_{n}[μ])) ⊂ H^{3} be the hyperbolic convex hull of the limit set of G_{n}[μ]. 𝒞 (Λ(G_{n}[μ])) is invariant under the action of G_{n}[μ] in H^{3} , and the quotient

(5.1)  𝒞 M[μ] = 𝒞 (Λ(G_{n}[μ]))/G_{n}[μ]

is the convex core of the orbifold M[μ] = H^{3}/G_{n}[μ].

The boundary of 𝒞 M[μ] is homeomorphic to ∂ M[μ]. Let ∂ 𝒞_{0} = ∂ 𝒞_{0}(Λ(G_{n})) ⊂ H^{3} be the component of the boundary of the convex hull of the limit set that lies above the invariant component Ω_{0}(G_{n}). The quotient surface 𝒮 = ∂ 𝒞_{0}/G_{n} is a punctured torus. 𝒮 inherits a geometric structure from the embedding of ∂ 𝒞_{0} in H^{3} , in which it is planar outside a geodesic lamination. We say that 𝒮 [μ] is pleated along this lamination, called the pleating locus on 𝒮 [μ]. The pleating locus carries a natural transverse measure, the bending measure that measures the bending along any interval transversal to the pleating locus. A good reference for this material is [5].

The Teichmüller space 𝒯 (G_{n}) is naturally isomorphic with 𝒯 (1, 1)/⟨Dehn_{α}^{n}⟩, and the natural space of laminations associated with 𝒯 (G_{n}) is 𝒫ℳℒ _{n}(1, 1) as defined at the end of Section 4. The pleating map associated with the family {G_{n}[μ]}_{μ∈ℳ_{n}} is

(5.2)  pl: ℳ_{n} → 𝒫ℳℒ _{n}(1, 1),

where pl(μ) is the projective class in 𝒫ℳℒ _{n}(1, 1) of the pleating lamination of ℳ_{n}[μ]. Keen and Series [12] showed that the pleating map is continuous.

5.1. Remark. For the groups G_{n}[μ], μ ∈ ℳ_{n}, the convex core 𝒞 M[μ] has a singular locus homeomorphic to an interval, that connects the two cone points on the sphere component of ∂ 𝒞 M[μ]. Two simple closed curves are considered to be homotopic on M[μ] (or on 𝒞 M[μ]) if they can be mapped to each other by a sequence of operations consisting of homotopies of the underlying space of M fixing the singular locus of M, and of “tightening” an arc that winds around the singular line exactly n times in a small neighborhood. See Figure 3. We define 𝒫ℳℒ _{n}(1, 1) to be the space of geodesic measured laminations on ∂ (H^{3}/G_{n}[μ]) with this definition of homotopy on the orbifold. Another way to describe the simple closed curves in this space is to note that if the regular covering space of a surface (in our case the invariant component Ω_{0}(G[μ])) is not simply connected, it is natural to consider as equivalent all the simple closed curves that are represented by the same covering transformation.

Figure 3. Tightening an arc that winds three times around a singular line where the local structure is the product of an interval with D/z ↦ z^{3} . The figure shows a projection where the singular line is represented by the dot in the middle.

5.2. Definition. A geodesic measured lamination is rational if it is supported on a simple closed curve. In particular, the laminations in 𝒫ℳℒ (G_{n}), n = 3, 4, …, ∞, that correspond to words W_{m/1} , m ∈ Z are called integral laminations.

5.3. Definition. Let n = 3, 4, …, ∞, and let λ ∈ 𝒫ℳℒ (G_{n}). The set
(5.3)  𝒫_{λ}^{n} = {μ ∈ ℳ_{n} | pl(μ) = λ}

is the λ-pleating ray in ℳ_{n}. Let p/q ∈ Q, and let γ_{p/q} ∈ 𝒫ℳℒ _{n} be the projective equivalence class represented by the transformation W_{p/q}[A_{n}, C] ∈ G_{n}[μ].

(5.4)  𝒫_{p/q}^{n} = 𝒫_{γ_{p/q}}^{n}

is a rational pleating ray.

The trace of the transformation W_{p/q}[A_{n}, C[μ]] is not uniquely defined as an analytical function of μ ∈ ℳ_{n} if n is finite. However, the ambiguity of the choice of the branch of the square root does not affect the Möbius transformation, in particular, whether the transformation is hyperbolic or not. See Figure 2 for a collection of pleating rays.

5.4. Definition. Let n = 3, 4, …, ∞. The real locus of W_{p/q}[A_{n}, C[μ]] is

(5.5)  𝒱_{p/q}^{n} = {μ ∈ ℳ_{n} : tr W_{p/q}[A_{n}, C[μ]] ∈ R}.

5.5. Proposition. 𝒫_{p/q}^{n} is a union of connected components of 𝒱_{p/q}^{n} ∩ ℳ_{n}.

Proof. For n = ∞ this is Proposition 5.4 of [10]. The same proof applies for finite n; one needs to observe that the proofs of Lemma 4.6 and Lemma A.2 of [10] do not use their assumption that the groups under consideration in [10] are free.

In Section 6 we observe, analogously to the case n = ∞, that 𝒫_{p/q}^{n} is a single component of 𝒱_{p/q}^{n}.

6. Circle chains

In this section we generalize the definition of circle chains used in [10] and [31] for groups with elliptic elements. For n = ∞ the chains defined here are the same as in [10] and [31]. For finite values of n they retain most of the useful properties of the case n = ∞, the difference being the fact that the chains will consist of a finite number of disks. We will sometimes refer to these cases as finite chains. In order to simplify notation, we will use the following convention: the notation i ∈ Z mod nq should be read i ∈ Z if n = ∞.

6.1. Definition. Let n = 3, 4, …, ∞, and μ ∈ C. Let G[μ] = ⟨F_{n}, C_{n}[μ]⟩ be a group of Möbius transformations (not necessarily discrete), where F_{n} = ⟨A_{n}, B_{n}⟩ as earlier, and C is a transformation satisfying

(6.1)  C^{−1}A_{n}C = B_{n}^{−1}

as in Section 2. Let W_{p/q} = W_{p/q}[A_{n}, C] be the p/q -word defined in Section 4. A collection {δ_{i}}, i ∈ Z mod nq of closed, round disks is a combinatorial p/q -chain for G_{n} (with generators A_{n} and C) if it satisfies the following conditions:

(1) δ_{0} is tangent to Λ(F), the limit set of F, at the fixed point of the parabolic element

(6.2)  K_{n} = W_{r/s}^{−1}W_{p/q}^{−1}W_{r/s}W_{p/q},

(2) W_{p/q}δ_{0} = δ_{0} ,

(3) C(δ_{j}) = δ_{j+p} for all j = 0, …, q, and

(4) A(δ_{j}) = δ_{j+q} for all j ∈ Z.

If a circle chain {δ_{i}} satisfies (1)–(4), we say {δ_{i}} connects fix K to A(fix K). The chain is tangent if δ_{i} and δ_{i+1} are tangent and the interiors int δ_{i} and int δ_{i+1} are disjoint for every i ∈ Z. If we know that the group is discrete, we say that the chain is proper if

(1) the interiors of the disks δ_{i} are contained in Ω(G) for all i,

(2) the interiors of adjacent disks δ_{i} and δ_{i+1} intersect for all i, and

(3) int δ_{i} ∩ int δ_{j} = ∅ for |i − j| > 1.

Figure 4. A tangent \frac{3}{5} circle chain for G_{4} .

6.2. Remarks. (1) The combinatorial p/q chain is associated to a group with a fixed pair of generators. For a different choice of generators there may be a chain with different combinatorics. We use this fact in the proof of Theorem 9.4, see also McShane, Parker and Redfern [23].

(2) The disks δ_{0}, …, δ_{p+q−1} form a combinatorially interesting set: this part of the chain can be generated starting from δ_{0} by the following process:

(i) if 0 ≤ i < q, set δ_{i+p} = C(δ_{i}), (ii) if q ≤ i < p + q, set δ_{i−q} = A^{−1}(δ_{i}).

(3) Unlike Keen and Series [10] we do not require that the group G is discrete. In fact, in Section 9, we use the existence of special circle chains with some additional properties to prove that the group G[μ] defined in Section 2 is discrete for a number of the parameters μ found in Lemma 7.3.

(4) The dependence on n of the definition of the circle chain is a result of the fact that A_{n}^{n} = id. This implies that the chain is actually finite, consisting of nq circles. See Figure 4.

The results in this section are analogs of some of the results of [10, Sections 4 and 5]. The proofs are essentially the same. We only give some of the arguments and calculations for the case of finite n, and refer to [10] for the details.

6.3. Proposition. Let n = 3, 4, …, ∞, and μ ∈ C. The group G_{n}[μ] has a proper p/q chain if and only if μ ∈ 𝒫_{p/q}^{n}.

Proof. [10, Proposition 4.11].

6.4. Lemma. Let 0 ≤ m < n. The integral pleating ray 𝒫_{m/1}^{n} is the radial half line

(6.3)  𝒫_{m/1}^{n} = ℳ_{n} ∩ {μ = t^{2}e^{−i2πm/n} : t > 0}

in 𝒯 (G_{n}[μ]).

Proof. It is easy to check that

(6.4)  W_{m/1} = ( \begin{matrix}e^{iπm/n} sinh(d/2)\sqrt{μ} & ∗ \\ ∗ & −e^{−iπm/n} sinh(d/2)/\sqrt{μ} \end{matrix} ) ,

so if we set μ = t^{2}e^{2iφ}, we get

(6.5)  Im tr W_{m/1} = (t + \frac{1}{t}) sinh(d/2) sin(φ + \frac{πm}{n}).

Thus

(6.6)  𝒱_{m/1}^{n} ⊂ {μ ∈ ℳ_{n} : arg(μ) = −2πm/n}.

It remains to show the existence of an m/1 chain. Here we can follow the argument of [10, Lemma 5.2].

6.5. Proposition. Let n ∈ {3, 4, …}, and p/q ∈ Q \ Z. Then

(1) 𝒫_{p/q}^{n} ≠ ∅ for all p/q ∈ Q mod nZ.

(2) The pleating ray 𝒫_{p/q}^{n} is contained in the sector

(6.7)  {μ ∈ ℳ_{n} : −2π([p/q] + 1)/n ≤ arg μ ≤ −2π[p/q]/n}

bounded by the integral rays 𝒫_{[p/q]}^{n} and 𝒫_{[p/q]+1}^{n}, where [p/q] is the integral part of p/q.

(3) 𝒫_{p/q}^{n} is the component of 𝒱_{p/q}^{n} ∩ ℳ_{n} asymptotic to the line

(6.8)  {μ ∈ ℳ_{n} : arg μ = −2πp/(qn)}.

Proof. (1): By Lemma 6.4, the half-lines {μ ∈ ℳ_{n} : arg μ = 2πk/n}, k ∈ Z, are mapped to the integral values in Q/Z, and no other points are mapped to these values. Let S be a large circle in ℳ_{n} centered at 0. The restriction of the pleating map, pl |_{S} is a continuous map of degree 1 of a circle to a circle, in particular, a surjective map.

(2) and (3): The fact that the radial lines (6.3) contain all points of ℳ_{n} that are mapped to integral values implies that the nonintegral rays are contained in the sectors (6.7), because we assumed n ≥ 3.

Let μ = τ^{2} . An easy induction argument as in [10, Proposition 3.1] shows that

(6.9)  W_{p/q}[μ] = ( \begin{matrix}e^{p iπ/n}τ^{q} sinh(d/2)^{q} + O(|τ^{q−1}|) & O(|τ|^{q}) \\ O(|τ|^{q−2}) & O(|τ|^{q−2}) \end{matrix} )

as |τ| → ∞. When |τ| → ∞ the condition tr W_{p/q} ∈ R, that defines 𝒱_{p/q}^{n}, approaches

(6.10)  arg μ = arg τ^{2} = −2π(\frac{p}{nq} + \frac{k}{q} ),  k ∈ Z.

For large values of n there are several branches in the sector (6.7) that qualify as candidates for points in 𝒫_{p/q}^{n}. Clearly it is enough to consider the case 0 < p/q < 1. To rule out all but one of the branches of 𝒱_{p/q}^{n} we do an induction argument on q:

(a) Let p/q = 1/2. Then the branch of 𝒱_{1/2}^{n} asymptotic to the radial line with argument −π/n is the only one asymptotically contained in the sector between 𝒫_{0}^{n} and 𝒫_{1}^{n}. By Proposition 5.5 this implies that 𝒫_{1/2}^{n} equals this component.

(b) Assume that 𝒫_{k/q}^{n}, k = 0, 1, …, q satisfy the claims (2) and (3) of the theorem. Let p ∈ {1, 2, …, n−1} such that p and q+1 are relatively prime. Then, (6.10) and elementary properties of the rationals imply that there is exactly one component of 𝒱_{p/(q+1)}^{n} ∩ℳ_{n} contained between the correct rays of denominator q. Continuity of the pleating map now implies, as above, that this component is exactly the pleating ray.

Combining these observations with the analysis of Keen and Series [10, Section 5] we get the following theorem. Here we use the notation μ_{p/q,∞} = μ_{p/q}.

6.6. Theorem. Let n ∈ {3, 4, …, ∞}, and p/q ∈ Q. If n is finite, the pleating ray 𝒫_{p/q}^{n} coincides with the branch of 𝒱_{p/q}^{n} asymptotic with the half line of argument arg μ = −2πip/(qn). 𝒫_{p/q}^{∞} coincides with the vertical component of 𝒱_{p/q}^{∞}∩ℳ_{∞} asymptotic to the line {μ ∈ C : Im μ = 2p/q}. For all n ∈ {3, 4, …, ∞} the pleating ray 𝒫_{p/q}^{n} is homeomorphic to R, and \overline{𝒫} _{p/q}^{n} ∩ ∂ ℳ_{n} consists of a single point μ_{p/q,n} such that the transformation W_{p/q}[A_{n}, C[μ_{p/q,n}]] is parabolic. Furthermore, the group G_{n}[μ_{p/q,n}] is geometrically finite, and it has a tangent circle chain connecting fix K_{n} to A_{n} fix K_{n}.

Proof. One can copy the arguments of [10, Theorem 5.1], with only insignificant changes resulting from the fact that the trace function in [10] is a polynomial in μ, and for finite n it is a polynomial in \sqrt{μ} and 1/\sqrt{μ}. We refer to [10] for details.

7. Extensions of pleating rays and disk-preserving subgroups

In this section we consider an extension of a rational pleating ray across the boundary of ℳ . We will use the fact that the rational pleating rays coincide with the asymptotically vertical components of the real locus of the trace of the corresponding special word in ℳ .

7.1. Definition. 𝒫p/q^{+} , the connected component of

(7.1)  \overline{𝒫_{p/q}} ∪ \{μ ∈ 𝒱_{p/q} ∩ (H \\overline{ℳ} )\}

containing 𝒫_{p/q} is the extended p/q ray.

7.2. Remarks. (1) The trace function, tr W_{p/q}: C → C, may have a critical point close to the boundary, even at μ_{p/q}, that is, 𝒫p/q^{+} might have a branching point. The somewhat awkward definition of the extended ray is used in this form so that we only get the correct ray corresponding to the pleating ray inside ℳ , and we avoid discussing the issue of critical points. In particular, the extended ray is not necessarily homeomorphic to R.

(2) Similarly, one could define extended rays for ℳ_{n}, n = 3, 4, …. However, the methods used in Sections 8 and 9 to study the discreteness of groups in H \\overline{ℳ} cannot be directly applied to treat the corresponding cases for finite n.

Figure 5. Parts of the real loci 𝒱_{0/1} and 𝒱_{2/7} . The cusped, nonsmooth curve in the figure is an approximation part of the boundary of ℳ , ℳ is the region above this boundary. 𝒫0/1^{+} = 𝒱_{0/1} = {μ ∈ H : Re μ = 0}, and the curve passing through the point labeled 2/7 is contained in the extended ray 𝒫2/7^{+} . The third smooth curve in the figure is part of 𝒱_{2/7} \ 𝒫2/7^{+} .

7.3. Lemma. There are points μ_{p/q}(n) ∈ 𝒫p/q^{+} for all large values of n such that W_{p/q}[μ_{p/q}(n)] is elliptic of order n, and

(7.2)  lim_{n→∞} μ_{p/q}(n) = μ_{p/q}.

Proof. The function μ ↦ tr W_{p/q}[μ] is holomorphic. Thus, the image of a small neighborhood of μ_{p/q} covers a small neighborhood of tr W_{p/q} = ±2, thus for large n the values tr W_{p/q}[μ] = ±2 cos(π/n) are attained.

Let p/q and r/s > p/q be Farey neighbors, μ ∈ C. Let

(7.3)  F = F_{p/q}[μ] = ⟨W_{p/q}, K⟩ = ⟨W_{p/q}, W_{r/s}^{−1}W_{p/q}^{−1}W_{r/s}⟩.

In most of what follows we aim at proving that for big n the groups G[μ_{p/q}(n)] are discrete. The method is based on a ‘reconstruction’ of G[μ] starting from a Fuchsian subgroup. In Section 2 we presented G[μ] as the HNN-extension of the torsion-free Fuchsian group ⟨S,S̃ ⟩. In Section 9 we will use a different set of generators, in terms of which the group is the HNN-extension of F_{p/q} for μ ∈

𝒫p/q^{+} .

Keen and Series, [10, Proposition A.1], show that if μ ∈ 𝒫_{p/q}, then F_{p/q} is a Fuchsian group of the second kind, that represents a punctured cylinder with boundary geodesics of equal length. For parameters in the extension of the p/q ray we have:

7.4. Lemma. If μ ∈ 𝒫p/q^{+} and tr W_{p/q} = 2 cos(π/n) for some n ∈ N, n ≥ 3, then F = F_{p/q}[μ] is a triangle group of signature (n, n, ∞).

Proof. The elements W_{p/q}^{−1} , W_{r/s}^{−1}W_{p/q}W_{r/s} and their product
(7.4)  W_{r/s}^{−1}W_{p/q}W_{r/s}W_{p/q}^{−1} = K

all have real traces. It follows from [6, Lemma 5.23] that F preserves a disk D. The quadrilateral with vertices at the fixed points of W_{p/q}, W_{r/s}^{−1}W_{p/q}W_{r/s}, K and

(7.5)  K̃ =K̃_{p/q} = W_{p/q}^{−1}KW_{p/q} = W_{p/q}^{−1}W_{r/s}^{−1}W_{p/q}W_{r/s}

in \overline{D} satisfies the conditions of Poincaré’s theorem.

7.5. Lemma. If μ ∈ 𝒫p/q^{+} and tr W_{p/q} ∈ (−2, 2) \ {2 cos(π/n) : n ∈ N, n ≥ 2}, then F = F_{p/q}[μ] is not discrete.

Proof. If W_{p/q} is an infinite order elliptic, the group is trivially non-discrete, so we can assume W_{p/q} is a non-primitive elliptic. Let W_{prim} be a primitive generator of ⟨W_{p/q}⟩. F has parabolic elements, so any fundamental polygon has cusps extending to the circle at infinity. The commutator relation implies that the isometric circles of W_{p/q} and W_{r/s}^{−1}W_{p/q}^{−1}W_{r/s} are tangent at the fixed points of K and \overline{K}. Clearly the isometric circles of W_{prim}^{±1} and W_{r/s}^{−1}W_{prim}^{∓1}W_{r/s} intersect inside D. The Ford fundamental region of F would thus be bounded away from the circle at infinity, which is impossible because there are parabolic elements in the group.

Lemma 7.5 has the following immediate corollary.

7.6. Proposition. Let μ ∈ 𝒫p/q^{+} \\overline{ℳ} . If G[μ] is discrete, then | tr W_{p/q}| = 2 cos(π/n) for some n = 2, 3, ….

This argument uses the disk-preserving subgroup F_{p/q} in an essential way. We can only find a subgroup like this for rational rays, and do not know how to treat groups corresponding to parameters that are not contained in an extended rational ray. However, we have the following result as an application of Jørgensen’s inequality [7] which implies that the discrete groups we can find in H \ ℳ are isolated. A Kleinian group is elementary if its limit set has fewer than two points.

7.7. Proposition. For each μ_{p/q}(n) there is an open neighborhood U_{p/q}(n) ⊂ C such that if μ ∈ U_{p/q}(n) \ {μ_{p/q}(n)}, then G[μ] is not discrete.

Proof. Consider the subgroup

(7.6)  H_{p/q}(n) = ⟨K, W_{p/q}^{n}⟩,

where

(7.7)  K = ( \begin{matrix}1 & 2 \\ −2 & −3 \end{matrix} ) .

By Jørgensen’s inequality, [7, Lemma 1], we know that if

(7.8)  τ(μ) = | tr(KW_{p/q}[μ]^{n}K^{−1}W_{p/q}[μ]^{−n} − 2| < 1,

then either H_{p/q}(n) is elementary or it is not discrete. If μ = μ_{p/q}(n), then KW_{p/q}[μ]^{n}K^{−1}W_{p/q}[μ]^{−n} is the identity, τ(μ) = 0, and H_{p/q}(n) = ⟨K⟩ is discrete. If W_{p/q}[μ] is loxodromic, H_{p/q}(n) cannot be elementary. Also, if W_{p/q}[μ] is elliptic and H_{p/q}(n) is elementary, then it has to be of order 2, 3, 4, or 6 by the classification of elementary groups (see [21, Section V.D]). This can only happen for a finite number of values of μ. The combination of these observations gives the desired neighborhood.

8. Tangent circle chains and discreteness

In this section we prove Theorem 8.3, a technical result used in the proof of Theorem 9.4. We show how to use the existence of a tangent circle chain to establish the discreteness of a group of the form

(8.1)  G_{n} = ⟨F_{n}, C⟩,

where F_{n} = ⟨A, B⟩ is a triangle group of signature (n, n, ∞), and C conjugates the canonical generators A and B of F_{n} as in Section 2. A tangent circle chain with an additional property (8.3) will be used to construct the closed topological disks required in Maskit’s fourth version of his second combination theorem, [22, Theorem II], Theorem 8.1 below. The full statement of the combination theorem is quite long and general, and we will only state a much weaker version that will be sufficient for our application. Condition (B) in the theorem below is a restatement of the condition “cyclic stabilizers are parabolic” of [22, Theorem II]. When we replace the original condition by (B), we restrict to the case when new parabolics are created by the combination of F and C. This is exactly the situation we have in Section 9.

Let H be a nonelementary Kleinian group. The set of discontinuity Ω(H) has a natural hyperbolic metric of curvature −1. We will denote the hyperbolic area of Ω(H)/H by area(H).

8.1. Theorem. Let F = ⟨A_{n}, B_{n}⟩ be a Fuchsian group of signature (n, n, ∞), n ≥ 3, with canonical generators A_{n} and B_{n}, and let J_{1} = ⟨A_{n}⟩ and J_{2} = ⟨B_{n}⟩ be finite cyclic subgroups of F. Let W_{p/q} be the word defined in Section 4. Assume the following (A) For m = 1, 2, there is a J_{m} -invariant closed topological disk B_{m}, with bound-

ary loop W_{m} ⊂ Ω(F); there is a finite J_{m} -invariant set of points Θ_{m} ⊂ W_{m} such that W_{m} is locally circular at points of Θ_{m}; and there is a Möbius transformation, C, mapping the exterior of B_{1} onto the interior of B_{2} , so that (A1) for all g ∈ F \ J_{m} and all points x ∈ W_{m} ∩ g(W_{m}), both x and g^{−1}(x)

are points of Θ_{m}, (A2) if there is an x ∈ B_{1} ∩ g(B_{2}) for some g ∈ F then x ∈ Θ_{1} ∩ g(Θ_{2}), and (A3) C conjugates J_{1} onto J_{2} and C(Θ_{1}) = Θ_{2} .

(B) W_{p/q}[A_{n}, C] is parabolic, and (C) Ĉ\ (B_{1} ∪ B_{2}) ≠ ∅. Then,

(1) G = ⟨F, C⟩ = F ∗_{C},

(2) G is discrete,

(3) every element of G that is not a conjugate of an element of F and is not a conjugate of W_{p/q}[A_{n}, C] is loxodromic,

(4) if D_{0} is a fundamental set of F such that D_{0} ∩ B_{m} is a fundamental set for the action of J_{m} in B_{m}, for m = 1, 2, then the set

(8.2)  D = D_{0} \ (int B_{1} ∪ B_{2} ∪ G(fix W_{p/q}))

is a fundamental set of G, and

(5) area(G) = area(F) + 4π/n.

Clearly, there can be non-discrete groups G_{n} that possess tangent circle chains. We will work with tangent circle chains that satisfy the additional property

(8.3)  δ_{i} ∩ δ_{j} = ∅  if |i − j| > 1 mod nq.

In an infinite tangent circle chain satisfying (8.3), only the circle δ_{i} intersects the two circles δ_{i−1} and δ_{i+1} , and for finite chains the same holds with the modifications caused by the fact that δ_{j} = δ_{j+nq} for all j ∈ Z.

In Proposition 9.2 we show that the circle chains of cusp groups G[μ_{p/q}] of ℳ satisfy (8.3). In fact, they have two disjoint circle chains with closely related combinatorics, see Proposition 9.1 and [23]. In the proof of Theorem 9.4 we produce circle chains satisfying (8.3) for parameters on 𝒫p/q^{+} by perturbing one of the chains of G[μ_{p/q}].

Let n = 3, 4, …, and let {δ_{i}}, i ∈ Z be a tangent p/q circle chain for G_{n} = ⟨A_{n}, C⟩ satisfying (8.3). We will now construct two closed topological disks 𝒟_{B} and 𝒟_{A} that will be the disks B_{1} and B_{2} required to apply Theorem 8.1. Let γ_{i} ⊂ δ_{i} be the unique circular arc orthogonal to ∂ δ_{i} connecting the points of tangency of δ_{i} with δ_{i−1} and δ_{i+1} , i ∈ Z mod nq. Let

(8.4)  𝒲_{A,0} = ⋃_{i∈Z} γ_{i},

and

(8.5)  𝒲_{B,0} = C^{−1}(𝒲_{A,0}).

See Figure 6 for the construction. If p/q ∉ Z, then

(8.6)  𝒲_{A,0} ∩ 𝒲_{B,0} = \overline{⋃_{i=1}^{q−1} γ_{i}}.

Figure 6. The construction of the Jordan curves 𝒲_{A,0} and 𝒲_{B,0} from a circle chain for G_{4} satisfying (8.3).

Condition (8.3) implies that 𝒲_{A,0} and 𝒲_{B,0} are Jordan curves. Next we will modify the Jordan curves so that they meet the requirements of Theorem 8.1. The boundaries of the disks in (8.1) are only allowed to meet at the points of tangency of the disks δ_{0}, …, δ_{q}, not along a union of circular arcs as above. If p/q ∈ Z, then 𝒲_{A,0} and 𝒲_{B,0} are round disks, tangent at exactly one point, and we do not change them.

Let p/q ∈ Q \ Z. In our application of Theorem 8.1, we are mainly interested in the discreteness of the group in question, so we can assume after a change of generators that 0 < p/q < 1.

Let γ be a directed geodesic in D connecting the points z_{1} and z_{2} on ∂ D, z a point on γ, and ε > 0. Let z(ε) be the point on the geodesic perpendicular to γ at z that lies the distance ε to the right from z. We denote by γ(ε) the union of the geodesic rays connecting z_{1} to z(ε) and z(ε) to z_{2} , including the end points z_{1}, z_{2} ∈ ∂ D.

Figure 7. The modification of 𝒲_{A} inside a disk of the circle chain.

Let i ∈ {1, 2, …, q}, and define V_{i} to be the word in A and C constructed as in 6.2(2) such that V_{i}(δ_{0}) = δ_{i}. Let k(i) be the number of C ’s in the word V_{i}. Set

(8.7)  \begin{aligned}γ_{i}^{A} & = γ_{i}(k(i) − 1)ε),  for i = 1, 2, …, q, \\ 𝒲_{A} & = ⋃_{j∈Z} ⋃_{i=1}^{q} A_{n}^{j}(γ_{i}^{A}),  and \\ 𝒲_{B} & = C^{−1}(𝒲_{A}). \end{aligned}

Let 𝒟_{A} be the component of Ĉ\𝒲_{A} contained in the same component of Ĉ\Λ(F) as 𝒲_{A}. Let 𝒟_{B} be the component of Ĉ \ 𝒲_{B} contained in the same component of Ĉ \ Λ(F) as 𝒲_{B}. Similarly, 𝒟_{A,0} and 𝒟_{B,0} denote the corresponding components of Ĉ\𝒲_{A,0} and Ĉ\𝒲_{B,0} . For use in Lemma 8.2 we extend k to Z as a q -periodic function.

Figure 8. An example of the modification of 𝒲_{A} and 𝒲_{B} in the case p/q = 1/2.

8.2. Lemma. If int 𝒟_{A,0} ∩ int 𝒟_{B,0} = ∅, then \overline{𝒟_{A}} ∩ \overline{𝒟_{B}} = ⋃_{i=1}^{q} δ_{i} ∩ δ_{i−1} .

Proof. The only thing to check is that the modification has separated \overline{𝒟_{A}} and \overline{𝒟_{B}} from each other inside the disks δ_{i}, i = 1, 2, …, q − 1. Let us denote by γ_{i}^{B} the part of 𝒲_{B} contained in δ_{i}. By definition,

(8.8)  \begin{aligned}γ_{i}^{A} & = γ((k(i) − 1)ε)  and, \\ γ_{i}^{B} & = C^{−1}(γ_{i+p}^{A}) = γ((k(i + p) − 1)ε). \end{aligned}

Clearly, k(i) = k(i + p) − 1, and this gives the claim.

8.3. Theorem. Let G_{n} = ⟨F, C⟩, where F = ⟨A_{n}, B_{n}⟩ is a triangle group of signature (n, n, ∞) (n ∈ N ∪ {∞}, n ≥ 3) with canonical generators A_{n} and B_{n}, and let C be a transformation satisfying

(8.9)  C^{−1}A_{n}C = B_{n}^{−1}.

If G_{n} has a tangent p/q -chain satisfying (8.3) such that

(i) the interiors of the disks 𝒟_{A} and 𝒟_{B}, as constructed above, are disjoint, and (ii) the transformations W_{p/q}[A_{n}, C] are parabolic, then

(1) G_{n} = F ∗_{C},

(2) G_{n} is discrete,

(3) G_{n} is geometrically finite,

(4) Ω(G_{n})/G_{n} = S_{1} ∪ S_{2} , where S_{1} has signature (0, 3; n, n, ∞), and S_{2} has signature (0, 3; ∞, ∞, ∞).

To prove this theorem we need to check that the existence of a chain satisfying (8.3) implies that the assumptions of Theorem 8.1 are satisfied. We use the following terminology in the proof:

8.4. Definition. Let G and H ⊂ G be Kleinian groups. A set E ⊂ Ĉ is precisely invariant under H in G, or precisely H -invariant, if h(E) = E for all h ∈ H and if g(E) ∩ E = ∅ for all g ∈ G \ H.

Proof of Theorem 8.3. Without loss of generality we can assume that F is normalized as in Section 2. Set

(8.10)  Θ_{A} = ⋃_{i∈Z} A^{i}(𝒲_{A} ∪ 𝒲_{B}),  Θ_{B} = ⋃_{i∈Z} B^{i}(𝒲_{A} ∪ 𝒲_{B}).

Note that the curves 𝒲_{A} and 𝒲_{B} are not locally circular at Θ_{A} and Θ_{B}, respectively. However, as remarked in [22, Section 0.2.3], this requirement can be weakened considerably. In our situation (see Figures 7 and 8) the curves 𝒲_{A} and 𝒲_{B} consist of circular arcs that are perpendicular to the boundaries of the disks δ_{i} and C^{−1}(δ_{i}) at the points of Θ_{A} and Θ_{B}. Also, we can make a small local perturbation of the curves 𝒲_{A} and 𝒲_{B} at their points of tangency such that the curves are locally circular at these points without introducing new points of tangency, or changing the mapping properties (8.7) of the disks 𝒟_{A} and 𝒟_{B}.

(B) of Theorem 8.1 is assumption (ii). Also, D ⊂ Ĉ \ (𝒟_{A} ∪ 𝒟_{B}), and thus (C) holds.

Let us check that assumption (A) of Theorem 8.1 is satisfied. The disks 𝒟_{A} and 𝒟_{B} are by definition invariant under the cyclic groups generated by A and B, respectively. The same applies to 𝒲_{A}, 𝒲_{B}, Θ_{A} and Θ_{B}. The transformation C maps 𝒲_{B} onto 𝒲_{A}. Also, C maps the fixed points of B onto the fixed points of A. The conjugation relation (8.9) is chosen in such a way that the fixed point of B contained in D is mapped to ∞ ∈ 𝒟_{A}. Thus,

(8.11)  C(int(Ĉ\ 𝒟_{B})) = int 𝒟_{A},

as required.

Let us now consider the conditions (A1)–(A3) of Theorem 8.1. (A3) is clear from (8.9) and (8.10). (A1) holds if and only if the set

(8.12)  𝒟′_{A} = 𝒟_{A} \ Θ_{A}

is precisely invariant under ⟨A⟩, and

(8.13)  𝒟′_{B} = 𝒟_{B} \ Θ_{B}

has the the corresponding property. (A2) is true if we can show that for all g ∈ F,

(8.14)  g(𝒟′_{A}) ⊂ Ĉ\ 𝒟′_{B}.

If, in addition to (8.12)–(8.14), we have for all g ∈ F,

(8.15)  g(𝒟′_{B}) ⊂ Ĉ\ 𝒟′_{A},

and

(8.16)  C(Ĉ\ 𝒟_{B}) ⊂ 𝒟′_{A}

and

(8.17)  C^{−1}(Ĉ\ 𝒟_{A}) ⊂ 𝒟′_{B},

then the triple (𝒟′_{B}, 𝒟′_{A},Ĉ\(𝒟_{A}∪𝒟_{B})) is called an interactive triple ([21, VII.D.7]). We will show that this is the case.

By Maskit [21, Propositions VII.E.3 and VII.E.4], it is enough to show that (i) 𝒟′_{A} is precisely ⟨A⟩-invariant, (ii) 𝒟′_{B} is precisely ⟨B⟩-invariant in F, and (iii) C maps Ĉ\ 𝒟′_{B} onto int 𝒟_{A} ∪ Θ_{A}.

Clearly, (iii) follows from (8.11) and the fact that C(Θ_{B}) = Θ_{A}. By [21, Proposition II.I.3], (i) and (ii) are equivalent with the existence of a fundamental set P for F such that P ∩ 𝒟′_{A} is a fundamental set for the action of ⟨A⟩ in 𝒟′_{A}, and P ∩ 𝒟′_{B} is a fundamental set for the action of ⟨B⟩ in 𝒟′_{B}.

Figure 9. The disks δ_{0}, …, δ_{229} of the circle chain for G_{∞}[μ_{21/229}]. δ_{21} = C(δ_{0}) is the disk tangent to the line Im z = Im μ_{21} ≈ 1.92. The curve 𝒲_{A} intersects the disk \{|z − \frac{1}{2} | < \frac{1}{2} \}. One obtains the fundamental set P by adding the set 𝒟_{A} ∩ \{|z − \frac{1}{2} | < \frac{1}{2} \} to the standard fundamental polygon P_{0} = \{z ∈ C : −1 ≤ Im z < 1, |z + \frac{1}{2} | ≥ \frac{1}{2} , |z − \frac{1}{2} | > \frac{1}{2} \} and by removing its image B(𝒟_{A} ∩ \{|z − \frac{1}{2} | < \frac{1}{2} \}).

We now construct this fundamental set. The property (8.11) implies

(8.18)  C(𝒟_{A}) ⊂ 𝒟_{A}.

By the A-invariance of 𝒟_{A}

(8.19)  A^{−1}C(𝒟_{A}) ⊂ 𝒟_{A}.

The basic combinatorial properties of circle chains (see Definition 6.1) imply

(8.20)  A^{−1}C(δ_{q−1}) = δ_{p−1}.

Thus, C^{−1}A^{−1}C(δ_{q−1}) is a disk that meets 𝒲_{B}, but not 𝒲_{A}. Similarly, we see that for all i = 1, …, q − 1 the part of the boundary of 𝒟_{A} inside δ_{i}, namely γ_{i}^{A}, is mapped by B into the part of 𝒲_{B} not touching 𝒲_{A}. For B^{−1} we have the corresponding result by the same kind of reasoning: for i = 1, …, q − 1, we have

(8.21)  AC(δ_{i}) = δ_{p+q+i}.

As i > 0, the combinatorial properties of the p/q chain imply that the part of 𝒲_{A} inside δ_{i} is mapped into the part of 𝒲_{B} not touching 𝒲_{A}. Thus, the images B^{±1}(int 𝒟_{A}) are disjoint from 𝒟_{A}. We can now modify the ‘standard’ Ford fundamental polygon of the triangle group F to obtain the required fundamental set (see Figure 6 for an illustration, and Figure 9 for a complicated case for n = ∞).

Now, claims (1)–(3) follow directly from Theorem 8.1.

Proof of claim (4). Theorem 8.1 gives

(8.22)  area(G) = 2 · 2π(1 − \frac{2}{n}) + 4π/n = 2π(1 − \frac{2}{n}) + 2π.

By (1) we know that the parabolics A, AB, and W_{p/q} define different conjugacy classes in G. Theorem 8.1(4) implies that the action of the stabilizer of D in G is the same as the action of F on D. Thus, the quotient Ω(G)/G contains one sphere with a puncture and two cone points of order n, contributing 2π(1 − 2/n) to the area (8.22). Furthermore, the fundamental set D given by Theorem 8.1(4) shows that AB corresponds to one puncture, and W_{p/q} to two distinct punctures on the remainder of the quotient. The area of a surface with genus g and n punctures is 2π(2g − 2 + n). There are at least three punctures in (Ω(G)/G) \ (D/ Stab(D)). (8.22) now implies that the quotient is as in claim (4).

9. The discreteness and conjugacy theorems

In this section we study a tangent combinatorial circle chain {Δ_{i}[μ]}_{i∈Z} for G_{∞}[μ], such that the disks in the chain are translates of H^{∗}. Keen and Series [10] and Wright [31] showed that there is an infinite p/q -combinatorial circle chain \{δ_{i}[μ]\}_{i∈Z} for G[μ], μ ∈ \overline{𝒫_{p/q}} ⊂\overline{ℳ} , with δ_{0}[μ] ⊂ H one of the invariant disks of

(9.1)  F_{p/q}[μ] = ⟨W_{p/q}[μ],W̃_{p/q}[μ]⟩.

Here p/q, r/s ∈ Q are Farey neighbors and we use the notation

(9.2)  W̃_{p/q} = W_{r/s}^{−1}W_{p/q}^{−1}W_{r/s}.

If μ ∈ 𝒫_{p/q}, then the chain {δ_{i}[μ]} is proper, and for the cusp parameter μ_{p/q} it is a tangent chain. The chains {Δ_{i}} are “dual” to the chains {δ_{i}}_{i∈Z} in a sense that is made precise in Proposition 9.1.

In this section the roles of the subgroups F_{p/q} and Γ_{0} = Stab H^{∗} are interchanged and we view the group G as an extension of F_{p/q} by W_{p/q}. This has the following advantage: the chains {Δ_{i}[μ]}_{i∈Z} constructed below are tangent for all μ ∈ C (see Proposition 9.2), and for large values of n we can use these chains and Theorem 8.3 to prove the discreteness of the groups G[μ_{p/q}(n)].

Recall that at the cusp μ_{p/q}, F_{p/q}[μ_{p/q}] is a torsion-free triangle group. The quotient Ω(G[μ_{p/q}])/G[μ_{p/q}] is the union of two thrice punctured spheres, one corresponding to H^{∗}/⟨S,S̃ ⟩, the other to δ_{0}/F_{p/q}. McShane, Parker and Redfern [23] made the following useful observation:

9.1. Proposition. G[μ_{p/q}] is the HNN extension of the torsion-free triangle group F_{p/q} by W_{r/s}^{−1} . In particular, if pp′ = 1 mod q, then G[μ_{p/q}] is the p′/q cusp in the deformation space of terminal b-groups of the form F_{p/q}∗_{T′} representing a punctured torus on their invariant component.

Proof. [23, Proposition 7.1].

Proposition 9.1 involves writing the group element S (that is parabolic for all μ) in terms of the new generators W_{p/q} and U = W_{r/s}^{−1} . Note that the conjugates of W_{p′/q}[W_{p/w}, U] in G[μ], being conjugates of S, are parabolic for all μ. This turns out to be quite useful for our purposes.

Proposition 9.1 implies that, in addition to a tangent p/q -chain {δ_{i}}_{i∈Z} for the generators S and T, there is an infinite tangent p′/q combinatorial circle chain {Δ_{i}}_{i∈Z} for ⟨W_{p/q}[μ_{p/q}], U[μ_{p/q}]⟩, where Δ_{0} = H^{∗}. The two chains {δ_{i}} and {Δ_{i}} do not intersect. Thus,

(9.3)  Δ_{1} = T (H^{∗}) = H + μ,

Figure 10. A schematic picture of how the two circle chains fit together for a cusp group (on the left) and for a parameter μ_{p/q}(5). For the cusp group, the grey disks form part of the chain {Δ_{i}} that accumulates at the fixed point of W_{p/q}.

see Figure 10. When the parameter μ moves on 𝒫p/q^{+} \ ℳ away from the cusp, Proposition 9.2 below shows that the chain {Δ_{i}} remains tangent. On the other hand, the transformation W_{p/q}[μ] is elliptic for these parameters, and it cannot stabilize two tangent circles. Thus, the circle chain {δ_{i}} separates, as in Figure 10. We will use the chain {Δ_{i}} and Theorem 8.3 to prove Theorem 9.4.

Let Γ_{i} be the stabilizer of Δ_{i} in G[μ_{p/q}]. The map μ ↦ ⟨S, T [μ]⟩ = G[μ] induces a homomorphism from G[μ_{p/q}] (isomorphic to the free group on two generators) onto any G[μ], μ ∈ C. Let us denote by Γ_{i}[μ] the image of Γ_{i} = Γ_{i}[μ_{p/q}] by this homomorphism. As the group

(9.4)  Γ_{0}[μ] = Stab_{G[μ]} H^{∗} = ⟨S,S̃ ⟩ = Γ_{0}

does not depend on the parameter μ, the groups Γ_{i}[μ], as conjugates in G[μ] of Γ_{0} , are discrete for all μ ∈ C. Thus, there is a naturally defined disk Δ_{i}[μ] for each μ ∈ H, bounded by the limit set of Γ_{i}[μ]. In particular,

(9.5)  Δ_{i}[μ_{p/q}] = Δ_{i}.

This defines a p′/q -combinatorial circle chain {Δ_{i}[μ]}_{i∈Z} for μ ∈ 𝒫p/q^{+} as a perturbation of the chain {Δ_{i}}_{i∈Z} .

9.2. Proposition. (1) Δ_{i}[μ] and Δ_{i+1}[μ] are tangent for any μ ∈ H.

(2) If |i − j| > 1, then Δ_{i}[μ_{p/q}] ∩ Δ_{j}[μ_{p/q}] = ∅, that is, the circle chain {Δ_{i}}_{i∈Z} satisfies (8.3).

Proof. (1) Δ_{i}[μ] and Δ_{i+1}[μ] are stabilized by the same parabolic transformation conjugate in G[μ] to W_{p′/q}[W_{p/q}, U], which is conjugate to S^{±1} in G[μ], and therefore parabolic for all μ. Thus the disks are tangent at the fixed point of this element.

(2) Assume j > i + 1 and that there is a point x ∈ Δ_{i}[μ_{p/q}] ∩ Δ_{j}[μ_{p/q}]. By Maskit [20, Theorem 3]:

(9.6)  Λ(Γ_{i}) ∩ Λ(Γ_{j}) = Λ(Γ_{i} ∩ Γ_{j}).

Δ_{i}[μ_{p/q}] and Δ_{j}[μ_{p/q}] are round disk components of G[μ_{p/q}], so they can intersect in at most one point. Thus Γ_{i}∩Γ_{j} = ⟨P ⟩, where P is a parabolic element fixing x. There are exactly three conjugacy classes of parabolic elements in G[μ_{p/q}]. Thus P has to be conjugate to one of W_{p/q}, K or S. All of these elements correspond to punctures on H^{∗}/Γ_{q} or δ_{0}/F_{p/q}. Two translates of H^{∗} = Δ_{0} meet at x ∈ Ĉ, the fixed point of P, implying that the transformation P can only correspond to punctures on Δ_{i}/Γ_{i}. This implies that K and W_{p/q} are not possible because they correspond to at least one puncture on δ_{0}/F_{p/q}. The only remaining possibility is that the disks Δ_{i}[μ_{p/q}] and … Δ_{j}[μ_{p/q}] are tangent at the fixed point of a parabolic P [μ_{p/q}], where P [μ_{p/q}] is conjugate in G[μ_{p/q}] to S. The homomorphism induced by μ ↦ G[μ] induces an isomorphism for parameters μ ∈\overline{ℳ} . Thus, there is a uniquely determined parabolic transformation

(9.7)  P [μ] ∈ Γ_{i}[μ] ∩ Γ_{j}[μ] ⊂ G[μ]

for all μ ∈\overline{ℳ} . But then, because the groups Γ_{i}[μ] = Stab(Δ_{i}[μ]) are Fuchsian groups of the first kind for all μ ∈ C, there is a closed curve

(9.8)  𝒞 ⊂ ⋃ ∂Δ_{i}[μ] ⊂ Λ(G[μ])

that divides Ĉ\𝒞 into two disjoint open sets E and E′ such that Λ(G[μ])∩E ≠ ∅, and Λ(G[μ]) ∩ E′ ≠ ∅. But this is a contradiction; if μ ∈ ℳ , then G[μ] has a simply connected invariant component Ω_{0} , and ∂Ω_{0} = Λ(G[μ]).

We need to show that for μ ∈ 𝒫p/q^{+} \ ℳ close to the cusp μ_{p/q}, the disks Δ_{0}[μ], …, Δ_{q}[μ] still satisfy (8.3). This will be a consequence of the following lemma. If K and K′ are compact sets, we denote by d(K, K′) the Hausdorff distance of K and K′, that is,

(9.9)  d(K, K′) = inf{r > 0 : K ⊂ K′_{r} and K′ ⊂ K_{r}},

where

(9.10)  K_{r} = {z ∈ C : |z − z_{0}| < r for some z_{0} ∈ K}.

9.3. Lemma. Let p/q ∈ Q, and ε > 0. There is an η > 0 such that if |μ − μ_{p/q}| < η, then for all i = 1, 2, …, q,

(9.11)  d(Λ(Γ_{i}[μ_{p/q}]), Λ(Γ_{i}[μ])) < ε.

Proof. Let p = p(μ_{p/q}) ∈ Λ(Γ_{i}[μ_{p/q}]) be a fixed point of a hyperbolic transformation of Γ_{i}[μ_{p/q}], and let c = c(μ_{p/q}) be the center of Λ(Γ_{i}[μ_{p/q}]). c(μ) can be expressed as a continuous function of three distinct points on Λ(Γ_{i}[μ]). Thus, there is a neighborhood 𝒰 ⊂ C of μ_{p/q} such that the maps p: 𝒰 → C and c: 𝒰 → C induced by the map μ ↦ G[μ], are continuous. This implies that there is an η > 0 such that for |μ − μ_{p/q}| < η, |p(μ) − p(μ_{p/q})| < \frac{1}{3}ε, and |c(μ) − c(μ_{p/q})| < \frac{1}{3}ε. Now, using the fact that Λ(Γ_{i}[μ]) is a circle, we have

(9.12) \begin{aligned}|z − c(μ_{p/q})| & ≤ |z − c(μ)| + |c(μ) − c(μ_{p/q})| \\ & = |p(μ) − c(μ)| + |c(μ) − c(μ_{p/q})| \\ & = |p(μ) − p(μ_{p/q})| + |p(μ_{p/q}) − cμ_{p/q})| + 2|c(μ) − c(μ_{p/q})| \\ & < |p_{1}(μ_{0}) − c(μ_{0})| + ε \end{aligned}

for any point z ∈ Λ(Γ_{i}[μ]). The claim follows from this by another easy application of the triangle inequality, and by observing that the same holds for the difference |z − c(μ)| for points z ∈ Λ(Γ_{i}[μ_{p/q}]).

9.4. Theorem. On the extension 𝒫p/q^{+} of each rational pleating ray there is an open neighborhood 𝒰 of the cusp μ_{p/q} on 𝒫p/q^{+} such that if μ ∈ 𝒫p/q^{+} ∩ 𝒰 and | tr W_{p/q}| = 2 cos(π/n) for some n ∈ N, n ≥ 3, then

(9.13)  G[μ] = F ∗_{W_{r/s}[S,T [μ]]} .

For these values of μ, G[μ] is a Kleinian group representing a thrice punctured sphere and a sphere with a puncture and two branch points of order n on its ordinary set.

If μ ∈ (𝒫p/q^{+} ∩ 𝒰 ) \\overline{ℳ} is not of this form, then G[μ] is not discrete.

Proof. The idea of the proof is to show that we can apply Theorem 8.3 in our situation. The finite tangent circle chains are obtained by perturbing the dual chain {Δ_{i}}_{i∈Z} . At the cusp μ_{p/q} ∈ ∂ ℳ the tangent combinatorial (p′/q)-chain (Δ_{i})_{i∈Z} can be written as the collection

(9.14)  {W_{p/q}^{i}(Δ_{j}) : i ∈ Z, 0 ≤ j ≤ q − 1}.

By Proposition 9.2 we have

(9.14)  Δ_{i} ∩ Δ_{j} = ∅

if |i−j| > 1. By Lemma 9.3 the finite piece Δ_{0}, …, Δ_{q−1} is deformed continuously in μ, and no new intersections of the q first disks Δ_{0}[μ], …, Δ_{q−1}[μ] are produced if |μ_{p/q} − μ| is small. Thus, we can change μ in a small open set 𝒰 containing μ_{p/q} such that

(9.16)  Δ_{i} ∩ Δ_{j} = ∅

holds for all μ ∈ 𝒰 and all i = 0, …, q − 1. Let μ_{p/q}(n) be a parameter on 𝒫p/q^{+} such that W_{p/q} is primitive elliptic as in Lemma 7.3. If μ_{p/q}(n) ∈ 𝒰 ∩𝒫p/q^{+} , then the disks

(9.17)  {W_{p/q}[μ_{p/q}(n)]^{i}(Δ_{j}[μ_{p/q}(n)]) : i ∈ Z mod n, 0 ≤ j ≤ q − 1}

form a finite combinatorial (p′/q)-chain. Also, after possibly making 𝒰 slightly smaller, the interiors of the topological disks 𝒟_{A} and 𝒟_{B}, as constructed in Section 8, are disjoint.

In order to apply apply Theorem 8.3 on discreteness and circle chains we need that the word W_{p′/q}[W_{p/q}[S, T [μ]], W_{r/s}[S, T [μ]]] is parabolic for μ = μ_{p/q}(n). But this is so, because it is a conjugate of S.

Figure 11. The limit set the Kleinian group on the extended ray 𝒫1/2^{+} for the parameter μ such that W_{1/2} is an elliptic of order 4.

The proof of Corollary 9.6 is similar to that of Theorem III of Keen, Maskit and Series [8] on the uniqueness of maximally parabolic groups.

9.5. Theorem. Let G and G′ be geometrically finite Kleinian groups. If f: Ω(G) → Ω(G′) is a conformal homeomorphism that induces an isomorphism φ: G → G′, then f is a Möbius transformation.

Proof. Tukia [30, Theorem 4.2].

9.6. Corollary. Let μ_{p′/q,n} be the boundary point of ℳ_{n} on the ray 𝒫_{p′/q}^{n}. Then the group G[μ_{p/q}(n)] is conjugate to G_{n}[μ_{p′/q,n}] by a Möbius transformation.

Proof. Let n ∈ {3, 4, …} be such that Theorem 9.4 applies, and the group G[μ_{p/q}] is discrete. By Theorem 9.4, G[μ_{p/q}] is the HNN-extension of a triangle group of signature (n, n, ∞). Thus it is of the form G_{n}[μ_{0}] for some μ_{0} ∈ C \ \overline{D}. Furthermore, G_{n}[μ_{0}] has a tangent p′/q -circle chain. Using this fact, we can construct a conformal map P_{n}: Ω(G_{n}[μ_{0}]) → Ω(G_{n}[μ_{p′/q}]) that induces an isomorphism between these groups. By Theorem 9.5, P_{n} is the restriction of a Möbius transformation.

9.7. Remark. It is easy to check using (2.14) and (2.15) that for all n = 3, 4, …, the end points of the integral rays in ℳ_{n} are of the form

(9.18)  μ_{m,n} = e^{2πi/n}(\frac{1 + sin(π/n)}{cos(π/n)})^{2} = e^{2πi/n} \frac{1 + sin(π/n)}{1 − sin(π/n)}.

Now one can check Theorem 9.4 for integral values of p/q by conjugating the groups G_{n}[μ_{m,n}] in such a way that C[μ_{m,n}] is conjugated to S and A_{n} is conjugated to T [μ_{0}(n)], where A_{n}, C, S and T are as in Section 2 and

(9.19)  μ_{0}(n) = 2 cos(π/n)

is as in Lemma 7.3.

G[μ_{0}(n)] is discrete for all n ≥ 2: T [μ_{0}(2)](z) = G[0](z) = 1/z is an elliptic of order 2. Thus, G[0] ⊂ PSL(2, Z[i]) is a discrete group. It is easy to see that Ω(G[0])/G[0] consists of a single sphere with three punctures.

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Received 25 October 1996