The asymptotic directions

The asymptotic directions of pleating rays in the Maskit embedding.

Sara Maloni Mathematics Institute, University of Warwick
Coventry CV4 7AL
UK
S.Maloni@warwick.ac.uk
July 11, 2019
Abstract.

This article was born as a generalisation of the analysis made by Series in [22] where she made the first attempt to plot a deformation space of Kleinian group of more than 1 complex dimension. We use the Top Terms’ Relationship proved by the author and Series in [13] to determine the asymptotic directions of pleating rays in the Maskit embedding of a hyperbolic surface as the bending measure of the ‘top’ surface in the boundary of the convex core tends to zero. The Maskit embedding of a surface is the space of geometrically finite groups on the boundary of quasifuchsian space for which the ‘top’ end is homeomorphic to , while the ‘bottom’ end consists of triply punctured spheres, the remains of when the pants curves have been pinched. Given a projective measured lamination on , the pleating ray is the set of groups in for which the bending measure of the top component of the boundary of the convex core of the associated -manifold is in the class .

MSC classification: 30F40, 30F60, 57M50

1. Introduction

Let be a surface of negative Euler characteristic together with a pants decomposition . Kra’s plumbing construction endows with a projective structure as follows. Replace each pair of pants by a triply punctured sphere and glue, or ‘plumb’, adjacent pants by gluing punctured disk neighbourhoods of the punctures. The gluing across the pants curve is defined by a complex parameter . The associated holonomy representation gives a projective structure on which depends holomorphically on the . In particular, the traces of all elements , are polynomials in the .

In [13] the author and Series proved a formula, called Top Terms’ Relationship, which is Theorem 2.11 in Section 2.1.1, giving a simple linear relationship between the coefficients of the top terms of , as polynomials in the , and the Dehn–Thurston coordinates of relative to , see Section 2.1.1 for the definitions. This result generalises the previous results proved by Keen and Series in [11] in the case of the once punctured torus and by Series in [22] for the twice punctured torus . These formulas were used in the case to determine the asymptotic directions of pleating rays in the Maskit embedding of as the bending measure of the ‘top’ surface in the boundary of the convex core tends to zero, see Section 2 for the definitions. In the present article we will use the general Top Terms’ Relationship to generalise the description of asymptotic directions of pleating rays to the case of an arbitrary hyperbolic surface , see Theorem 3.7 in Section 3. The Maskit embedding of a surface is the space of geometrically finite groups on the boundary of quasifuchsian space for which the ‘top’ end is homeomorphic to , while the ‘bottom’ end consists of triply punctured spheres, the remains of when the pants curves have been pinched. As such representations vary in the character variety, the conformal structure on the top side varies over the Teichmüller space , see Section 2.4 for a detailed discussion.

Let , and suppose we have a geometrically finite free and discrete representation for which . Denote the complexity of the surface . Fix disjoint, non-trivial, non-peripheral and non-homotopic simple closed curves which form a maximal pants decomposition of . We consider groups for which the conformal end is a union of triply punctured spheres glued across punctures corresponding to , while is a marked Riemann surface homeomorphic to . Kra’s plumbing construction gives us an explicit parametrisation of a holomorphic family of representation such that, for certain values of the parameters, has the above geometry, see Section 2.2 for the definition of this construction. The Maskit embedding is the map which sends a point to the point for which the group has . Denote the image of this map by . Note that, with abuse of notation, we will also call Maskit embedding the image of the map just described.

We investigate using the method of pleating rays. Given a projective measured lamination on , the pleating ray is the set of groups in for which the bending measure of the top component of the boundary of the convex core of the associated -manifold is in the class . It is known that is a real 1-submanifold of . In fact we can parametrise this ray by , where , so that we associate to the group such that , see Theorem 6 in [21] for the case . Note that this result relies on Thurston’s bending conjecture which is solved for rational lamination by work of Otal and Bonahon and in the case of punctured tori by work of Series. For a general (irrational) lamination, anyway, we can only conjecture that the real dimension of the associated pleating ray is 1. Our main result is a formula for the asymptotic direction of in as the bending measure tends to zero, in terms of natural parameters for the representation space and the Dehn–Thurston coordinates of the support curves to relative to the pinched curves on the bottom side. This leads to a method of locating in .

We restrict to pleating rays for which is rational, that is, supported on closed curves, and for simplicity write in place of , although noting that depends only on . From general results of Bonahon and Otal [3], for any pants decomposition such that are mutually non-homotopic and fill up (see section 2.5 for the definitions), and any pair of angles , there is a unique group in for which the bending measure of is . (This extends to the case for provided fill up and also to the case .) Thus given , there is a unique group with bending measure for any sufficiently small .

Let denote the set of homotopy classes of multiple loops on , and let the pants curves defining be . The Dehn–Thurston coordinates of are , where is the geometric intersection number between and and is the twist of about . For a detailed discussion about this parametrisation see Section 2.1.1 below or Section 3 of [13]. If , the above condition of Bonahon and Otal on is equivalent to ask . We call such laminations admissible.

The main result of this paper is the following. We will state this result more precisely, as Theorem 3.7 in Section 3.

Theorem A.

Suppose that is admissible. Then, as the bending measure tends to zero, the pleating ray approaches the line

We should note that, in contrast to Series’ statement, we were able to dispense with the hypothesis ‘ non exceptional’ (see [22] for the definition), because we were able to improve the original proof. In addition, the definition of the line is different because we have corrected a misprint in [22].

One might also ask for the limit of the hyperbolic structure on as the bending measure tends to zero. The following result is an immediate consequence of the first part of the proof of Theorem A.

Theorem B.

Let be as above. Then, as the bending measure tends to zero, the induced hyperbolic structure of along converges to the barycentre of the laminations in the Thurston boundary of .

This should be compared with the result in [20], that the analogous limit through groups whose bending laminations on the two sides of the convex hull boundary are in the classes of a fixed pair of laminations , is a Fuchsian group on the line of minima of . It can also be compared with Theorem 1.1 and 1.2 in [6].

Finally, we wanted to underline that the result achieved in Theorem 2.4 about the relationship between the Thurston’s symplectic form and the Dehn–Thurston coordinated for the curves is very interesting in its own. It tells us that given two loops which belongs to the same chart of the standard train track, see Section 2.1.2 for the definition, then where the vector are the Dehn–Thurston coordinates of the curves .

The plan of the paper is as follows. Section 2 provides an overview of all the background material needed for understanding and proving the main results which we will prove in Section 3. In particular, in Section 2 we will discuss issues related to curves on surfaces (for example we will recall the Dehn–Thurston coordinates of the space of measured laminations, Thurston’s symplectic structure, and the curve and the marking complexes). In the same section we will also review Kra’s plumbing construction which endows a surface with a projective structure whose holonomy map gives us a group in the Maskit embedding, and we will discuss the Top Terms’ Relationship. Then we will recall the definition of the Maskit embedding and of the pleating rays. In Section 3, on the other hand, after fixing some notation, we will prove the three main results stated above. We will follows Series’ method [22]: we will state (without proof) the theorems which generalise straightforwardly to our case, but we will discuss the results which require further comments. In particular, many proofs become much more complicated when we increase the complex dimension of the parameter space from 1 or 2 to the general . It is worth noticing that, using a slightly different proof in Theorem A, we were able to extend Series’ result to the case of ‘non-exceptional’ laminations, see Section 3 for the definition. We also corrected some mistakes in the statement of Theorem A.

2. Background

2.1. Curves on surfaces

2.1.1. Dehn–Thurston coordinates

In this section we review Dehn–Thurston coordinates, which extend to global coordinates for the space of measure laminations . These coordinates are effectively the same as the canonical coordinates in [22]. We follow the description in [13]. First we need to fix some notation.

Suppose is a surface of finite type, let denote the set of free homotopy classes of connected closed simple non-boundary parallel curves on , and let be the set of multi-curves on , that is, the set of finite unions of pairwise disjoint curves in . For simplicity we usually refer to elements of as ‘curves’ rather than ‘multi-curves’, in other words, a curve is not required to be connected. The geometric intersection number between is the least number of intersections between curves representing the two homotopy classes, that is

Given a surface of finite type and negative Euler characteristic, choose a maximal set of homotopically distinct and non-boundary parallel loops in called pants curves, where is the complexity of the surface. These connected curves split the surface into three-holed spheres , called pairs of pants. (Note that the boundary of may include punctures of .) We refer to both the set , and the set , as a pants decomposition of .

Now suppose we are given a surface together with a pants decomposition as above. Given , define for all . Notice that if are pants curves which together bound a pair of pants whose interior is embedded in , then the sum of the corresponding intersection numbers is even. The are usually called the length parameters of .

To define the twist parameter of about , we first have to fix a marking on . (See D. Thurston’s preprint [24] for a detailed discussion about three different, but equivalent ways of fixing a marking on .) A way of specifying the marking is by choosing a set of curves , each one dual to a pants curve , see next paragraph for the definition. Then, after isotoping into a well-defined standard position relative to and to the marking, the twist is the signed number of times that intersects a short arc transverse to . We make the convention that if , then is the number of components in freely homotopic to .

Each pants curve is the common boundary of one or two pairs of pants whose union we refer to as the modular surface associated to , denoted . Nota that if is adjacent to exactly one pair of pants, is a one holed torus, while if is adjacent to two distinct pairs of pants, is a four holed sphere. A curve is dual to the pants curve if it intersect minimally and is completely contained in the modular surface .

Remark 2.1 (Convention on dual curves).

We shall need to consider dual curves to . The intersection number of such a connected curve with is if a one-holed torus and if it is a four-holed sphere. We adopt a useful convention introduced in [24] which simplifies the formulae, in such a way as to avoid the need to distinguish between these two cases. Namely, for those for which is , we define the dual curve to be two parallel copies of the connected curve intersecting once, while if is we take a single copy. In this way we always have, by definition, . See Section 2 of [13] for a deeper discussion.

There are various ways of defining the standard position of , leading to differing definitions of the twist. In this paper we will always use the one defined by D. Thurston [24] (which we will denote ), but we refer to our previous article [13] for a further discussion about the different definitions of the twist parameter and for the precise relationship between them (Theorem 3.5 [13]). With either definition, a classical theorem of Dehn [5], see also [19] (p 12), asserts that the length and twist parameters uniquely determine . This result was described by Dehn in a 1922 Breslau lecture [5].

Theorem 2.2.

(Dehn’s theorem, 1922)
Given a marking on , the map which sends to
is an injection. The point
is in the image of (and hence corresponds to a curve) if and only if:

  1. if , then , for each .

  2. if are pants curves which together bound a pair of pants whose interior is embedded in , then the sum of the corresponding intersection numbers is even.

One can think of this theorem in the following way. Suppose given a curve , whose length parameters necessarily satisfy the parity condition (ii), then the uniquely determine for each pair of pants , , in accordance with the possible arrangements of arcs in a pair of pants, see for example [19]. Now given two pants adjacent along the curve , we have points of intersection coming from each side and we have only to decide how to match them together to recover . The matching takes place in the cyclic cover of an annular neighbourhood of . The twist parameter specifies which of the possible choices is used for the matching.

In 1976 William Thurston rediscovered Dehn’s result and extended it to a parametrisation of (Whitehead equivalence classes of) measured foliation of , see Fathi, Laudenbach and Poénaru [8] or Penner with Harer [19] for a detailed discussion. Penner’s approach for parametrising is through train tracks. Using them, Thurston also defined a symplectic form on , called Thurston’s symplectic form. Since it will be useful later, we will recall its definition and some properties in the next section.

2.1.2. Thurston’s symplectic form

We will focus on Penner’s approach, following Hamenstad’s notation [9]. We will define train tracks and some other related notions, so as to be able to define the symplectic form. Then we will present an easy way to calculate it.

A train track on the surface is an embedded 1–complex whose edges (called branches) are smooth arcs with well–defined tangent vectors at the endpoints. At any vertex (called a switch) the incident edges are mutually tangent. Through each switch there is a path of class which is embedded in and contains the switch in its interior. In particular, the branches which are incident on a fixed switch are divided into “incoming” and “outgoing” branches according to their inward pointing tangent vectors at the switch. Each closed curve component of has a unique bivalent switch, and all other switches are at least trivalent. The complementary regions of the train track have negative Euler characteristic, which means that they are different from discs with 0, 1 or 2 cusps at the boundary and different from annuli and once-punctured discs with no cusps at the boundary. A train track is called generic if all switches are at most trivalent. Note that in the case of a trivalent vertex there is one incoming branch and two outgoing ones.

Denote the set of branches of . Then a function (resp. ) is a transverse measure (resp. weighting) for if it satisfies the switch condition, that is for all switches , we want where the are the incoming branches at and are the outgoing ones.

A train track is called recurrent if it admits a transverse measure which is positive on every branch. A train track is called transversely recurrent if every branch is intersected by an embedded simple closed curve which intersects transversely and is such that does not contain an embedded bigon, i.e. a disc with two corners on the boundary. A recurrent and transversely recurrent train track is called birecurrent. A geodesic lamination or a train track is carried by a train track if there is a map of class which is isotopic to the identity and which maps to in such a way that the restriction of its differential to every tangent line of is non–singular. A generic transversely recurrent train track which carries a complete geodesic lamination is called complete, where we define a geodesic lamination to be complete if there is no geodesic lamination that strictly contains it.

Given a generic birecurrent train track , we define to be the collection of all (not necessary nonzero) transverse measures supported on and let be the vector space of all assignments of (not necessary nonnegative) real numbers, one to each branch of , which satisfy the switch conditions. By splitting, we can arrange to be generic. Since is oriented, we can distinguish the right and left hand outgoing branches, see Figure 1. If are weightings on (representing points in ), then we denote by the weights of the left hand and right hand outgoing branches at respectively. The Thurston product is defined as

In Theorem 3.1.4 of Penner [19] it is proved that, if the train track is complete, then the interior of for a complete train track can be thought of as a chart on the PIL manifold of rational measured laminations, that is laminations supported on multi-curves. (PIL is short for piecewise–integral–linear, see [19, Section 3.1] for the definition.) In addition, in this case, we can identify with the tangent space to at a point in . The Thuston product defined above allows us to define a symplectic structure on the PIL manifold .

Figure 1. Weighted branches at a switch.

It is interesting to note that if is oriented, then there is a natural map , see Section 3.2 [19], which is related to the Thurston product by the following result. For a generalisation of this result to the case of an arbitrary (not necessarily orientable) track , see Section 3.2 [19].

Proposition 2.3 (Lemma 3.2.1 and 3.2.2 [19]).

For any train track , is a skew-symmetric bilinear pairing on . In addition, if is connected, oriented and recurrent, then for any , is the homology intersection number of the classes and .

In Proposition 4.3 of [22], Series relates Thurston product to the Dehn–Thurston coordinates described above, but her proof works only for the case , since she uses a particular choice of train tracks, called canonical train tracks. Our idea was to use the standard train tracks, as defined by Penner [19] in Section 2.6. The Dehn-Thurston coordinates, using Penner’s twist , give us a choice of a standard model and of specific weights on each edge of the track. Then one can calculate the Thurston’s product, using the definition above, for a pair of curves supported on a common standard train track. Finally, using the relationship between Penner’s and D. Thurston’s twist, as described by Theorem 3.5 by Maloni and Series [13], one can prove the following result, which will be very important in the proof of our main theorems. In particular, the standard train track are of two types: the tracks in the annuli around the pants curves and the tracks in the pair of pants. The sum of the Thurston’s product in the annuli give us , using Penner’s twists, while the sum of the pairs of pants give us some terms, so that the total sum give us the results that we want, that is the product , where we use D. Thurston’s twist. We should notice that this result, although we proved it because we need it in our last section, is really interesting in its own and it is possible much more can be said from it.

Theorem 2.4.

Suppose that loops belongs to the same chart (and so are supported on a common standard train track) and they are represented by coordinates . Then

In addition, if are disjoint, then .

Notice that this symplectic form induces a map defined by such that

where is the usual inner product on To understand the meaning of the vector better, we should recall the last Proposition of Section 3.2 of [19] and some notation from Bonahon’s work (see his survey paper [1] for a general introduction to the argument and for other further references). After rigorously defining the tangent space with , Bonahon proved in [2] that we can interpret any tangent vector as a geodesic lamination with a transverse Hölder distribution. Note that the space can be seen as the space of Hölder distributions on the track since it is defined to be the vector space of all assignments of not necessary nonnegative real numbers, one to each branch of , which satisfy the switch conditions. He also characterised which geodesic laminations with transverse distributions correspond to tangent vectors to . Notice that if the lamination is carried by the track , we can locally identify with which is isomorphic to .

Theorem 2.5 (Theorem 3.2.4 [19]).

For any surface , the Thurston product is a skew-symmetric, nondegenerate, bilinear pairing on the tangent space to the PIL manifold .

2.1.3. Complex of curves and marking complex

In this section, we review the definitions of the complex of curves and of the marking complex. We will use this language in the last Section where we will prove our main Theorems. While it is not essential to use this language, we believe most readers will already be familiar with these definitions and will find easier to understand the ideas of our proofs. In addition, these tools will shorten the proofs. We summarise briefly the definition of simplicial complex and few related definitions which we will need later on, and we refer to Hatcher [10] for a complete discussion.

Definition 2.6.

Given a set (of vertices), then , where is the power set of , is a simplicial complex if

  1. ;

Given , we define the link of to be the set .

Definition 2.7.

Given a surface , let be the set of isotopy classes of essential, nonperipheral, simple closed curves in . Then we define the complex of curves as the simplicial complex with vertex set and where multicurves gives simplices. In particular –simplices of are –tuples of distinct nontrivial free homotopy classes of simple, nonperipheral closed curves, which can be realised disjointly.

Note that this complex is obviously finite–dimensional by an Euler characteristic argument, and is typically locally infinite. If , then the dimension is

Note that the cases of lower complexity, which are called sporadic by Masur and Minsky [16], require a separate discussion. In particular if with , then is empty. If , using this definition, is disconnected (in fact, it is just an infinite set of vertices). So we slightly modify the definition, in such a way that edges are placed between vertices corresponding to curves of smallest possible intersection number ( for the tori, for the sphere). Finally also in the case of an annulus, that is , needs to be defined in a different way, which we do not not discuss here as it is not needed. We refer the interested to Masur and Minsky [17] for a detailed discussion.

We define now the marking complex. Before defining it, we need to give few additional definitions.

Definition 2.8.

Given a surface , a complete clean marking on is a pants decomposition , called the base of the marking, together with the choice of dual curves for each such that for any .

There are two types of elementary moves on a complete clean marking:

  1. Twist: Replace a dual curve by another dual curve obtained from by a Dehn–twist or an half–twist around .

  2. Flip: Exchange a pair with a new pair and change the dual curves with so that they will satisfy the property described in the Definition 2.8. This operation is called cleaning the marking and it is not uniquely defined.

We will only need to use the base of the markings, so we will not describe these operations more deeply. The interested reader can refer to [17] for a deeper analysis on this topic.

Definition 2.9.

Given a surface , let be the set of complete clean markings in . Then we define the marking complex as the simplicial complex with vertex set and where two vertices are connected by an edge if the two markings are connected by an elementary move.

2.2. Plumbing construction

In this section we review the plumbing construction which gives us the complex parameters for the Maskit embedding. The idea of the plumbing construction is to manufacture by gluing triply punctured spheres across punctures. There is one triply punctured sphere for each pair of pants , and the gluing across the pants curve is implemented by a specific projective map depending on a parameter . The will be the parameters of the resulting holonomy representation with .

More precisely, we first fix an identification of the interior of each pair of pants to a standard triply punctured sphere . We endow with the projective structure coming from the unique hyperbolic metric on a triply punctured sphere. Then the gluing is carried out by deleting open punctured disk neighbourhoods of the two punctures in question and gluing horocyclic annular collars round the resulting two boundary curves, see Figure 2.

Figure 2. Deleting horocyclic neighbourhoods of the punctures and preparing to glue.

2.2.1. The gluing

First recall (see for example [18] p. 207) that any triply punctured sphere is isometric to the standard triply punctured sphere , where

Fix a standard fundamental domain for , as shown in Figure 3, so that the three punctures of are naturally labelled . Let be the ideal triangle with vertices , and be its reflection in the imaginary axis. We sometimes refer to as the white triangle and as the black.

Figure 3. The standard fundamental domain for . The white triangle is unshaded.
Figure 4. The gluing construction when and .

With our usual pants decomposition , fix homeomorphisms from the interior of each pair of pants to . This identification induces a labelling of the three boundary components of as in some order, fixed from now on. We denote the boundary labelled by . The identification also induces a colouring of the two right angled hexagons whose union is , one being white and one being black. Suppose that the pants are adjacent along the pants curve meeting along boundaries and . (If then clearly .) The gluing across will be described by a complex parameter with , called the plumbing parameter of the gluing.

Let be the ideal ‘white’ triangle with vertices . Notice that there is a unique orientation preserving symmetry of which sends the vertex to :

As described in Figure 4, first we use the maps to reduce to the case . In that case, we first need to reverse the direction in the left triangle , by the map which is a rotation about the origin of an angle , and then we should translate it, by the map where

The gluing map between the pants is then described by

For a general discussion, we refer to Section 4 and 5 of [13]. The recipe for gluing two pants apparently depends on the direction of travel across their common boundary. Lemma 4.2 in [13] shows that, in fact, the gluing in either direction is implemented by the same recipe and uses the same parameter .

Remark 2.10 (Relationship with Kra’s construction).

As explained in detail in Section 4.4 of [13], Kra’s plumbing construction (see Kra [12]) is essentially identical to our construction. The difference is that we implement the gluing in the upper half space without first mapping to the punctured disk . In particular the precise relationship between our plumbing parameter and Kra’s one is given by

2.3. Top Terms’ Relationship

We can now state the main result of our previous work which will be fundamental for the proof of our main theorems. The plumbing construction described in Section 2.2 endows with a projective structure whose associated holonomy representation depends holomorphically on the plumbing parameters . In particular, the traces of all elements , are polynomials in the . Theorem A of [13] is a very simple relationship between the coefficients of the top terms of , as polynomials in the , and the Dehn–Thurston coordinates of relative to .

Theorem 2.11 (Top Terms’ Relationship).

Let be a connected simple closed curve on ,such that its Dehn–Thurston coordinates are . If not parallel to any of the pants curves , then is a polynomial in whose top terms are given by:

where

  • ;

  • represents terms with total degree in at most and of degree at most in the variable ;

  • is the total number of -arcs in the standard representation of relative to , see below.

If , then for some , is parabolic, and .

The non-negative integer is defined as follows. The curve is first arranged to intersect each pants curve minimally. In this position, it intersects a pair of pants in a number of arcs joining the boundary curves of . We call one of these an -arc (short for same-(boundary)-component-connector) if it joins one boundary component to itself, and denote by the total number of -arcs, taken over all pants in . Note that some authors call the scc-arcs waves.

Remark 2.12.

As noted in Section 4.2 [13], with our convention the base point for the gluing construction is when . It would be more natural to have, as base point, . That can be achieved by changing the fundamental domain for the standard triply punctured sphere. In particular, one should have as the white triangle the set . This new parameter, equal the old one minus 1, would also make the formula above neater. In fact the formula, with this new parameter, also called call , becomes:

From now on we will use this new parameter which is equal the parameter in [13] minus 1.

2.4. Maskit embedding

In this section we recall the definition of the Maskit embedding of , following Series’ article [22], see also [15]. Let be the set of representations modulo conjugation in . Let be the subset of representations for which:

  1. the group is discrete (Kleinian) and is an isomorphism,

  2. the images of , , are parabolic,

  3. all components of the regular set are simply connected and there is exactly one invariant component ,

  4. the quotient has components (where if ), is homeomorphic to and the other components are triply punctured spheres.

In this situation, see for example Section 3.8 of Marden [14], the corresponding –manifold is topologically . Moreover is a geometrically finite cusp group on the boundary (in the algebraic topology) of the set of quasifuchsian representations of . The ‘top’ component of the conformal boundary may be identified to and is homeomorphic to . On the ‘bottom’ component , identified to , the pants curves have been pinched, making a union of triply punctured spheres glued across punctures corresponding to the curves . The conformal structure on , together with the pinched curves , are the end invariants of in the sense of Minsky’s ending lamination theorem. Since a triply punctured sphere is rigid, the conformal structure on is fixed and independent of , while the structure on varies. It follows from standard Ahlfors–Bers theory, using the Measurable Riemann Mapping Theorem (see again Section 3.8 of [14]), that there is a unique group corresponding to each possible conformal structure on . Formally, the Maskit embedding of the Teichmüller space of is the map which sends a point to the unique group for which has the marked conformal structure .

2.4.1. Relationship between the plumbing construction and the Maskit embedding

In the Section 2.2, given a pants decomposition of , we constructed a family of projective structures on , to each of which is associated a natural holonomy representation . Proposition 4.4 of [13] proves that our plumbing construction described above, for suitable values of the parameters, gives exactly the Maskit embedding of .

Proposition 2.13 (Proposition 4.4 [13]).

Suppose that is such that the associated developing map is an embedding. Then the holonomy representation is a group isomorphism and .

2.5. Three manifolds and pleating rays

Let be a hyperbolic 3–manifold, that is a complete 3-dimensional Riemannian manifold of constant curvature such that the fundamental group is finitely generated. We exclude the somewhat degenerate case has an abelian subgroup of finite index, that is is an elementary Kleinian group. An important subset of is its convex core which is the smallest, non-empty, closed, convex subset of . The boundary of this convex core is a surface of finite topological type whose geometry was described by W. Thurston [23]. Note that given a hyperbolic –manifold , we can also define the convex core as the quotient where is the convex hull of the limit set of , see [7] for a detailed discussion on the pleated structure of the boundary of the convex core. If is geometrically finite, then there is a natural homeomorphism between each component of and each component of the conformal boundary of . Each component of inherits an induced hyperbolic structure from . Thurston also proved such each component is a pleated surface, that is a hyperbolic surfaces which is totally geodesic almost everywhere and such that the locus of points where it fails to be totally geodesic is a geodesic lamination. Formally a pleated surface is defined in the following way.

Definition 2.14.

A pleated surface with topological type in a hyperbolic 3–dimensional manifold is a map such that:

  • the path metric obtained by pulling back the hyperbolic metric of by is a hyperbolic metric on ;

  • there is an -geodesic lamination such that sends each leaf of to a geodesic of and is totally geodesic on .

In this case, we say that the pleated surface admits the geodesic lamination as a pleated locus and is called the bending lamination and the images of the complementary components of are called the flat pieces (of the pleated surface).

The bending lamination of each component of carries a natural transverse measure, called the bending measure (or pleating measure). In the case , there are two components and of and we will denote the respective pleating measure on each one of them.

We will deal with manifolds for which the bending lamination is rational, that is, supported on closed curves. The subset of rational measured laminations is denoted and consists of measured laminations of the form , where the curves are disjoint and non-homotopic, , and denotes the transverse measure which gives weight to each intersection with . If is the bending measure of a pleated surface , then is the angle between the flat pieces adjacent to , also denoted . In particular, if and only if the flat pieces adjacent to are in a common totally geodesic subset of . We take the term pleated surface to include the case in which a closed leaf of the bending lamination maps to the fixed point of a rank one parabolic cusp of . In this case, the image pleated surface is cut along and thus may be disconnected. Moreover the bending angle between the flat pieces adjacent to is . See discussion in [22] or [4].

An important result, due to Bonahon and Otal, about the existence of hyperbolic manifolds with prescribed bending laminations is the following. Recall that a set of curves in a surface fills the surface if for any there exist such that .

Theorem 2.15 (Theorem 1 of [3]).

Suppose that is –manifold homeomorphic to , and that . Then there exists a geometrically finite group such that and such that the bending measures on the two components of equal respectively, if and only if for all and fill up (i.e. if for every ). If such a structure exists, it is unique.

Specialising now to the Maskit embedding , let where be a representation such that the image . The boundary of the convex core has components, one facing and homeomorphic to , and triply punctured spheres whose union we denote . The induced hyperbolic structures on the components of are rigid, while the structure on varies. We recall that we denoted the bending lamination of . Following the discussion above, we view as a single pleated surface with bending lamination , indicating that the triply punctured spheres are glued across the annuli whose core curves correspond to the parabolics .

Corollary 2.16.

A lamination is the bending measure of a group if and only if . If such a structure exists, it is unique.

We call admissible if .

2.5.1. Pleating rays

Denote the set of projective measured laminations on by and the projective class of by . The pleating ray of is the set of groups for which . To simplify notation we write for and note that depends only on the projective class of , also that is non-empty if and only if is admissible. In particular, we write for the ray . As increases, limits on the unique geometrically finite group in the algebraic closure of at which at least one of the support curves to is parabolic, equivalently so that with . We write .

The following key lemma is proved in Proposition 4.1 of Choi and Series [4], see also Lemma 4.6 of Keen and Series [11]. The essence is that, because the two flat pieces of on either side of a bending line are invariant under translation along the bending line, the translation can have no rotational part.

Lemma 2.17.

If the axis of is a bending line of , then .

Notice that the lemma applies even when the bending angle along vanishes. Thus if , where , we have for any whose axis projects to a curve , .

In order to compute pleating rays, we need the following result which is a special case of Theorems B and C of [4], see also [11]. Recall that a codimension- submanifold is called totally real if it is defined locally by equations , where are local holomorphic coordinates for . As usual, if is a bending line we denote its bending angle by . Recall that the complex length of a loxodromic element is defined by , see e.g. [4] for details. By construction, .

Theorem 2.18.

The complex lengths are local holomorphic coordinates for in a neighbourhood of . Moreover is connected and is locally defined as the totally real submanifold of . Any –tuple , where is either the hyperbolic length