Annales Academiæ Scientiarum Fennicæ Mathematica Volumen 26, 2001, 267–304

MATRICES FOR FENCHEL–NIELSEN COORDINATES

Bernard Maskit

The University at Stony Brook, Mathematics Department Stony Brook NY 11794-3651, U.S.A.; bernie@math.sunysb.edu

Abstract. We give an explicit construction of matrix generators for finitely generated Fuchsian groups, in terms of appropriately defined Fenchel–Nielsen (F-N) coordinates. The F-N coordinates are defined in terms of an F-N system on the underlying orbifold; this is an ordered maximal set of simple disjoint closed geodesics, together with an ordering of the set of complementary pairs of pants. The F-N coordinate point consists of the hyperbolic sines of both the lengths of these geodesics, and the lengths of arc defining the twists about them. The mapping from these F-N coordinates to the appropriate representation space is smooth and algebraic. We also show that the matrix generators are canonically defined, up to conjugation, by the F-N coordinates. As a corollary, we obtain that the Teichmüller modular group acts as a group of algebraic diffeomorphisms on this Fenchel–Nielsen embedding of the Teichmüller space.

1. Introduction

There are several different ways to describe a closed Riemann surface of genus at least 2; these include its representation as an algebraic curve; its representation as a period matrix; its representation as a Fuchsian group; its representation as a hyperbolic manifold, in particular, using Fenchel–Nielsen (F-N) coordinates; its representation as a Schottky group; etc. One of the major problems in the overall theory is that of connecting these different visions. Our primary goal in this paper is to construct a bridge between F-N coordinates, for an arbitrary 2-orbifold with finitely generated fundamental group, and matrix generators for the corresponding Fuchsian group.

The usual view of F-N coordinates is that they consist of the lengths and twists about a maximal number of disjoint simple closed geodesics, here called coordinate geodesics (for a closed surface of genus g, this maximal number is 3g−3). We start with these geodesics being undirected, and we use the hyperbolic sines of these lengths (the twists can also be described as lengths of geodesic arcs) as coordinates. It is obvious that the lengths of the geodesics are intrinsic on the surface. Fenchel and Nielsen, in their original unpublished manuscript [6] showed that this space of coordinates is naturally homeomorphic to the Teichmüller space. It follows that the twists cannot be intrinsic on the surface, but it is generally known that the twists can be canonically defined on the surface marked with a basis for the fundamental group; a proof of this fact appears in Section 9.

1991 Mathematics Subject Classification: Primary 30F10; Secondary 32G15.

Research supported in part by NSF Grant DMS 9500557.

One of our results is that the exponential map is universal in the following sense. Following Wolpert [18], we choose the twist to be independent of the length of the geodesic we are twisting about. Then, for any maximal set of simple disjoint geodesics on a hyperbolic orbifold S_{0} , with finitely generated fundamental group, a quasiconformal deformation of S_{0} has all its corresponding F-N coordinates (real) algebraic if and only if the corresponding (appropriately normalized) Fuchsian group is a discrete subgroup of some PSL(2, k), where k is a (real) number field.

A corollary of the above is that the Teichmüller modular group acts on any such space of F-N coordinates as a group of algebraic diffeomorphisms.

An F-N system consists of a hyperbolic base orbifold, S_{0} , with finitely generated fundamental group, together with a maximal set, L_{1}, …, L_{p}, of simple disjoint geodesics, none parallel to the boundary. These coordinate geodesics divide S_{0} into pairs of pants, P_{1}, …, P_{q}. We also assume that the geodesics and pairs of pants are given in a particular order; see Section 2.1. It is well known (see [1]) that, for any given F-N system, the space of F-N coordinates is real-analytically equivalent to the appropriate (reduced) Teichmüller space.

Our first major goal is to write down formulae for matrix generators for the Fuchsian group described by a point in the given F-N coordinate space. The necessary information concerning the topology is encoded in the signature and in the pairing table, defined in Section 8. In the first step, in Section 6, we explicitly describe a set of hyperbolic isometries which generate the corresponding Fuchsian group; then, in Section 7, we give formulae for these isometries as matrices in PSL(2, R). For the first step, we follow the procedure of Fenchel and Nielsen [6]; we start with Fuchsian groups representing pairs of pants, these are orbifolds of genus 0 with three boundary components, including orbifold points (matrices representing generators for these groups are constructed in Section 5), and then use combination theorems to glue these pants groups together.

Our construction yields a well-defined set of hyperbolic isometries, which depend only on the signature of the base orbifold, the pairing table, which describes the topology of the F-N system, and the point in the corresponding coordinate space; these isometries are independent of the conformal or hyperbolic structure on the base orbifold.

The formulae for our matrix generators are sufficiently explicit for us to immediately observe that the entries in the matrices are (real) algebraic functions of the F-N coordinates. In fact, as functions of the parameters, the entries in the matrices are obtained by taking a finite number of degree two field extensions of the field of rational functions of the parameters. We also immediately observe that the arguments of these square roots are bounded away from zero, so that the mapping from F-N coordinates to the space of discrete faithful representations of the fundamental group is smooth. We make these observations here, and again in the recapitulation, but will not repeat them at each stage of the process.

We write our result in the form of an explicit algorithm, which is stated in Section 8. For any given F-N system, the algorithm yields a set of explicit formulae for the entries in the matrix generators, where these entries depend on the particular point in the F-N coordinate space.

We also need to reverse the above process. In Section 9, we start with a set of matrices, generating a discrete group G; we assume these have been defined by the above process. We show that these matrices uniquely define the signature of the underlying orbifold, the pairing table describing the F-N system, and the coordinate point in this system. It follows that our map from F-N coordinates to an appropriate space of discrete faithful representations of the fundamental group is injective, and that the twists are canonically defined on the Teichmüller space.

In Section 10, we give precise statements of our results, which include a new version of the original Fenchel–Nielsen theorem.

In Section 11, we explicitly work through the algorithm for one case of a closed surface of genus 3. In this case, the F-N system has one dividing geodesic and 5 non-dividing geodesics.

The algorithm, as stated, yields matrices that could be simpler, even for genus 2; in Section 12, we give a variation of this algorithm which yields simpler matrices in most cases, and then, in Section 13, we work out this algorithm for the case of three non-dividing geodesics on a closed surface of genus 2. The other case of a closed surface of genus 2, with one dividing geodesic and two non-dividing geodesics, appears in [14].

We also present a second variation of the algorithm in Section 14. In this second variation, the zero twist coordinate position for the handle closing generators is always given by the common orthogonal between two geodesics in the universal covering lying over the same coordinate geodesic. However, this second variation is in some sense less explicit, in that, for any given F-N system, the entries in the matrices are defined algorithmically, rather than being given by explicit formulae.

For orbifolds of dimension 3, our results here extend almost immediately to quasifuchsian groups of the first kind. There are also related results for other classes of Kleinian groups; these will be explored elsewhere.

Some of the ideas and computations used here, as well as various versions of the negative trace theorem, have appeared in print. References for these include Abikoff [1], Fenchel [5], Fenchel and Nielsen [6], Fine and Rosenberger [7], Fricke and Klein [8], Gilman and Maskit [10]; Jörgensen [11], Rosenberger [16], Seppälä and Sorvali [17], Wolpert [18]; see also [13] and [15].

This work was in part inspired by the work of Buser and Silhol [3], who worked out explicit F-N coordinates for certain algebraic curves. The author also wishes to thank Irwin Kra and Dennis Sullivan for informative conversations.

2. Topological preliminaries

We assume throughout that all orbifolds are complete, orientable, of dimension 2, and have non-abelian, finitely generated (orbifold) fundamental group. We denote the hyperbolic plane by H^{2} ; we will usually regard H^{2} as being the upper half-plane endowed with its usual hyperbolic metric, so that the group of all orientation-preserving isometries of H^{2} is canonically identified with PSL(2, R).

Let S be a hyperbolic orbifold; that is, there is a finitely generated Fuchsian group F so that S = H^{2}/F. Topologically, S is a surface of genus g with some number of boundary elements; there are also some number of orbifold points. Geometrically, we regard the orbifold points as boundary elements, so that there are three types of boundary elements. The punctures or parabolic boundary elements, are in natural one-to-one correspondence with the conjugacy classes of maximal parabolic cyclic subgroups of F; the orbifold points, or elliptic boundary elements, are in natural one-to-one correspondence with the conjugacy classes of maximal elliptic cyclic subgroups of F; the order of a puncture is ∞; the order of an orbifold point is the order of a corresponding maximal elliptic cyclic subgroup; and the holes, or hyperbolic boundary elements, are in natural one-to-one correspondence with the conjugacy classes of hyperbolic boundary subgroups of F.

A boundary subgroup H ⊂ F is a maximal hyperbolic cyclic subgroup, whose axis, the boundary axis, bounds a half-plane that is precisely invariant under H in F. The elements of a boundary subgroup are called boundary elements; the boundary axis projects to the corresponding boundary geodesic on S, which is parallel to the boundary. The size of the corresponding boundary element is half the length of this boundary geodesic; that is, if a is a generator of the boundary subgroup H, then its size σ is given by 2 cosh(σ) = | tr(a)|. If a generates a boundary subgroup of G, then the corresponding axis A separates H^{2} into two half-planes. The boundary half-plane is precisely invariant under ⟨a⟩^{1} in G. The other half-plane, which is not precisely invariant (unless G is elementary), is called the action half-plane.

The orbifold S is completely described, up to quasiconformal deformation, by its genus g; the number of boundary elements n that are either punctures or elliptic orbifold points; the orders α_{1}, …, α_{n} of these points; and the number m of holes.

As usual, we encode this information in the signature

(g, n, m; α_{1}, …, α_{n}).

When we do not need to know the actual values of the α_{i}, we write the signature as simply (g, n+m). Since we require S to be hyperbolic, there are some well-known restrictions on these numbers.

1 The group generated by a, … is denoted by ⟨a, …⟩.

It is well known that there are at most p = 3g − 3 + n + m simple disjoint geodesics L_{1}, …, L_{p} on an orbifold of signature (g, n + m), where none of the L_{i} is parallel to the boundary. There are also m boundary geodesics, which we label as L_{p+1}, …, L_{p+m}.

For the remainder of this section, we will consider a geodesic to be defined modulo orientation; that is, we do not distinguish between a geodesic and its inverse.

2.1. F-N systems. An F-N system on S is an ordered set of p + m simple disjoint geodesics—this is the maximal possible number of such geodesics, together with an ordering of the other n boundary elements, where the ordering satisfies the conditions below. We write the F-N system either as L_{1}, …, L_{p+m}, b_{1}, …, b_{n} or as L_{1}, …, L_{p}, b_{1}, …, b_{n+m}, or as L_{1}, …, L_{p+n+m}. It will always be clear from the context which system of notation we are using.

Except in Section 4, we will assume throughout that S is not a pair of pants; that is, p > 0.

None of the first p geodesics of an F-N system are parallel to the boundary; they are the coordinate geodesics. The coordinate geodesics divide S into q = 2g − 2 + n + m pairs of pants, P_{1}, …, P_{q}, each of which is a hyperbolic orbifold of signature (0, n_{0} + m_{0}), n_{0} + m_{0} = 3. Each coordinate geodesic is either a boundary element of two distinct pairs of pants, or corresponds to two boundary elements of the same pair of pants.

There is in general no canonical way to order and direct the coordinate geodesics, and to order the pairs of pants they divide the surface into. From here on, we assume that the coordinate geodesics and boundary elements, and also the pairs of pants, have been ordered in accordance with the following set of rules.

2.1.1. Rules for order.

(i) If there is a dividing coordinate geodesic, then L_{1} is dividing; in any case, if q ≥ 2, then L_{1} lies between P_{1} and P_{2} .

(ii) If q ≥ 3, then L_{2} lies between P_{1} and P_{3} . (iii) The first q − 1 coordinate geodesics, and the q pairs of pants, P_{1}, …, P_{q},

are ordered so that, for every j = 3, …, q − 1, there is an i = i(j), with 1 ≤ i(j) < j, so that L_{j} lies on the common boundary of P_{i} and P_{j}. The coordinate geodesics L_{1}, …, L_{q−1} are called the attaching geodesics; the coordinate geodesics L_{q}, …, L_{p} are called the handle geodesics.

(iv) The hyperbolic boundary elements b_{1}, …, b_{m} precede the parabolic boundary elements, which, in turn precede the elliptic boundary elements. Also, the elliptic boundary elements are in decreasing order. From here on, we reserve the indices, m, n, p and q, for the meanings given

above.

2.2. F-N coordinates. Let G_{0} be a given finitely generated Fuchsian group, and let S_{0} = H^{2}/G_{0} . A (quasiconformal) deformation of G_{0} is a discrete faithful representation ψ of G_{0} into PSL(2, R), where there is a quasiconformal homeomorphism f: H^{2} → H^{2} inducing ψ. Two such deformations, ψ and ψ′ are equivalent if there is an element a ∈ PSL(2, R) so that ψ(g) = aψ′(g)a^{−1} for all g ∈ G_{0} .

Let ψ: G_{0} → G be a quasiconformal deformation. The F-N coordinates of (the equivalence class of) ψ are given by the following vector:

Φ = (s_{1}, …, s_{p+m}, t_{1}, …, t_{p}) ∈ (R^{+})^{p+m} × R^{p}.

The geodesics L_{1}, …, L_{p}, L_{p+1}, …, L_{p+m} are well defined on S = H^{2}/G. The length of L_{i} on S, i = 1, …, p + m, is 2σ_{i}, where s_{i} = sinh σ_{i}. Also, for i = 1, …, p, the twist about L_{i} is 2τ_{i}, where t_{i} = sinh τ_{i}; this will be explained in Section 6.

2.3. Pairs of pants. Each pair of pants P has three boundary elements; in most cases, the ordering of the coordinate geodesics and boundary elements of S_{0} induces an ordering of the boundary elements of P. There are two exceptional cases in which there are two boundary elements of P corresponding to just one coordinate geodesic of S_{0} .

In the first exceptional case, S_{0} is a torus with one boundary component, so two of the boundary elements of P are hyperbolic, necessarily of the same size, and the other boundary element can be of any type. Since the torus with one boundary component is elliptic (i.e., admits a conformal involution with 3 or 4 fixed points), one cannot tell the difference between the two boundary elements of P corresponding to the one coordinate geodesic on S_{0} . We make an arbitrary choice of which of these two boundary elements precedes the other; since the elliptic involution acts ineffectively on the Teichmüller space, it makes no difference which choice we make.

In the second exceptional case, all three boundary elements of P are necessarily hyperbolic. Here, one of the boundary elements of P corresponds to an attaching geodesic, which is also a dividing geodesic of S_{0} , while the other two boundary elements both correspond to the same non-dividing handle geodesic. In this case, b_{1} , the first boundary element, necessarily corresponds to the dividing geodesic. The other two boundary elements, b_{2} and b_{3} , are necessarily hyperbolic of the same size. As above, if S_{0} is elliptic or hyperelliptic, then the choice of which boundary element of P to call b_{2} and which to call b_{3} is arbitrary, and it does not matter which choice we make. If S_{0} is not elliptic or hyperelliptic, then one can make a canonical choice; this will be done in Section 6. Until then, we leave it that this choice is made somehow.

We now return to the general case. Between any two boundary elements of P, b_{i} and b_{j}, there is a unique simple orthogonal geodesic arc N_{ij} ⊂ P. That is, if b_{i} is parabolic, then N_{ij} has infinite length, with an infinite endpoint at the parabolic puncture; if b_{i} is elliptic, then N_{ij} has one endpoint at this elliptic orbifold point; if b_{i} is hyperbolic, then N_{ij} is orthogonal to the corresponding boundary geodesic.

In the case that b_{i} is hyperbolic, with boundary geodesic L_{i}, then the two common orthogonals to the two other boundary elements of P meet L_{i} at two distinct points of L_{i}; these two points divide L_{i} into two arcs of equal length.

We will use the following notation throughout. The boundary elements of the pair of pants, P_{i}, are labeled as b_{i,1}, b_{i,2}, b_{i,3} , in the order given above.

2.4. Directing the coordinate and boundary geodesics. We need to specify a direction for each coordinate geodesic. In general, we direct L_{1} so that P_{1} lies on the right as we traverse L_{1} in the positive direction. In the exceptional cases that S_{0} is elliptic or hyperelliptic, this choice of a first direction is necessarily arbitrary; however, as mentioned above, it is irrelevant which choice is made.

We say that two geodesics on the boundary of some pair of pants P are consistently oriented with respect to P, if P lies on the right as we traverse either geodesic in the positive direction, or if P lies on the left as we traverse either geodesic in the positive direction.

Assume that L_{1}, …, L_{j}, j ≥ 1, have been directed. If L_{j+1} lies on the boundary of two distinct pairs of pants, or is a boundary geodesic, then there is a lowest index i, so that L_{j+1} lies on the boundary of P_{i}. Since j + 1 > 1, L_{j+1} corresponds to either b_{i,2} or b_{i,3} , for b_{i,1} must correspond to some attaching geodesic, L_{j′}, j′ ≤ j. We direct L_{j+1} so that L_{j+1} and L_{j′} are consistently oriented as boundary elements of P_{i}.

If L_{j+1} is a handle geodesic, with the same pair of pants, P_{i}, on both sides of L_{j+1} , then the direction of L_{j+1} is more complicated. As above, we will see in Section 6 that this choice can be made canonically; for the moment, we assume that this choice has been made somehow.

3. SL(2, R) and PSL(2, R)

There is a canonical identification of PSL(2, R) with the group of orientationpreserving isometries of H^{2} ; each such transformation has two representatives in SL(2, R). There is likewise a canonical identification of PGL(2, R) with the group of all plane hyperbolic isometries; each such isometry has two representatives in S^{±}L(2, R), the group of real 2 × 2 matrices with determinant ±1.

We will use the following convention throughout. If ã is a matrix in S^{±}L(2, R), then the corresponding hyperbolic isometry is denoted by a.

We will usually use this notation in reverse; that is, given the isometry a, we will choose a representative matrix ã ∈ S^{±}L(2, R). Also, all hyperbolic isometries (i.e., all transformations) that are not explicitly identified as reflections, are assumed to be orientation-preserving.

A Fuchsian group F is algebraic if there is a (real) number field k so that F ⊂ PSL(2, k). Correspondingly, the orbifold S = H^{2}/F is algebraic if F is algebraic.

We remark that a Fuchsian group F, with generators a_{1}, …, a_{i} …, is algebraic if and only if the numbers, a_{i}(0), a_{i}(1), a_{i}(∞), are all algebraic.

For our purposes, from here on, a Fuchsian group is a non-Abelian finitely generated discrete subgroup of PSL(2, R); it is elementary if it contains an Abelian subgroup of finite index, and non-elementary otherwise. Unless explicitly stated otherwise, all Fuchsian groups will be assumed to be non-elementary^{2} .

4. Reflections and geometric generators

It will often be convenient to have an order among the different kinds of hyperbolic isometries. We say that hyperbolic transformations are higher than the parabolic ones, which in turn are higher than the elliptic ones; further, elliptic transformations of higher (finite) order are higher than elliptic transformations of lower order.

An elliptic transformation a of order α is primitive if | tr(a)| = 2 cos(π/α); that is, a is a geometrically primitive rotation.

4.1. Transformations with disjoint axes. Let a_{1} , a_{2} and a_{3} = (a_{1}a_{2})^{−1} be elements of PSL(2, R). If a_{i} is hyperbolic, then its axis A_{i} is, as usual, the hyperbolic line connecting its fixed points. If a_{i} is parabolic or elliptic, then its axis A_{i} is its fixed point, which lies either on the circle at infinity or is an interior point of H^{2} .

We will use the following conventions throughout: If a_{α}^{β} is a given element of PSL(2, R), then its axis is denoted by A_{α}^{β}.

In general, if X ⊂ H^{2} , then we denote the Euclidean closure of X by \overline{X}. Also, in general, two lines, L and L′, are disjoint if L̄ ∩ L̄′ = ∅, in particular, the axes of a_{1} and a_{2} are disjoint if Ā_{1} ∩ Ā_{2} = ∅.

If a is elliptic or parabolic, then we say that the line M is orthogonal to A if M passes through A, or ends at A. We now have that, independent of the type of a_{i} and a_{j}, if a_{i} and a_{j} have disjoint axes, then these axes have a unique common orthogonal.

4.2. Reflections in lines. For every hyperbolic line M, there is a welldefined reflection r, whose fixed point set is equal to M.

Let M_{1} and M_{2} be distinct lines; denote reflection in M_{i} by r_{i}, and let a = r_{1}r_{2} . Then a is hyperbolic, respectively, parabolic, respectively, elliptic, if M_{1} and M_{2} are disjoint, respectively, meet at the circle at infinity, respectively, cross inside H^{2} . If a is hyperbolic, then | tr(a)| = 2 cosh λ, where λ is the distance between M_{1} and M_{2} ; if a is elliptic, then | tr(a)| = 2| cos θ|, where θ is the angle between M_{1} and M_{2} .

2 Among Fuchsian groups, the (2, 2, ∞)-triangle group is uniquely elementary but not Abelian; it shares many important properties with the non-elementary Fuchsian groups.

4.3. Matrices for reflections. The matrices in S^{±}L(2, R) of determinant −1 and trace 0 correspond to reflections in lines. One can regard the choice of a matrix for a reflection as being equivalent to a choice of a direction on the fixed line (see Fenchel [5]). More precisely, we choose the matrix

r̃ = \frac{1}{x − y} ( \begin{matrix}x + y & −2xy \\ 2 & −x − y \end{matrix} )

to correspond to the reflection in the upper half-plane with fixed line ending at x and y, where x is the positive endpoint of this line. Then by continuity, the reflection with matrix

r̃ = ( \begin{matrix}1 & −2y \\ 0 & −1 \end{matrix} )

has its positive fixed point at ∞ and its negative fixed point at y.

If M_{1} and M_{2} are disjoint directed hyperbolic lines, then we say that the positive endpoints of M_{1} and M_{2} are adjacent to mean that both negative endpoints lie on the same arc of the circle at infinity between these positive endpoints.

Easy observations now show the following.

Proposition 4.1. Let r̃_{1}, r̃_{2} ∈ S^{±}L(2, R) represent reflections in the disjoint directed lines M_{1} , M_{2} , respectively. Then tr(r̃_{1}r̃_{2}) > 0 if and only if the positive endpoints of M_{1} and M_{2} are adjacent.

Proposition 4.2. Let r̃_{1}, r̃_{2} ∈ S^{±}L(2, R) represent reflections in the directed lines M_{1} , M_{2} , respectively, where M_{1} and M_{2} have exactly one endpoint on the circle at infinity in common. Then tr(r̃_{1}r̃_{2}) = +2 if and only if the common endpoint is either the positive endpoint, or the negative endpoint, of both lines.

Proposition 4.3. Let r̃_{1}, r̃_{2} ∈ S^{±}L(2, R) represent reflections in the directed lines M_{1} , M_{2} , respectively, where M_{1} and M_{2} intersect at an interior point of H^{2} . Then tr(r̃_{1}r̃_{2}) = 2 cos θ, where θ is the angle between the positive endpoints of M_{1} and M_{2} .

4.4. Hyperbolic triangles. In Euclidean geometry, a triangle is completely determined by three lines, no two of which are parallel; in hyperbolic geometry, the situation is somewhat more complicated. For our purposes, a triangle D is the intersection of the three closed half-planes, R_{1}, R_{2}, R_{3} , bounded by the three distinct lines, M_{1}, M_{2}, M_{3} , respectively, provided that, for i = 1, 2, 3, M_{i} ∩ D contains a non-trivial open arc of M_{i}. This arc, M_{i} ∩ D, is called a side of D.

Every pair of these lines, M_{i} and M_{j}, defines a vertex, v_{i,j}, which is the common orthogonal of M_{i} and M_{j}; this vertex is hyperbolic, respectively, parabolic, respectively, elliptic, if M_{i} and M_{j} are disjoint, respectively, meet at the circle at infinity, respectively, meet at an interior point of H^{2} .

We remark that the triangle D is not necessarily uniquely determined by the three lines, M_{1}, M_{2}, M_{3} .

We say that D is a Poincaré triangle if the interior angle at every elliptic vertex is of the form, π/α, α ∈ Z, α ≥ 2.

A triangle D is degenerate if two of the bounding lines are each orthogonal to the third. The orientation preserving half of the group generated by reflections in the three sides of a degenerate triangle is elementary.

Let r_{i} denote reflection in M_{i}, i = 1, 2, 3; set a_{1} = r_{2}r_{3} , a_{2} = r_{3}r_{1} , and a_{3} = r_{1}r_{2} .

Poincaré’s polygon theorem asserts that if D is a Poincaré triangle, then the group Ĵ = ⟨r_{1}, r_{2}, r_{3}⟩ is discrete; D is a fundamental polygon for Ĵ; and Ĵ has the following presentation:

Ĵ = ⟨r_{1}, r_{2}, r_{3} : r_{1}^{2} = r_{2}^{2} = r_{3}^{2} = (r_{1}r_{2})^{α_{3}} = (r_{2}r_{3})^{α_{1}} = (r_{3}r_{1})^{α_{2}} = 1⟩,

where the statement (r_{i}r_{j})^{α_{k}} = a_{k}^{α_{k}} = 1 has its usual meaning if the vertex v_{i,j} is elliptic of order α_{k}; it means that a_{k} is parabolic if v_{i,j} is parabolic; and it has no meaning if v_{i,j} is hyperbolic. That is, a_{k} is hyperbolic, respectively, parabolic, respectively, elliptic of order α_{k}, if and only if the vertex v_{i,j} is hyperbolic, respectively, parabolic, respectively, elliptic of order α_{k}.

We note that A_{k} is the common orthogonal to M_{i} and M_{j}. Further, if a_{k} is hyperbolic, then | tr(a_{k})| = 2 cosh λ_{k}, where λ_{k} is the distance between M_{i} and M_{j}; if a_{k} is parabolic, then | tr(a_{k})| = 2, and if a_{k} is elliptic of order α_{k}, then | tr(a_{k})| = 2 cos π/α_{k}.

Let J be the orientation-preserving half of Ĵ. If D is a Poincaré triangle, then H^{2}/J is a pair of pants, where the boundary element b_{i} is the projection of A_{i}.

If D is a Poincaré triangle, then we say that a_{1} = r_{2}r_{3} and a_{2} = r_{3}r_{1} are geometric generators of a pants group. On course, in this case, a_{2} and a_{3} , or a_{3} and a_{1} , are also geometric generators of the same pants group.

It is well known that every pants group, including the triangle groups, has a set of geometric generators; in fact, these generators are unique up to conjugation in the pants group, up to orientation, and up to a choice of which of the three generators to call a_{1} , and which to call a_{2} .

4.5. Appropriate orientation. Let a_{1} and a_{2} be hyperbolic isometries with disjoint axes.

If a_{1} and a_{2} are both hyperbolic, then A_{1} and A_{2} are naturally directed. If the region between these two axes lies on the left as one traverses one of these axes in the positive direction, and it lies on the right as one traverses the other axis in the positive direction, then a_{1} and a_{2} are not appropriately oriented.

If either a_{1} or a_{2} is elliptic of order 2, then a_{1} and a_{2} are appropriately oriented.

Every parabolic element imparts a natural direction to the circle at infinity, as does every elliptic element of order at least 3. If a_{1} and a_{2} are both either parabolic or elliptic of order at least 3, then they are appropriately oriented if they impart the same direction to the circle at infinity.

Suppose a_{1} is hyperbolic and a_{2} is either parabolic or elliptic of order at least 3. Let H be the half-plane bounded by A_{1} , where \overline{H} ⊃ A_{2} , and let S be the arc of the circle at infinity on \overline{H} . Then a_{1} and a_{2} are appropriately oriented if they impart the same direction to S.

4.6. The negative trace theorem.

Theorem 4.1. Let ã_{1} and ã_{2} be matrices in SL(2, R), where a_{2} is not higher than a_{1} .

A. If A_{1} and A_{2} are not disjoint, then a_{1} and a_{2} are geometric generators of a pants group if and only if a_{1} is hyperbolic and a_{2} is elliptic of order 2.

B. If A_{1} and A_{2} are disjoint, then a_{1} and a_{2} are geometric generators of a pants group if and only if the following hold:

(i) T = tr(ã_{1}) tr(ã_{2}) tr(ã_{1}ã_{2}) ≤ 0; and (ii) if any of a_{1} , a_{2} or a_{1}a_{2} is elliptic, then it is primitive.

Proof. All cases of two transformations with non-disjoint axes are well known. The group G = ⟨a_{1}, a_{2}⟩ is discrete only in the case above, and in various cases of two hyperbolic generators with crossing axes. In these latter cases, either H^{2}/G has signature (1, 1), or has signature (0, 3), but a_{1} and a_{2} are not geometric generators.

Now assume that the axes of a_{1} and a_{2} are disjoint. Let L_{3} be the common orthogonal to A_{1} and A_{2} . One easily finds lines, M_{1} and M_{2} , so that, denoting reflection in M_{i} by r_{i}, a_{1} = r_{2}r_{3} and a_{2} = r_{3}r_{1} .

Let M_{1} , M_{2} , M_{3} , be any three distinct directed lines. Let r̃_{i} ∈ S^{±}L(2, R) be the matrix representing reflection in M_{i}, with the given orientation. Let ã_{1} = r̃_{2}r̃_{3} , ã_{2} = r̃_{3}r̃_{1} and ã_{3} = r̃_{1}r̃_{2} .

Observe that T is unchanged if we replace any r̃_{i} by −r̃_{i}.

There are five cases to consider; we do not need to consider the sixth case, where all three lines meet at a point, for we assume that Ā_{1}∩Ā_{2} = ∅. In each case, we draw three lines, and direct them somehow; the sign of T does not depend on which direction we choose. Then we use Propositions 4.1–4.3, to compute the sign of T.

Case 1. If the three lines are pairwise disjoint, and one of the lines separates the other two inside H^{2} , then a_{1} and a_{2} are not geometric generators, and T > 0.

Case 2. If the lines have no points of intersection inside H^{2} , and the three lines bound a common region, then a_{1} and a_{2} are geometric generators, and T < 0.

Case 3. If exactly two of the lines meet inside H^{2} , then there is exactly one of the five regions cut out by these three lines that can be a triangle. The angle at the one elliptic vertex is acute if and only if T < 0; that angle is a right angle if and only if T = 0.

Case 4. If say M_{1} meets both M_{2} and M_{3} , but M_{2} ∩ M_{3} = ∅, then these three lines separate H^{2} into six regions, of which at most one can be a triangle with all angles ≤ π. There is such a triangle if and only if T ≤ 0.

Case 5. If M_{1} , M_{2} and M_{3} form a compact triangle, then T ≤ 0 if and only if none of the angles are obtuse.

5. Fully normalized pants groups

We assume that we are given three numbers λ_{1} , λ_{2} and λ_{3} , where either λ_{i} ≥ 0, or λ_{i} = iπ/α, α ∈ Z, α ≥ 2. We need to write down matrices, ã_{1} and ã_{2} , corresponding to geometric generators for a pants group, where | tr(ã_{1})| = 2 cosh λ_{1} , | tr(ã_{2})| = 2 cosh λ_{2} and | tr(ã_{3})| = | tr(ã_{1}ã_{2})^{−1}| = 2 cosh λ_{3} . We can assume without loss of generality that the λ_{i} are given so that the a_{i} are in non-increasing order; we can also assume that tr(ã_{1}) ≥ 0 and tr(ã_{2}) ≥ 0.

Since the normalizations are different, we will take up separately the different cases according to the types of a_{1} , a_{2} and a_{3} = (a_{1}a_{2})^{−1} .

5.1. Standard normalizations. If a_{1} is hyperbolic, then A_{1} is the imaginary axis, pointing towards ∞. If a_{2} and a_{3} are both elliptic of order 2, then A_{2} is the point i. Otherwise, A_{1} and A_{2} are disjoint, in which case A_{2} lies in the right half-plane and M_{3} , the common orthogonal between A_{1} and A_{2} , lies on the unit circle.

If a_{1} is parabolic, then A_{1} is the point at infinity, and M_{3} lies on the imaginary axis. If a_{2} is also parabolic, then A_{2} is necessarily at 0; if a_{2} is elliptic, then A_{2} is at the point i.

If a_{1} is elliptic, then we change our point of view; regard H^{2} as being the unit disc; place A_{1} at the origin, and place A_{2} on the positive real axis.

5.2. Three hyperbolics. In this case, λ_{i} > 0, i = 1, 2, 3. We need to find matrices ã_{1} and ã_{2} , with tr(ã_{1}) = 2 cosh(λ_{1}); tr(ã_{2}) = 2 cosh(λ_{2}); and tr(ã_{1}ã_{2}) = −2 cosh(λ_{3}).

We need a_{1} and a_{2} to be appropriately oriented; hence we write our matrices so that the repelling fixed point of a_{2} is greater than 1, while the attracting fixed point is less than 1.

We define μ by:

(1)  coth μ = \frac{cosh λ_{1} cosh λ_{2} + cosh λ_{3}}{sinh λ_{1} sinh λ_{2}} ,  μ > 0.

We write:

ã_{1} = ( \begin{matrix}e^{λ_{1}} & 0 \\ 0 & e^{−λ_{1}} \end{matrix} );  ã_{2} = \frac{1}{sinh μ} ( \begin{matrix}sinh(μ − λ_{2}) & sinh λ_{2} \\ − sinh λ_{2} & sinh(μ + λ_{2}) \end{matrix} ) .

Then a_{1} is as desired, and a_{2} has its attracting fixed point at e^{−μ}, its repelling fixed point at e^{μ}, and tr(ã_{2}) = 2 cosh λ_{2} . Equation (1) yields that tr(ã_{1}ã_{2}) = −2 cosh(λ_{3}).

We will also need a different matrix representation for a_{3} = a_{2}^{−1}a_{1}^{−1} . We recall that M_{3} , the common orthogonal between A_{1} and A_{2} meets A_{1} at i. Then, since a pair of pants is hyperelliptic, M_{2} , the common orthogonal of A_{1} and A_{3} , meets A_{1} halfway between i and a_{1}(i). Hence M_{2} , lies on the circle |z| = e^{λ_{1}} .

We define ν by the following.

(2)  coth ν = \frac{cosh λ_{1} cosh λ_{3} + cosh λ_{2}}{sinh λ_{1} sinh λ_{3}} ,  ν > 0.

Using the above remark, together with the definition of ν, it is easy to see that we can write

ã_{3} = −ã_{2}^{−1}ã_{1}^{−1} = \frac{1}{sinh ν} ( \begin{matrix}sinh(ν − λ_{3}) & e^{λ_{1}} sinh λ_{3} \\ −e^{−λ_{1}} sinh λ_{3} & sinh(ν + λ_{3}) \end{matrix} ) .

Remark 5.1. One easily sees that μ is related to δ, the distance between A_{1} and A_{2} , by coth μ = cosh δ, or, equivalently, sinh μ sinh δ = 1. The RHS of equation (1) is the well-known formula for the hyperbolic cosine of the length of one side of a hexagon with all right angles, given the lengths of three non-adjacent sides. Similar remarks hold for equation (2).

5.3. Closing a handle. The case that S_{0} is a torus with one hole needs to be treated separately. In this case, S_{0} has one coordinate geodesic, necessarily non-dividing, and one boundary geodesic. We change our usual order, and label the boundary geodesic of the one pair of pants P as b_{1} , and label the other two boundary elements, corresponding to the coordinate geodesic, as b_{2} and b_{3} .

We proceed exactly as above, and construct ã_{1} , ã_{2} and ã_{3} , with λ_{2} = λ_{3} . We need to find a matrix for the handle-closer d, which maps the action half-plane of a_{2} onto the boundary half-plane of a_{3} , while twisting by 2τ in the positive direction along A_{2} .

We can write d = rr_{2}e_{τ}, where e_{τ} is the hyperbolic motion (or the identity) with the same fixed points as a_{2} and with trace equal to 2 cosh τ, where a_{2} and e_{τ} have the same attracting fixed point if τ > 0, and have opposite attracting fixed points if τ < 0; r_{2} is the reflection in A_{2} ; and r is the reflection in the line halfway between A_{2} and A_{3} .

Using ã_{2} as a model, we already know how to find ẽ_{τ}:

ẽ_{τ} = \frac{1}{sinh μ} ( \begin{matrix}sinh(μ − τ) & sinh τ \\ − sinh τ & sinh(μ + τ) \end{matrix} ) .

The line halfway between A_{2} and A_{3} is the circle centered at the origin of radius exp(\frac{1}{2}λ_{1}). Hence we can write

r̃ = ( \begin{matrix}0 & exp(\frac{1}{2}λ_{1}) \\ exp(−\frac{1}{2}λ_{1}) & 0 \end{matrix} ) .

To find a matrix for r_{2} , observe that if

ã = ( \begin{matrix}α & β \\ γ & δ \end{matrix} ) ∈ SL(2, R),

where a is hyperbolic and βγ ≠ 0, then we can write the matrix for the reflection r_{A} in A as

(3)  r̃_{A} = \frac{1}{\sqrt{(α + δ)^{2} − 4}} ( \begin{matrix}α − δ & 2β \\ 2γ & δ − α \end{matrix} ) .

In our case, we obtain

r̃_{2} = \frac{1}{sinh μ} ( \begin{matrix}− cosh μ & 1 \\ −1 & cosh μ \end{matrix} ) .

Hence we can write

(4)  d̃= r̃r̃_{2}ẽ_{τ} .

5.4. Two hyperbolics, one parabolic or elliptic. As above, there is one special case, where S_{0} has signature (1, 1); we take up that case below. Here we only assume that a_{1} and a_{2} are hyperbolic, and that a_{3} is parabolic or primitive elliptic. As above, our normalization yields that a_{1}(z) = e^{2λ_{1}}z, and a_{2} has its fixed points at e^{±μ}, μ > 0. Since a_{1} and a_{2} need to be appropriately oriented, we place the repelling fixed point of a_{2} at e^{+μ}. Then, if a_{3} is parabolic, it has its fixed point at e^{λ_{1}} . If a_{3} is primitive elliptic of order α, then it has its fixed point in the first quadrant on the circle |z| = e^{λ_{1}} . We write

ã_{1} = ( \begin{matrix}e^{λ_{1}} & 0 \\ 0 & e^{−λ_{1}} \end{matrix} );  ã_{2} = \frac{1}{sinh μ} ( \begin{matrix}sinh(μ − λ_{2}) & sinh λ_{2} \\ − sinh λ_{2} & sinh(μ + λ_{2}) \end{matrix} ) ,

where μ is defined by

(5)  coth μ = \frac{cosh λ_{1} cosh λ_{2} + cosh λ_{3}}{sinh λ_{1} sinh λ_{2}} ,  μ > 0.

Remark 5.2. The formula here for μ is the same as that in equation (1); the geometric meaning is the same in both cases.

5.5. The torus with one puncture or orbifold point. In this case we change our standard normalization, and require a_{1} to be parabolic or primitive elliptic. Then λ_{2} = λ_{3} > 0. We normalize so that A_{1} lies on the positive imaginary axis (A_{1} is the point at infinity if a_{1} is parabolic), and so that the unit circle is the common orthogonal between A_{2} and A_{3} , where the fixed points of a_{3} are positive, with the repelling fixed point larger than the attracting one, and the fixed points of a_{2} are negative. Then r_{0} , the reflection in the imaginary axis, conjugates a_{2} into a_{3}^{−1} . We can write the fixed points of a_{2} as −e^{±μ} and the fixed points of a_{3} as e^{±μ}. We write

ã_{2} = \frac{1}{sinh μ} ( \begin{matrix}sinh(μ + λ_{2}) & sinh λ_{2} \\ − sinh λ_{2} & sinh(μ − λ_{2}) \end{matrix} ) ,
ã_{3} = \frac{1}{sinh μ} ( \begin{matrix}sinh(μ − λ_{2}) & sinh λ_{2} \\ − sinh λ_{2} & sinh(μ + λ_{2}) \end{matrix} ) .

Since we require tr(ã_{2}ã_{3}) = −2 cosh λ_{1} , easy computations show that

(6)  sinh^{2}(μ) = \frac{2 sinh^{2} λ_{2}}{cosh λ_{1} + 1}.

Remark 5.3. The formula for μ given in equation (6) is different from that given in equations (1) and (5) because the underlying geometry is different. We still have coth μ = cosh δ, but here δ is the distance from A_{2} to the imaginary axis, which is the common orthogonal of A_{3} with the common orthogonal of A_{1} and A_{2} .

As in Section 5.3 we also need a matrix representing the handle-closer d, which conjugates a_{2} onto a_{3}^{−1} while introducing a twist of 2τ along A_{2} . We write d = r_{0}f_{τ} r_{2} , where r_{2} is the reflection in A_{2} , f_{τ} is the twist by 2τ along the imaginary axis, and r_{0} is the reflection in the imaginary axis. The corresponding matrices are given by

r̃_{0} = ( \begin{matrix}1 & 0 \\ 0 & −1 \end{matrix} ) , r̃_{2} = \frac{1}{sinh μ} ( \begin{matrix}cosh μ & 1 \\ −1 & − cosh μ \end{matrix} ) , f̃_{τ} = ( \begin{matrix}e^{τ} & 0 \\ 0 & e^{−τ} \end{matrix} ) .

5.6. Exactly one hyperbolic and at least one parabolic. Here λ_{1} > 0, λ_{2} = 0, and either λ_{3} = 0 or λ_{3} = iπ/α. We revert to our standard normalization, so that a_{1}(z) = e^{2λ_{1}}z, and a_{2} has its fixed point at +1. Then a_{3} has its fixed point in the right half-plane on the circle |z| = e^{λ_{1}} .

We write the matrices

ã_{1} = ( \begin{matrix}e^{λ_{1}} & 0 \\ 0 & e^{−λ_{1}} \end{matrix} ) ,  ã_{2} = ( \begin{matrix}1 + β & −β \\ β & 1 − β \end{matrix} ) .

Using Theorem 4.1, easy computations show that

β = −\frac{cosh λ_{1} + cosh λ_{3}}{sinh λ_{1}} .

5.7. The elementary pants groups. In the special case that λ_{1} > 0 and λ_{2} = λ_{3} = 0, we write

ã_{1} = ( \begin{matrix}e^{λ_{1}} & 0 \\ 0 & e^{−λ_{1}} \end{matrix} ) ,  ã_{2} = ( \begin{matrix}0 & −1 \\ 1 & 0 \end{matrix} ) .

The group generated by these is of course elementary.

5.8. Exactly one hyperbolic and no parabolics. Here λ_{1} > 0, λ_{2} = iπ/α_{2} , and λ_{3} = iπ/α_{3} . We have a_{1}(z) = e^{2λ_{1}}z, and a_{2} has its fixed point on the unit circle in the (open) right half-plane. Then as above, a_{3} has its fixed point in the right half-plane on the circle |z| = e^{λ_{1}} . Denote the fixed point of a_{2} by e^{μ} = e^{iθ}, 0 < θ < \frac{1}{2}π.

As in [12, p. 6], we can write

ã_{1} = ( \begin{matrix}e^{λ_{1}} & 0 \\ 0 & e^{−λ_{1}} \end{matrix} ) ,  ã_{2} = \frac{1}{sinh μ} ( \begin{matrix}sinh(μ − λ_{2}) & sinh λ_{2} \\ − sinh λ_{2} & sinh(μ + λ_{2}) \end{matrix} ) .

As above, we obtain

(7)  coth μ = \frac{cosh λ_{1} cosh λ_{2} + cosh λ_{3}}{sinh λ_{1} sinh λ_{2}} .

Remark 5.4. Here, coth μ = −i sinh δ, where δ is the distance from A_{1} to A_{2} . This gives a known formula for one side of a quadrilateral with two adjacent right angles, in terms of the other two angles, and the distance between the two right angles (see [5]).

5.9. The classical triangle groups. For the sake of completeness, since this form seems not to be known in the literature—although a related form can be found in [9], we write down matrices for geometric generators for the general Fuchsian (α_{1}, α_{2}, α_{3})-triangle group.

The case of three parabolics is well known and needs no further discussion. If λ_{1} = λ_{2} = 0, and λ_{3} = iπ/α_{3} , then we normalize so that a_{1}(z) = z + 1,

and a_{2} has its fixed point at 0; we write

ã_{1} = ( \begin{matrix}1 & 1 \\ 0 & 1 \end{matrix} ) ,  ã_{2} = ( \begin{matrix}1 & 0 \\ −2 − 2 cosh λ_{3} & 1 \end{matrix} ) .

If λ_{1} = 0, λ_{2} = iπ/α_{2} and λ_{3} = iπ/α_{3} , then we normalize so that a_{1} has its fixed point at ∞, with a_{1}(0) > 0, and so that a_{2} has its fixed point at i. Then, since a_{1} and a_{2} are appropriately oriented, a_{2}(∞) < 0. We write

ã_{1} = ( \begin{matrix}1 & (2 cosh λ_{2} + 2 cosh λ_{3})/(−i sinh λ_{2}) \\ 0 & 1 \end{matrix} ) , ã_{2} = ( \begin{matrix}cosh λ_{2} & −i sinh λ_{2} \\ i sinh λ_{2} & cosh λ_{2} \end{matrix} ) .

Finally, if λ_{j} = iπ/α_{j}, j = 1, 2, 3, then we change our view of H^{2} , which we now regard as being the unit disc, and we normalize so that a_{1}(z) = e^{2πi/α_{1}}z = e^{2λ_{1}}z, and a_{2} has its fixed point at e^{−μ}, μ > 0. We write

ã_{1} = ( \begin{matrix}e^{λ_{1}} & 0 \\ 0 & e^{−λ_{1}} \end{matrix} );  ã_{2} = \frac{1}{sinh μ} ( \begin{matrix}sinh(μ + λ_{2}) & − sinh λ_{2} \\ sinh λ_{2} & sinh(μ − λ_{2}) \end{matrix} ) ,

where μ is given by

(8)  coth μ = −\frac{cosh λ_{1} cosh λ_{2} + cosh λ_{3}}{sinh λ_{1} sinh λ_{2}} .

Remark 5.5. Here, as in equation (1), coth μ = cosh δ, where δ is the distance from A_{1} to A_{2} . This equation for δ is one of the two hyperbolic laws of cosines (see [2]).

Note that we can solve equation (8) for μ > 0 if and only if the right-hand side is > 1, which occurs if and only if

\frac{π}{α_{1}} + \frac{π}{α_{2}} + \frac{π}{α_{3}} < π.

6. From F-N systems to homotopy bases

We now assume we are given an F-N system on the hyperbolic orbifold S_{0} , where the coordinate geodesics L_{1}, …, L_{p}, the pairs of pants, P_{1}, …, P_{q}, and the boundary elements, b_{1}, …, b_{n+m}, are ordered as in 2.1.1, and the coordinate geodesics are directed as in 2.4. We also assume we are given a point

Φ = (s_{1}, …, s_{p+m}, t_{1}, …, t_{p}) ∈ (R^{+})^{p+m} × R^{p},

in the corresponding space of F-N coordinates.

In this section, we give a canonical procedure for writing down a set of generators for the corresponding Fuchsian group, where these are described as hyperbolic isometries; we find explicit matrices for these generators in the next section.

We write the generators in the following order. The first 2p−q +1 generators are hyperbolic; their axes project, in order, to the p coordinate geodesics, followed by the p − q + 1 handle closers. The axes of the remaining n + m generators project, in order, to the boundary elements of S_{0} . We note that the total number of generators, d = 2q + 1 = 4g − 3 + 2n + 2m, is in general far from minimal.

6.1. Normalization. Our normalization is somewhat unusual in that we require more than can normally be achieved by conjugation by an orientationpreserving isometry. We achieve this by including the possibility of replacing a_{1} by a_{1}^{−1} and/or replacing a_{2} by a_{2}^{−1} .

We will use the following standard normalization throughout. Let a_{1}, …, a_{d} be a set of generators for a Fuchsian group. For our purposes, we can assume that a_{1} and a_{2} are both hyperbolic with disjoint axes. We require that A_{1} lie on the imaginary axis, pointing towards ∞; A_{2} lies in the right half-plane, with the attracting fixed point smaller than the repelling one; and the common orthogonal between A_{1} and A_{2} lies on the unit circle.

6.2. Cutting and extending holes. Let S = H^{2}/G be an orbifold with a hole. Let H be an open half-plane in H^{2} lying over the hole, and let K̂ be the complement of the union of the translates of H. Then S′ = K̂/G is the orbifold obtained from S by cutting off the infinite end of the hole.

There is an obvious process that reverses the above; we say that S is obtained from S′ by completing the hole.

6.3. Basic building blocks. The cases where q = 1 have already been dealt with; we assume q > 1. Each P_{i} has three distinct boundary elements, which are labeled as b_{i,1}, b_{i,2}, b_{i,3} . Except for the cases where we have not yet distinguished between b_{i,2} and b_{i,3} , the order of these boundary elements is determined by the order of the coordinate geodesics and boundary elements of the F-N system. Further, the size of each hyperbolic boundary element is specified by Φ.

Let P_{i} be one of the pairs of pants of our F-N system, and let P′_{i} be P_{i} with its incomplete holes completed—these are the holes corresponding to the coordinate geodesics. For each i = 1, …, q, there is a unique fully normalized pants group H_{i} representing P′_{i}. That is, H^{2}/H_{i} = P′_{i}; H_{i} has three distinguished generators a_{i,1}, a_{i,2}, a_{i,3} , where a_{i,1}a_{i,2}a_{i,3} = 1, and A_{i,k} projects onto b_{i,k}, k = 1, 2, 3. Even in the cases where we have not yet distinguished between b_{i,2} and b_{i,3} , the fully normalized pants group H_{i}, with its three distinguished generators, is uniquely determined.

6.4. Base points. We will need a canonical base point on each A_{i,k}. For i = 1, …, q, the point i is the canonical base point on A_{i,1} ; for i = 1, …, q, and for k = 2, 3, the base point on A_{i,k} is the point of intersection of A_{i,k} with the common orthogonal between A_{i,1} and A_{i,k}.

6.5. The primary chain. For j = 1, …, q − 1, we define the suborbifold Q_{j} as the interior of the closure of the union of the P_{i}, i ≤ j. In general, Q_{j} is incomplete, let Q′_{j} be the orbifold obtained by completing the incomplete holes of Q_{j}.

Each Q′_{j} has a naturally defined F-N system, where the coordinate geodesics are L_{1}, …, L_{j−1} , and the pairs of pants are P_{1}, …, P_{j}. The order of the boundary elements of Q′_{j} will be described below.

Let H_{1} = J_{1} be the fully normalized pants group representing P_{1} ; then the imaginary axis, which is the axis of a_{1,1} , projects to L_{1} ; the positive direction of A_{1,1} projects to the positive direction of L_{1} .

Let H_{2} be the fully normalized pants group representing P_{2} . Let c_{2} be the hyperbolic isometry which maps the right half-plane onto the left half-plane, while introducing a twist of 2τ_{1} in the positive direction on L_{1} ; that is, c_{2}(0) = ∞, c_{2}(∞) = 0, and c_{2}(i) = e^{2τ_{1}}i.

Let Ĥ_{2} = c_{2}H_{2}c_{2}^{−1} . Then the action half-plane of a_{1,1} = â_{1,1} is equal to the boundary half-plane of â_{2,1} = c_{2}a_{2,1}c_{2}^{−1} . It follows that one can use the AFP combination theorem (First combination theorem in [12]) to amalgamate Ĥ_{2} to H_{1} . Set J_{2} = ⟨H_{1},Ĥ_{2}⟩. The subgroups Ĥ_{1} = H_{1} and Ĥ_{2} are the distinguished subgroups of J_{2} . The following now follow from the AFP combination theorem.

(i) J_{2} is Fuchsian. (ii) J_{2} is generated by a_{1,1}, a_{1,2}, a_{1,3}, â_{2,2}, â_{2,3} ; in addition to the defining rela-

tions of H_{1} and H_{2} , these satisfy the one additional relation: a_{1,1} = â_{2,2}â_{2,3} . (iii) H^{2}/J_{2} has signature (0, 4); the corresponding boundary subgroups are gen-

erated by the above four generators.

It is clear that one can canonically identify H^{2}/J_{2} with Q′_{2} . This imposes a new order on the boundary elements of H^{2}/J_{2} , as follows. If b and b′ are boundary elements of Q_{2} , where b precedes b′ as coordinate geodesics or as boundary elements of S_{0} , then b precedes b′ as a boundary element of Q_{2} . If b and b′ both correspond to the same coordinate geodesic on S_{0} , and b corresponds to a boundary geodesic on P_{1} , while b′ corresponds to a boundary geodesic on P_{2} , then, as boundary elements of Q_{2} , b precedes b′. Finally, if b and b′ both correspond to boundary elements of either P_{1} or P_{2} , then b precedes b′ if b corresponds to A_{i,2} and b′ corresponds to A_{i,3} .

In the case that P_{i} has two boundary elements corresponding to the same handle geodesic, L, we now direct L so that the positive direction of A_{i,2} projects onto the positive direction of L. We note that we have now given an order to the boundary elements of Q_{2} , and directed them.

We introduce a new ordered set of generators for J_{2} as a_{1}^{2}, …, a_{5}^{2} , where a_{1}^{2} = a_{1,1} = â_{2,1}^{−1} , and a_{2}^{2}, …, a_{5}^{2} , are the generators a_{1,2}, a_{1,3}, â_{2,2} , â_{2,3} , where these have been rearranged so as to be in proper order; i.e., A_{j}^{2} projects onto b_{j−1}^{2} . We remark that those A_{j}^{2} that are hyperbolic are all directed so that the attracting fixed point of a_{j}^{2} is smaller than the repelling fixed point.

Each A_{j}^{2} has a canonical base point on it. In the case that a_{j}^{2} = a_{1,i}, the base point is the canonical base point for a_{1,i}; in the case that a_{j}^{2} = c_{2}a_{2,i}c_{2}^{−1} , then the canonical base point is the c_{2} image of the canonical base point for a_{2,i}.

We now iterate the above process. Assume that we have found J_{k}, with distinguished subgroups, Ĥ_{1}, …,Ĥ_{k}, representing Q′_{k}, where k < q. Assume that we have ordered the boundary elements of Q_{k}, and that we have found the distinguished generators, a_{1}^{k}, …, a_{2k+1}^{k}, for J_{k}, where each distinguished generator lies in some distinguished subgroup, so that, for i = 1, …, k, A_{i}^{k} projects onto the coordinate geodesic L_{i}, and, for i = k + 1, …, 2k + 1, A_{i}^{k} projects onto the boundary element b_{k−i}^{k} of Q_{k}. We assume that we have assigned a canonical base point on each of the geodesics A_{i}^{k}, i = k+1, …, 2k+1, and we also assume that all the hyperbolic boundary generators of J_{k} are directed so that the attracting fixed point is smaller than the repelling fixed point. Then there is some j so that A_{k+j}^{k} projects onto L_{k+1} . We need to renormalize H_{k+1} so that the renormalized A_{k+1,1} agrees with A_{k+j}^{k}, but with the opposite orientation, and with an appropriate twist.

We define the conjugator c_{k+1} to be the unique orientation-preserving hyperbolic isometry mapping the left half-plane onto the action half-plane of A_{k+j}^{k}, while mapping the base point i onto the point whose distance from the base point on A_{k+j}^{k}, measured in the positive direction along A_{k+j}^{k}, is exactly 2τ_{k}. As above, the AFP combination theorem assures us that J_{k+1} = ⟨J_{k}, c_{k+1}H_{k+1}c_{k+1}^{−1}⟩ satisfies the following.

(i) J_{k+1} is Fuchsian. (ii) J_{k+1} = ⟨a_{1}^{k} …, a_{2k+1}^{k}, c_{k+1}a_{k+1,1}c_{k+1}^{−1}, c_{k+1}a_{k+1,2}c_{k+1}^{−1}, c_{k+1}a_{k+1,3}c_{k+1}^{−1}⟩; these

satisfy the defining relation(s) of J_{k}, the defining relation(s) of H_{k+1} , together with the additional defining relation: a_{k+j}^{k} = (c_{k+1}a_{k+1,1}c_{k+1}^{−1})^{−1} .

(iii) H^{2}/J_{k+1} has signature (0, 2k + 1).

The distinguished subgroups of J_{k+1} are the distinguished subgroups of J_{k}, together with Ĥ_{k+1} = c_{k+1}H_{k+1}c_{k+1}^{−1} .

As above, we rewrite the generators of J_{k+1} as a_{1}^{k+1}, …, a_{2k+3}^{k+1} , where the first k generators correspond in order to the k coordinate geodesics of Q_{k+1} , and the remaining generators correspond to the boundary elements of Q_{k+1} in the following order. Each boundary generator a_{i} of J_{k+1} corresponds to either a coordinate geodesics L_{j} on S_{0} , or it corresponds to a boundary element b_{j} of S_{0} ; pulling back the order of the coordinate geodesics and boundary elements from S_{0} imposes a partial order on the boundary generators of J_{k+1} . The only ambiguities occur when the generators a_{i} and a_{i′} both correspond to the same coordinate geodesic. If a_{i}, respectively, a_{i′}, lies in the distinguished subgroup Ĥ_{j}, respectively, Ĥ_{j′}, where j < j′, then a_{i} precedes a_{i′}; if a_{i} and a_{i′} both lie in the same distinguished subgroup, Ĥ_{j}, then â_{j,2} precedes â_{j,3} .

The hyperbolic boundary generators of J_{k} all have distinguished base points; we assign the c_{k} image of the distinguished base point on a_{k+1,2} and a_{k+1,3} as the distinguished base point on the new boundary generators of J_{k+1} . We also observe that the hyperbolic boundary generators of J_{k+1} are all directed so that their attracting fixed points are smaller than their repelling fixed points.

When k = q − 1, we reach the group J_{q}, representing Q′_{q}. Note that, as part of the above process, we have ordered and directed those boundary geodesics of Q_{q} that correspond to handle geodesics on S_{0} , where the same pair of pants appears on both sides of the handle geodesic.

6.6. Closing the handles. We rename the group J_{q}, and now call it K_{0} . We also rename its ordered set of generators and call them, in order, a_{1}^{0}, …, a_{2q+1}^{0} . The first q−1 of these generators correspond to the attaching coordinate geodesics of S_{0} ; the next 2(p−q +1) generators correspond to the boundary elements of Q_{q} that are handle geodesics on S_{0} ; the remaining generators correspond to boundary elements of both Q_{q} and S_{0} .

We define the first handle-closer d_{1} as the orientation-preserving hyperbolic isometry mapping the action half-plane of a_{q}^{0} onto the boundary half-plane of a_{q+1}^{0} , while mapping the point at distance −2τ_{q} from the base point on A_{q}^{0} to the base point on A_{q+1}^{0} . Note that d_{1} conjugates a_{q}^{0} onto (a_{q+1}^{0})^{−1} .

Set K_{1} = ⟨K_{0}, d_{1}⟩. The following follow from the second combination theorem (HNN extension):

(i) K_{1} is Fuchsian. (ii) K_{1} is generated by a_{1}^{0} …, a_{2q+1}^{0}, d_{1} ; these satisfy the defining relations of

K_{1} , together with the additional relation: a_{q+1}^{0} = d_{1}(a_{q}^{0})^{−1}d_{1}^{−1} . (iii) H^{2}/K_{1} is an orbifold of signature (1, q).

The distinguished subgroups of K_{1} are the distinguished subgroups of K_{0} . The generators of K_{1} are the generators of K_{0} , in the same order, but with a_{q+1}^{0} deleted, and d_{1} added to the list. In the list of generators for K_{1} , d_{1} appears after all the generators corresponding to coordinate geodesics on S_{0} , and before the first generator corresponding to a boundary element of S_{0} .

There is no difficulty (other than notation) in iterating the above process. In the next iteration, we eliminate the generator a_{q+3}^{0} , and the new generator d_{2} appears immediately after d_{1} .

After g iterations, we reach the discrete group G = K_{g}, where H^{2}/G = S_{0} . Further, G has q distinguished subgroups, representing in order the q pairs of pants, P_{1}, …, P_{q}; and G has 2q + 1 distinguished generators, a_{1}, …, a_{2q+1} . For i = 1, …, p, the axis A_{i} projects onto the coordinate geodesic L_{i}; the generators a_{p+1}, …, a_{2p−q+1} are handle closers; and the axes of the remaining generators project, in order, onto the boundary elements of S_{0} .

6.7. Summary. We started with an F-N system on S_{0} , together with the point Φ in the corresponding coordinate space, and constructed from these a set of generators for the Fuchsian group representing the deformation of S_{0} determined by Φ. It is easy to observe that if Φ, respectively, Φ′, are points in this F-N coordinate space, and a_{1}, …, a_{d}, respectively, a′_{1}, …, a′_{d}, are the corresponding sets of generators, then there is a quasiconformal deformation of the hyperbolic plane conjugating each a_{i} onto the corresponding a′_{i}.

We note that, among other things, we have shown the following.

Proposition 6.1. Let L_{1}, …, L_{p} be an F-N system on the hyperbolic orbifold S_{0} . Then there is a canonical procedure for choosing a basis for the (orbifold) fundamental group of S_{0} , so that, for i = 1, …, p, L_{i} is the shortest geodesic in the free homotopy class of the i-th generator.

Remark 6.1. The above definition of the conjugators has the unfortunate consequence that the untwisted handle-closers do not in general preserve the common orthogonal between the axes of the two generators they conjugate. There is a relatively easy way to solve this problem, but this entails the loss of the explicit formulae for the entries in the matrices; see Section 14.

7. Explicit matrices

7.1. Reduction to primitive conjugators. Let H_{i} and H_{j+1} be the fully normalized pants groups representing the pairs of pants, P_{i} and P_{j+1} , respectively. We assume that i < j+1, and that there is a k, 1 ≤ k ≤ 3, so that a_{j+1,1} and a_{i,k} represent the same attaching geodesic, L_{j} on S_{0} , but with reverse orientations. Then | tr(ã_{j+1,1})| = | tr(ã_{i,k})|. The elementary conjugator e_{i,k} maps the left half-plane (the boundary half-plane of a′ = a_{j+1,1} ) onto the action half-plane of a = a_{i,k}, while introducing a twist of 2τ_{j}; that is, e_{i,k} maps the base point on A_{j+1,1} to the point on A_{i,k} at distance 2τ_{j} from the base point on A_{i,k}; since i < j + 1, this is the positive direction on the projection of these axes. The untwisted elementary conjugator, e^{0} = e_{i,k}^{0} , which is independent of the index j, also maps the left half-plane onto the action half-plane of a, but maps the base point on A_{j+1,1} (this is the point i) to the base point on A_{i,k}.

Once we have found matrices for the elementary conjugators, then we can inductively find matrices for all the conjugators. If P_{i} and P_{j+1} are adjacent pairs of pants in the F-N system on S_{0} , with i < j + 1, and, as above, b_{j+1,1} attached to b_{i,k}, then, once we have found the matrix for the conjugator c_{i}, and we have found the matrix for the elementary conjugator, e_{i,k}, the matrix for the conjugator c_{j+1} is given by

(9)  c̃_{j+1} = c̃_{i}ẽ_{i,k}.

For each j = q, …, p, the handle closer d_{j} conjugates one distinguished boundary generator of K_{0} onto the inverse of another distinguished boundary generator. These two boundary generators either lie in the same distinguished subgroup, or they lie in different distinguished subgroups.

If the two boundary generators lie in the same distinguished subgroup, Ĥ_{i}, then there is a conjugator c_{i}, and there is a fully normalized pants group H_{i}, so that Ĥ_{i} = c_{i}H_{i}c_{i}^{−1} . In this case, we can choose the matrix for the handle closer as

(10)  d̃_{j} = c̃_{i}d̃c̃_{i}^{−1},

where d̃ is given in equation (4).

If these two boundary generators lie in distinct distinguished subgroups, say Ĥ_{i} and Ĥ_{i′}, where i < i′, where the first boundary generator corresponds to â_{i,k}, and the second corresponds to â_{i′,k′} then we write d_{j} as a product of four transformations: first we twist along the axis Â_{i,k}; then we map the boundary half-plane of a_{i,k} onto the right half-plane; then we interchange left and right half-planes; and last, we map the right half-plane onto the boundary half-plane of â_{i′,k′}.

The first transformation preserves both sides of Â_{i,k} and maps the point on Â_{i,k}, whose distance from the base point is −2τ_{j}, to the base point on Â_{i,k}. We can write the matrix for this transformation as

(11)  (c̃_{i}ẽ_{i,k}^{0})f̃_{τ_{j}}(c̃_{i}ẽ_{i,k}^{0})^{−1},

where f_{τ} is the universal twist map, f_{τ} (z) = e^{−2τ} z.

The second transformation maps the boundary half-plane of â_{i,k} onto the right half-plane, and maps the base point on Â_{i,k} to the base point on the imaginary axis. We can write the matrix for this transformation as

(12)  (c̃_{i}ẽ_{i,k}^{0})^{−1}.

The interchange transformation g interchanges left and right half-planes and preserves the base point on the imaginary axis; that is, g(z) = −1/z.

The final transformation maps the left half-plane onto the boundary half-plane of â_{i′,k′}; the matrix for this transformation can be written as

(13)  c̃_{i′}ẽ_{i′,k′}^{0}.

Combining equations (11)–(13), we obtain

(14)  d̃_{j} = c̃_{i′}ẽ_{i′,k′}^{0} g̃f̃_{τ_{j}}(c̃_{i}ẽ_{i,k}^{0})^{−1}.

Hence, in this case as well, we can write the matrix for the handle closer, once we know the sizes of the attaching geodesics, and we have the matrices for the twisted and untwisted elementary conjugators.

7.2. The universal twist map and interchange. We represent the universal twist map, f_{τ}, which twists by −2τ in the positive direction along the imaginary axis, and the interchange transformation g, as follows:

f̃_{τ} = ( \begin{matrix}e^{−τ} & 0 \\ 0 & e^{τ} \end{matrix} );  g̃ = ( \begin{matrix}0 & −1 \\ 1 & 0 \end{matrix} ) .

7.3. The elementary conjugator for a_{i,1} . We first take up the case that the attaching coordinate geodesic L_{j} corresponds to both a′ = a_{j+1,1} and a = a_{i,1} . This case occurs only for i = j = 1. We note that the left half-plane is the boundary half-plane for both a′ and a.

The interchange transformation g interchanges the right and left half-planes in H^{2} , and fixes the point i, which is the base point on both A and A′. Hence, we can choose g̃ as the matrix for the untwisted elementary conjugator ẽ_{2,1}^{0} .

Then the matrix for e_{2,1} , is given by

ẽ_{2,1} = ẽ_{2,1}^{0}f̃_{τ_{1}} = ( \begin{matrix}0 & −e^{τ_{1}} \\ e^{−τ_{1}} & 0 \end{matrix} ) .

7.4. The elementary conjugator for a_{i,2} . We next take up the case that k = 2. Since the base point of A = A_{i,2} is the point of intersection of A with the common orthogonal of A and A_{i,1} , we can choose e_{i,2}^{0} to be the composition of the reflection r_{0} in A_{i,1} , followed by the reflection r_{12} in the line halfway between A_{i,1} and A_{i,2} .

We have already found the matrix for r_{0} as

r̃_{0} = ( \begin{matrix}1 & 0 \\ 0 & −1 \end{matrix} ) .

Since we can locate the fixed points of a at e^{±μ_{i}}, easy computations show that we can choose

r̃_{12} = \frac{1}{\sqrt{2 sinh μ_{i}}} ( \begin{matrix}exp(\frac{1}{2}μ_{i}) & − exp(−\frac{1}{2}μ_{i}) \\ exp(−\frac{1}{2}μ_{i}) & − exp(\frac{1}{2}μ_{i}) \end{matrix} ) ,

where μ_{i} is defined by equation (1) or (5) or (7), depending on the type of a_{i,3} . We can also write the above as

(15)  r̃_{12} = \frac{1}{\sqrt{2}} ( \begin{matrix}\sqrt{coth μ_{i} + 1} & −\sqrt{coth μ_{i} − 1} \\ \sqrt{coth μ_{i} − 1} & −\sqrt{coth_{i} μ + 1} \end{matrix} ) .

Remark 7.1. Up to this point, the entries in all our matrices were rational functions of hyperbolic sines and cosines of lengths of closed geodesics, or of geodesic arcs. We see from equation (15) that, for Fuchsian groups representing sufficiently complicated surfaces or orbifolds, one must introduce square roots of these hyperbolic sines and cosines.

We can now write

(16)  ẽ_{i,2}^{0} = r̃_{12}r̃_{0} = \frac{1}{\sqrt{2 sinh μ_{i}}} ( \begin{matrix}exp(\frac{1}{2}μ_{i}) & exp(−\frac{1}{2}μ_{i}) \\ exp(−\frac{1}{2}μ_{i}) & exp(\frac{1}{2}μ_{i}) \end{matrix} ) ,

and

(17) ẽ_{i,2} = ẽ_{i,2}^{0}f̃_{τ_{j}} = \frac{1}{\sqrt{2 sinh μ_{i}}} ( \begin{matrix}exp(\frac{1}{2}μ_{i} − τ_{j}) & exp(−\frac{1}{2}μ_{i} + τ_{j}) \\ exp(−\frac{1}{2}μ_{i} − τ_{j}) & exp(\frac{1}{2}μ_{i} + τ_{j}) \end{matrix} ) .

7.5. The primitive conjugator for a_{3} . For k = 3, the base point on A_{i,3} is at the point where A_{1,3} meets the common orthogonal with A_{i,1} . However, the base point on A_{i,1} is the point where it meets the common orthogonal with A_{i,2} . We write e_{i,3}^{0} = r_{13}r_{0}f_{−λ_{i}}, where r_{0} is as above, r_{13} is the reflection in the line halfway between A_{i,1} , and A_{i,3} , and λ_{i} is the size of a_{i,1} ; that is, | tr(a_{i,1})| = 2 cosh λ_{i}.

We have already observed that the common orthogonal between A_{i,1} and A_{i,3} lies on the circle of radius e^{λ_{i}}. Also, the fixed points of a = a_{i,3} are at e^{λ_{i}±ν_{i}}, where ν_{i} is defined by equation (2). Hence, we can choose

r̃_{13} = \frac{1}{\sqrt{2 sinh ν_{i}}} ( \begin{matrix}exp(\frac{1}{2}(ν_{i} + λ_{i})) & exp(\frac{1}{2}(−ν_{i} + λ_{i})) \\ exp(\frac{1}{2}(−ν_{i} − λ_{i})) & exp(\frac{1}{2}(ν_{i} − λ_{i})) \end{matrix} ) .

We now write

(18)  ẽ_{i,3}^{0} = r̃_{13}r̃_{0}f̃_{−λ_{i}} = \frac{1}{\sqrt{2 sinh ν_{i}}} ( \begin{matrix}exp(\frac{1}{2}ν_{i} + λ_{i}) & exp(−\frac{1}{2}ν_{i}) \\ exp(−\frac{1}{2}ν_{i}) & exp(\frac{1}{2}ν_{i} − λ_{i}) \end{matrix} ) ,

and

(19) ẽ_{i,3} = ẽ_{i,3}^{0}f̃_{τ_{j}} = \frac{1}{\sqrt{2 sinh ν_{i}}} ( \begin{matrix}exp(\frac{1}{2}ν_{i} + λ_{i} − τ_{j}) & exp(−\frac{1}{2}ν_{i} + τ_{j}) \\ exp(−\frac{1}{2}ν_{i} − τ_{j}) & exp(\frac{1}{2}ν_{i} − λ_{i} + τ_{j}) \end{matrix} ) .

8. The algorithm

Assume we are given an explicit F-N system on a hyperbolic orbifold; this geometric information is given as the signature (g, n, m; α_{1}, …, α_{n}) of S_{0} , and the pairing table, which has q rows, one for each pair of pants P_{i}, and three columns, one for each boundary element of P_{i}. The entry in the i-th row and k -th column identifies the boundary element b_{i,k} as corresponding to either a coordinate geodesic L_{j}, or a boundary element b_{j} of S_{0} .

We also assume we are given the point Φ in the appropriate coordinate space. Step 1. For each i = 1, …, q, and for k = 1, 2, 3, we read off from the pairing

table whether b_{i,k} corresponds to a coordinate geodesic or to a boundary element. If b_{i,k} corresponds to L_{j}, then we read off the size of a_{i,k} from the j -th entry in Φ. Each b_{i,1} corresponds to an attaching geodesic; we write the size of this geodesic as λ_{i}. If b_{i,k} corresponds to a boundary element, then we read off the type of this boundary element from the signature of S_{0} ; if the type is hyperbolic, then we read off the size of a_{i,k} from Φ; if the type is parabolic or elliptic, we read off the order from the signature.

We use the constructions of Section 5 to write down the matrices ã_{i,k}. We remark that, in practice, we will not need all of these.

Step 2. The first conjugator c_{1} is the identity. We have already constructed the matrix for the second conjugator; it is the elementary conjugator ẽ_{2,1} .

Continuing inductively, assume that we have found matrices for the conjugators, c_{1}, …, c_{j−1} , j ≤ q. The attaching geodesic L_{j} appears in the pairing table once as b_{j+1,1} , and once as some b_{i,k}, i < j + 1, k > 1. We have already constructed c̃_{i}; equation (9) then gives the formula for c̃_{j}.

Step 3. Write the matrices for the first q − 1 generators. These are the generators corresponding to the attaching geodesics. We write:

ã_{1} = ã_{1,1}, ã_{2} = c̃_{3}ã_{3,1}^{−1}c̃_{3}^{−1} , …, ã_{q−1} = c̃_{q}ã_{q,1}^{−1}c̃_{q}^{−1}.

Step 4. Find matrices for the generators corresponding to the handle geodesics; these are the generators a_{q}, …, a_{p}.

Each L_{j}, q ≤ j ≤ p, appears twice in the pairing table; either there is some i so that L_{j} appears as both b_{i,2} and b_{i,3} , or there are two distinct rows, i < i′, so that L_{j} appears as b_{i,k} and as b_{i′,k′}.

In the first case, we write the matrix for the handle generator as ã_{j} = c̃_{i}ã_{i,2}c̃_{i}^{−1} .

In the second case, we write the matrix for the handle generator as ã_{j} = c̃_{i}ã_{i,k}c̃_{i}^{−1} .

Step 5. Find matrices for the handle closing generators, a_{p+1}, …, a_{2p−q+1} . We need to consider separately the same two possibilities as in the previous step.

If there is some j, q ≤ j ≤ p, so that L_{j} appears as both b_{i,2} and b_{i,3} in the pairing table, then we write ã_{p+j} = d̃_{j}, where the formula for d̃_{j} is given by equation (10).

For q ≤ j ≤ p, if the two entries of L_{j} in the pairing table appear as b_{i,k} and b_{i′,k′}, where i < i′, then we write ã_{p+j} = d̃_{j}, where the formula for d̃_{j} is given by equation (14). Note that, in order to use this equation, we need the quantities λ_{i} and λ_{i′}, defined in Step 1, and matrices for the untwisted elementary conjugators, e_{i,k}^{0} and e_{i′,k′}^{0} ; these matrices are obtained by appropriate use of equations (16) and (18).

Step 6. Write down the matrices corresponding to the boundary elements of S_{0} . Each b_{j}, j = 1, …, m + n, appears exactly once in the pairing table. If b_{j} corresponds to b_{i,k}, then the matrix for the corresponding generator is given by ã_{2p+q+j} = c̃_{i}ã_{i,k}c̃_{i}^{−1} .

9. From matrices to F-N coordinates

In this section, we start with a finite set of matrices, ã_{1}, …, ã_{d} ∈ SL(2, R), which we regard as Möbius transformations. We assume that these are a fully normalized set of distinguished generators defined by a coordinate point Φ for some F-N system on some orbifold S. We give a procedure for determining the signature of S; the pairing table of the F-N system; and the coordinate point Φ that these generators represent.

Remark 9.1. We could start with an arbitrary finite set of matrices, ã_{1}, …, ã_{d}, and write down necessary and sufficient conditions for the corresponding Möbius transformations to be a (not necessarily normalized) distinguished set of generators corresponding to some coordinate point in some F-N system. The conditions are easy to derive from the rules in 2.1.1, together with the rules for directing the coordinate and boundary geodesics. This would yield a set of sufficient conditions for the corresponding transformations to generate a discrete group.

9.1. Recovering the signature and the pairing table. We know that the transformations a_{1}, …, a_{d}, are in non-increasing order, so the last n of them are not hyperbolic. Also, d = 2q + 1, so q is also determined. We write S = H^{2}/G, where G = ⟨a_{1}, …, a_{d}⟩.

We find the endpoints of the axes of the hyperbolic generators, and compute which pairs of these axes cross each other, and which pairs are disjoint. If the axes of the hyperbolic generators are not all disjoint, then there is a largest index p so that A_{1}, …, A_{p} are all disjoint. In this case, since g = p − q + 1, g is determined. We now have that the projections of A_{1}, …, A_{q−1} are the attaching geodesics, and the projections of A_{q}, …, A_{p} are the handle geodesics. Then a_{p+1}, …, a_{2p−q+1} are the handle closers, and the remaining generators correspond to boundary elements of S. We can use the traces of these last m + n generators to find m, and to find the orders of the elliptic and parabolic generators. Hence, in this case, we know the signature (g, m, n; α_{1}, …, α_{n}).

If the axes of the hyperbolic generators are all disjoint, then g = 0, p = n + m − 3, and q = n + m − 2. Hence p = q − 1 and m = q − n + 2. So in this case as well, we know the signature of G.

We also need the (unordered) set of basic generators. These are the generators other than the handle-closers, together with the handle generators conjugated by the corresponding handle-closing generators; that is, the basic generators are:

a_{1}, …, a_{p}, a_{2p−q}, …, a_{d}, a_{p+1}a_{q}a_{p+1}^{−1}, …, a_{2p−q+1}a_{2p−q}a_{2p−q+1}^{−1}.

The axes of the hyperbolic basic generators divide the hyperbolic plane into regions. Each of these regions either contains three axes of distinguished generators in its closure, in which case the three corresponding basic generators generate one of the distinguished pants subgroups, or the region contains exactly one axis of a distinguished generator in its closure, in which case the corresponding generator is boundary hyperbolic. For j = 1, …, q, the attaching geodesic L_{j} appears in the pairing table as both b_{j+1,1} and as some b_{i,k}, i < j + 1. Since we know that a_{j} = â_{j+1,1} , j = 1, …, q, we can order the above regions as corresponding to P_{1}, …, P_{q}. If the other two axes on the boundary of the region corresponding to P_{j} are translates of A_{i} and A_{i′}, where i < i′, then L_{i}, respectively, L_{i′}, is the entry in the second, respectively, third, column of the j -th row of the pairing table. If i = i′, then L_{i} is the entry in both the second and third column of this row. This completes the reconstruction of the pairing table.

9.2. Recovering the F-N coordinates. The F-N coordinates

Φ = (s_{1}, …, s_{p+m}, t_{1}, …, t_{p})

can now be read off as follows.

The numbers s_{1}, …, s_{p} are almost immediate. For i = 1, …, p, we have s_{i} = sinh σ_{i}, where cosh σ_{i} = 2| tr(ã_{i})|. Similarly, for i = p + 1, …, p + m, s_{i} = 2 sinh σ_{i}, where cosh σ_{i} = 2| tr(ã_{2p−q+1+i})|.

Each of the twists about an attaching geodesic is of the following form. We have three basic generators, a, a′ and a′′, with disjoint axes, where A separates A′ from A′′. The twist 2τ is the distance between the point of intersection on A of the common orthogonal, N′, of A with A′, and the point of intersection on A of the common orthogonal, N′′, of A with A′′. We can now compute 2|τ|, for N′ is the axis of the Fenchel–Jørgensen commutator, r′ = aa′ − a′a, and N′′ is the axis of r′′ = aa′′ − a′′a. These are both half-turns; their product, when represented by a matrix in SL(2, R), has trace equal to ±2 cosh τ. The sign of τ is also determined; although we do not have an explicit formula for it, for τ > 0 if and only if a and r′′r′ have the same attracting fixed point.

Once we know the twists about the attaching geodesics, we can reconstruct the conjugators, for each conjugator is a product of primitive conjugators, and each primitive conjugator is determined by the pairing table, the lengths of the coordinate geodesics, and the twists about the attaching geodesics.

Once we have the conjugators and the pairing table, we can reconstruct the elementary conjugators. It is then an exercise to reconstruct the untwisted elementary conjugators from the twisted ones.

For the twist about a handle geodesic, we note (see equations (10) and (14)) that the handle-closer is of the form cf_{τ} (c′)^{−1} , where c and c′ are products of primitive conjugators. Since we are given matrices for the handle closing generators, and we can reconstruct these products of primitive conjugators, we can compute f_{τ}; hence sinh τ is determined.

10. Recapitulation—statements of results

In this section, we combine the results of the preceding sections, and formally state the theorems we have proven.

10.1. The algorithm. Given a hyperbolic orbifold S_{0} , with finitely generated fundamental group, given L_{1}, …, L_{p}, an F-N system on S_{0} , and given a point Φ = s_{1}, …, s_{p+m}, t_{1}, …, t_{p} ∈ (R^{+})^{p+m} × R^{m}, we have given in Section 8 an algorithm yielding explicit formulae for a set of matrices ã_{1}, …, ã_{d}, so that G = ⟨a_{1}, …, a_{d}⟩ is the Fuchsian group uniformizing the deformation of S_{0} determined by the coordinate point Φ. As part of our procedures, we have written the entries in these matrices as smooth algebraic functions of Φ.

10.2. The Fenchel–Nielsen theorem. Combining our algorithm with the results in Section 9, we have a new proof of a somewhat stronger version of the original Fenchel–Nielsen theorem.

Let G_{0} be a finitely generated Fuchsian group. Let 𝒟ℱ (G_{0}) be the identity component of the space of discrete faithful representions of G_{0} into PSL(2, R), modulo conjugation. It is well known, assuming that G_{0} is of cofinite volume, that 𝒟ℱ (G_{0}) is real-analytically equivalent to the Teichmüller space of H^{2}/G_{0} (see Abikoff [1]); if G_{0} is not of cofinite volume, then 𝒟ℱ (G_{0}) is real-analytically equivalent to the corresponding reduced Teichmüller space (see Earle [4]).

Theorem 10.1. Let G_{0} be a finitely generated non-elementary Fuchsian group, with an F-N system defined on S_{0} = H^{2}/G_{0} . Then the algorithm for constructing matrix generators for deformations of G_{0} defines a canonical algebraic diffeomorphism of 𝒟ℱ (G_{0}) onto (R^{+})^{m+p} × R^{p}.

10.3. Algebraic orbifolds. We easily obtain the following corollaries.

Theorem 10.2. Let S be a hyperbolic orbifold with finitely generated fundamental group, with an F-N system defined on it; S is algebraic if and only if its F-N coordinates in this system are algebraic.

Theorem 10.3. Let S be a hyperbolic orbifold with finitely generated fundamental group. If the F-N coordinates of S are algebraic in one F-N system, then they are algebraic in any F-N system.

10.4. The action of the Teichmüller modular group. For our final observation, we use the well-known fact that if α is an element of the Teichmüller modular group, and if L_{1}, …, L_{p} is an F-N system on S_{0} = H^{2}/G_{0} , then α either preserves this F-N system, in which case it maps some F-N coordinate for S_{0} in this system onto another coordinate point for the same system, or α maps the given F-N system onto a different F-N system on S_{0} . We have proven the following.

Theorem 10.4. Let G_{0} be a finitely generated Fuchsian group, and let an F-N system on S_{0} = H^{2}/G_{0} be given. Let Mod denote the action of the (reduced) Teichmüller modular group on (R^{+})^{m+p} ×R^{p}, where this action is given using the canonical identification of Theorem 10.1. Then Mod acts as a group of algebraic diffeomorphisms.

11. Special case of genus 3

There are five topologically distinct F-N systems on a closed surface of genus 3, and there are many possible ways to order the geodesics in each of them. We choose one particular case, in which there is one dividing geodesic, necessarily L_{1} , and five non-dividing geodesics. Then L_{2} and L_{3} are necessarily attaching geodesics, while L_{4} , L_{5} and L_{6} are handle geodesics.

The pairing table for our particular F-N system is given below.

\begin{matrix}P_{1} & L_{1} & L_{2} & L_{4} \\ P_{2} & L_{1} & L_{5} & L_{5} \\ P_{3} & L_{2} & L_{3} & L_{6} \\ P_{4} & L_{3} & L_{4} & L_{6} \end{matrix}

We also assume that we are given the point in parameter space

Φ = s_{1}, …, s_{6}, t_{1}, …, t_{6}.

Then the quantities σ_{i} and τ_{i} are given by s_{i} = sinh σ_{i}, σ_{i} > 0, and t_{i} = sinh τ_{i}. We need to write down 9 matrices, ã_{1}, …, ã_{9} , so that, for i = 1, …, 6, A_{i} projects onto L_{i}, and for i = 7, 8, 9, a_{i} is the handle closer for L_{i−3} .

Step 1. The generating matrices for the four fully normalized pants groups, H_{1}, …, H_{4} , are as follows.

ã_{1,1} = ( \begin{matrix}e^{σ_{1}} & 0 \\ 0 & e^{−σ_{1}} \end{matrix} );
ã_{1,2} = \frac{1}{sinh μ_{1}} ( \begin{matrix}sinh(μ_{1} − σ_{2}) & sinh σ_{2} \\ − sinh σ_{2} & sinh(μ_{1} + σ_{2}) \end{matrix} ) ,

where μ_{1} is defined by

coth μ_{1} = \frac{cosh σ_{1} cosh σ_{2} + cosh σ_{4}}{sinh σ_{1} sinh σ_{2}} ,  μ_{1} > 0;
ã_{1,3} = −ã_{1,2}^{−1}ã_{1,1}^{−1} = \frac{1}{sinh ν_{1}} ( \begin{matrix}sinh(ν_{1} − σ_{4}) & e^{σ_{1}} sinh σ_{4} \\ −e^{−σ_{1}} sinh σ_{4} & sinh(ν_{1} + σ_{4}) \end{matrix} ) ,

where ν_{1} is defined by

coth ν_{1} = \frac{cosh σ_{1} cosh σ_{4} + cosh σ_{2}}{sinh σ_{1} sinh σ_{4}} , ν_{1} > 0.

For H_{2} , since both b_{2,2} and b_{2,3} correspond to the same geodesic, L_{5} , we have ν_{2} = μ_{2} .

ã_{2,1} = ( \begin{matrix}e^{σ_{1}} & 0 \\ 0 & e^{−σ_{1}} \end{matrix} );
ã_{2,2} = \frac{1}{sinh μ_{2}} ( \begin{matrix}sinh(μ_{2} − σ_{5}) & sinh σ_{5} \\ − sinh σ_{5} & sinh(μ_{2} + σ_{5}) \end{matrix} );
ã_{2,3} = \frac{1}{sinh μ_{2}} ( \begin{matrix}sinh(μ_{2} − σ_{5}) & e^{σ_{1}} sinh σ_{5} \\ −e^{−σ_{1}} sinh σ_{5} & sinh(μ_{2} + σ_{5}) \end{matrix} ) ,

Matrices for Fenchel–Nielsen coordinates 297

where μ_{2} is defined by

coth μ_{2} = coth σ_{5} \frac{cosh σ_{1} + 1}{sinh σ_{1}}, μ_{2} > 0.

The matrices for the generators of H_{3} are as follows.

ã_{3,1} = ( \begin{matrix}e^{σ_{2}} & 0 \\ 0 & e^{−σ_{2}} \end{matrix} );
ã_{3,2} = \frac{1}{sinh μ_{3}} ( \begin{matrix}sinh(μ_{3} − σ_{3}) & sinh σ_{3} \\ − sinh σ_{3} & sinh(μ_{3} + σ_{3}) \end{matrix} ) ,

where μ_{3} is defined by

coth μ_{3} = \frac{cosh σ_{2} cosh σ_{3} + cosh σ_{6}}{sinh σ_{2} sinh σ_{3}} ,  μ_{3} > 0;
ã_{3,3} = −ã_{3,2}^{−1}ã_{3,1}^{−1} = \frac{1}{sinh ν_{3}} ( \begin{matrix}sinh(ν_{3} − σ_{6}) & e^{σ_{2}} sinh σ_{6} \\ −e^{−σ_{2}} sinh σ_{6} & sinh(ν_{3} + σ_{6}) \end{matrix} ) ,

where ν_{3} is defined by

coth ν_{3} = \frac{cosh σ_{2} cosh σ_{6} + cosh σ_{3}}{sinh σ_{2} sinh σ_{6}} , ν_{3} > 0.

Finally, the matrices for the generators of H_{4} are as follows.

ã_{4,1} = ( \begin{matrix}e^{σ_{3}} & 0 \\ 0 & e^{−σ_{3}} \end{matrix} );
ã_{4,2} = \frac{1}{sinh μ_{4}} ( \begin{matrix}sinh(μ_{4} − σ_{4}) & sinh σ_{4} \\ − sinh σ_{4} & sinh(μ_{4} + σ_{4}) \end{matrix} ) ,

where μ_{4} is defined by

coth μ_{4} = \frac{cosh σ_{3} cosh σ_{4} + cosh σ_{6}}{sinh σ_{3} sinh σ_{4}} ,  μ_{4} > 0;
ã_{4,3} = −ã_{4,2}^{−1}ã_{4,1}^{−1} = \frac{1}{sinh ν_{4}} ( \begin{matrix}sinh(ν_{4} − σ_{6}) & e^{σ_{3}} sinh σ_{6} \\ −e^{−σ_{3}} sinh σ_{6} & sinh(ν_{4} + σ_{6}) \end{matrix} ) ,

where ν_{4} is defined by coth ν_{4} = \frac{cosh σ_{3} cosh σ_{6} + cosh σ_{4}}{sinh σ_{3} sinh σ_{6}} , ν_{4} > 0.

Step 2. The second conjugator, c_{2} , is the primitive conjugator e_{1,1} ; we obtain

c̃_{2} = ẽ_{1,1} = ( \begin{matrix}0 & −e^{τ_{1}} \\ e^{−τ_{1}} & 0 \end{matrix} ) .

The third conjugator c_{3} is the primitive conjugator e_{1,2} ; we obtain

c̃_{3} = ã_{1,2} = r̃_{12}r̃_{0}f̃_{τ_{2}} = \frac{1}{\sqrt{2 sinh μ_{1}}} ( \begin{matrix}exp(\frac{1}{2}μ_{1} − τ_{2}) & exp(−\frac{1}{2}μ_{1} + τ_{2}) \\ exp(−\frac{1}{2}μ_{1} − τ_{2}) & exp(\frac{1}{2}μ_{1} + τ_{2}) \end{matrix} ) .

The fourth conjugator, c_{4} , which conjugates H_{4} onto Ĥ_{4} , is a product of the primitive conjugator c_{3} = e_{1,2} , and the primitive conjugator e_{3,3} , which maps the left half-plane onto the action half-plane of a_{3,3} . We first write the matrix for e_{3,3} :

ẽ_{3,3} = r̃_{13}r̃_{0}f̃_{τ_{3}} = \frac{1}{\sqrt{2 sinh ν_{3}}} ( \begin{matrix}exp(\frac{1}{2}ν_{3} − τ_{3}) & exp(−\frac{1}{2}ν_{3} + σ_{2} + τ_{3}) \\ exp(\frac{1}{2}ν_{3} − σ_{2} − τ_{3}) & exp(\frac{1}{2}ν_{2} + τ_{3}) \end{matrix} ) .

Then we can write c̃_{4} = c̃_{3}ẽ_{3,3} .

Step 3. We write down the matrices corresponding to the attaching geodesics.

ã_{1} = ã_{1,1},  ã_{2} = ã_{1,2},  ã_{3} = c̃_{3}ã_{3,2}c̃_{3}^{−1}.

Step 4. We write down the matrices corresponding to the handle geodesics.

ã_{4} = ã_{1,3},  ã_{5} = c̃_{2}ã_{2,2}c̃_{2}^{−1},  ã_{6} = c̃_{3}ã_{3,3}c̃_{3}^{−1}.

Step 5. In order to write down the matrices for the handle closing geodesics, we need some untwisted elementary conjugators. For the first handle-closer, we need the untwisted elementary conjugators, e_{1,3}^{0} and e_{4,2}^{0} . Matrices for these are as follows.

ẽ_{1,3}^{0} = \frac{1}{\sqrt{2 sinh ν_{1}}} ( \begin{matrix}exp(\frac{1}{2}ν_{1}) & exp(−\frac{1}{2}ν_{1} + σ_{1}) \\ exp(−\frac{1}{2}ν_{1} − σ_{1}) & exp(\frac{1}{2}ν_{1}) \end{matrix} ) , and
ẽ_{4,2}^{0} = \frac{1}{\sqrt{2 sinh μ_{4}}} ( \begin{matrix}exp(\frac{1}{2}μ_{4}) & exp(−\frac{1}{2}μ_{4}) \\ exp(−\frac{1}{2}μ_{4}) & exp(\frac{1}{2}μ_{4}) \end{matrix} ) .

The matrix for the first handle-closer is then

ã_{7} = c̃_{4}ẽ_{4,2}^{0}f̃_{τ_{4}}(ẽ_{1,3}^{0})^{−1}.

For the second handle-closer, we need the untwisted elementary conjugators, e_{2,2}^{0} and e_{2,3}^{0} ; we write

ẽ_{2,2}^{0} = \frac{1}{\sqrt{2 sinh μ_{2}}} ( \begin{matrix}exp(\frac{1}{2}μ_{2}) & exp(−\frac{1}{2}μ_{2}) \\ exp(−\frac{1}{2}μ_{2}) & exp(\frac{1}{2}μ_{2}) \end{matrix} ) , and
ẽ_{2,3}^{0} = \frac{1}{\sqrt{2 sinh μ_{2}}} ( \begin{matrix}exp(\frac{1}{2}μ_{2}) & exp(\frac{1}{2}μ_{2} + σ_{1}) \\ exp(−\frac{1}{2}μ_{2} − σ_{1}) & exp(\frac{1}{2}μ_{2}) \end{matrix} ) .

Then we can write

ã_{8} = c̃_{2}ẽ_{2,3}^{0}f̃_{τ_{5}}(ẽ_{2,2}^{0})^{−1}c̃_{2}^{−1}.

For the third handle-closer, we need matrices for the untwisted elementary conjugators e_{3,3}^{0} and e_{4,3}^{0} ; these are as follows.

ẽ_{3,3}^{0} = \frac{1}{\sqrt{2 sinh ν_{3}}} ( \begin{matrix}exp(\frac{1}{2}ν_{3}) & exp(−\frac{1}{2}ν_{3} + σ_{2}) \\ exp(−\frac{1}{2}ν_{3} − σ_{2}) & exp(\frac{1}{2}ν_{3}) \end{matrix} ) , and
ẽ_{4,3}^{0} = \frac{1}{\sqrt{2 sinh ν_{4}}} ( \begin{matrix}exp(\frac{1}{2}ν_{4}) & exp(−\frac{1}{2}ν_{4} + σ_{3}) \\ exp(−\frac{1}{2}ν_{4} − σ_{3}) & exp(\frac{1}{2}ν_{4}) \end{matrix} ) .

Now we can write our final generator as

ã_{9} = c̃_{4}ẽ_{4,3}^{0}f̃_{τ_{6}}(ẽ_{3,3}^{0})^{−1}c̃_{3}^{−1}.

12. The first variation

For our first variation, we distinguish between left and right half-planes. If H_{i} is a fully normalized pants group, with generators a_{i,1} , a_{i,2} and a_{i,3} , then the corresponding fully normalized right pants group H_{i}^{R} = H_{i}, with generators a_{i,k}^{R} = a_{i,k}. The corresponding fully normalized left pants group H_{i}^{L} has generators a_{i,1}^{L} = a_{i,1}^{−1} , a_{i,2}^{L} = ra_{i,2}^{−1}r^{−1} , and a_{i,3}^{L} = ra_{i,3}^{−1}r^{−1} , where r denotes reflection in the imaginary axis. It is essentially immediate that H^{2}/H_{i}^{L} = H^{2}/H_{i}^{R}, but with reversed orientation. Since every pair of pants is symmetric; they are in fact indistinguishable.

By using both left and right pants groups, we can eliminate the elementary conjugators e_{i,1} and e_{i,1}^{0} . That is, the fully normalized pants group corresponding to P_{1} is H_{1}^{R}; the fully normalized pants group corresponding to P_{2} is H_{2}^{L}; the fully normalized pants group corresponding to P_{3} is H_{3}^{R}; the fully normalized pants group corresponding to P_{4} depends on the entries in the pairing table. If the geodesic appearing as b_{4,1} also appear in either the first or third row, then the corresponding fully normalized pants group is H_{4}^{R}; if this geodesic also appears in the second row, then the corresponding fully normalized pants group is H_{4}^{L}. Continuing inductively, the fully normalized pants group corresponding to P_{j} is H_{j}^{R} (H_{j}^{L}) if the fully normalized pants group corresponding to P_{i(j)} is H_{i}^{R} (H_{i}^{L}), where i(j) is the row in the pairing table containing the same coordinate geodesic as the entry in the (j, 1) place.

For those indices i for which the fully normalized pants group is H_{j}^{R}, the (right) conjugators are defined by e_{i,k}^{0R} = e_{i,k}^{0} and c_{i}^{R} = c_{i}.

For those indices i for which the fully normalized pants group is H_{i}^{L}, the corresponding untwisted elementary left conjugators are defined by e_{i,2}^{0L} = re_{i,2}^{0}r^{−1} , and e_{i,3}^{0} = re_{i,3}^{0}r^{−1} . As before, the conjugator c_{1} = c_{1}^{R} = 1. However, for i = 2, the second fully normalized pants group is necessarily H_{2}^{L}, and the corresponding conjugator is defined by c_{2}^{L} = f_{τ_{1}}^{−1} .

We define the left universal twist by f_{τ}^{L} = (f_{τ})^{−1} . Then the twisted left elementary conjugators are given by e_{i,k}^{L} = e_{i,k}^{0L}f_{τ}^{L}. Finally, the conjugator, c_{i}^{L} is defined inductively by c_{i}^{L} = c_{j(i)}^{L}e_{j(i),k}^{L}.

Now that we have defined the fully normalized pants groups corresponding to every P_{i}, and the corresponding twisted and untwisted conjugators, we can easily write down matrices for the generators and the conjugators, using the matrix,

r̃ = ( \begin{matrix}−1 & 0 \\ 0 & 1 \end{matrix} ) ,

except that we will have to change the sign of some matrices so as to have positive traces. Once we have made these modifications, the algorithm proceeds as above, with the following modifications.

For Step 1, we need right and left fully normalized pants groups.

For Step 2, one has to modify equation (9) to appropriately reflect right and left conjugators.

Similarly, in Step 5, one has to replace (14) by appropriately replacing the conjugators by left and right conjugators, and one has to adjust the reversing transformation g. We define g^{XY} to be the identity if X ≠ Y , and to be the reversing transformation g(z) = −1/z if X = Y . Then the new equation for the handle closer is:

(20)  d̃_{j} = c̃_{j′}^{X}ẽ_{j′,k′}^{0X}g^{XY} f̃_{τ}^{Y} (c̃_{j}^{Y} ẽ_{j,k}^{0Y})^{−1},

where X and Y both take on the values L and R, and are determined by the fact that, for every j, exactly one of c_{j}^{L} and c_{j}^{R} is defined.

13. Special case of genus 2

In this section, we illustrate the first variation of our algorithm in the case of a closed surface of genus 2, where all three coordinate geodesics of the F-N system are non-dividing.

In this case, the signature is (2, 0), and the pairing table is as follows:

\begin{matrix}P_{1} & L_{1} & L_{2} & L_{3} \\ P_{2} & L_{1} & L_{2} & L_{3} \end{matrix}

We assume we are given Φ = (s_{1}, s_{2}, s_{3}, t_{1}, t_{2}, t_{3}).

Step 1. The two pairs of pants, P_{1} and P_{2} , are necessarily isometric; we write down the generators for the fully normalized pants groups, H_{1} = H_{1}^{R} and H_{2} = H_{2}^{L}, as follows.

ã_{1,1}^{R} = ( \begin{matrix}e^{σ_{1}} & 0 \\ 0 & e^{−σ_{1}} \end{matrix} );
ã_{1,2}^{R} = \frac{1}{sinh μ} ( \begin{matrix}sinh(μ − σ_{2}) & sinh σ_{2} \\ − sinh σ_{2} & sinh(μ + σ_{2}) \end{matrix} ) ,

where μ is defined by

coth μ = \frac{cosh σ_{1} cosh σ_{2} + cosh σ_{3}}{sinh σ_{1} sinh σ_{2}} ,  μ > 0;
ã_{1,3}^{R} = \frac{1}{sinh ν} ( \begin{matrix}sinh(ν − σ_{3}) & e^{σ_{1}} sinh σ_{3} \\ −e^{−σ_{1}} sinh σ_{3} & sinh(ν + σ_{3}) \end{matrix} ) ,

where ν is defined by

coth ν = \frac{cosh σ_{1} cosh σ_{3} + cosh σ_{2}}{sinh σ_{1} sinh σ_{3}} ,  ν_{1} > 0;
ã_{1,1}^{L} = ( \begin{matrix}e^{−σ_{1}} & 0 \\ 0 & e^{σ_{1}} \end{matrix} );
ã_{1,2}^{L} = \frac{1}{sinh μ} ( \begin{matrix}sinh(μ − σ_{2}) & sinh σ_{2} \\ − sinh σ_{2} & sinh(μ + σ_{2}) \end{matrix} );
ã_{1,3}^{L} = \frac{1}{sinh ν} ( \begin{matrix}sinh(ν − σ_{3}) & e^{σ_{1}} sinh σ_{3} \\ −e^{−σ_{1}} sinh σ_{3} & sinh(ν + σ_{3}) \end{matrix} ) .

Step 2. The first conjugator c_{1} = 1; the second conjugator, c_{2}^{L} = f_{τ_{1}}^{−1} . Then

c̃_{2}^{L} = ( \begin{matrix}e^{τ_{1}} & 0 \\ 0 & e^{−τ_{1}} \end{matrix} ) .

Step 3. We write down the matrix corresponding to the attaching geodesic; we obtain

ã_{1} = ã_{1,1}^{R} = ( \begin{matrix}e^{σ_{1}} & 0 \\ 0 & e^{−σ_{1}} \end{matrix} ) .

Step 4. We write down the matrices corresponding to the handle geodesics.

ã_{2} = ã_{1,2}^{R} = \frac{1}{sinh μ} ( \begin{matrix}sinh(μ − σ_{2}) & sinh σ_{2} \\ − sinh σ_{2} & sinh(μ + σ_{2}) \end{matrix} ) ,
ã_{3} = ã_{1,3}^{R} = \frac{1}{sinh ν} ( \begin{matrix}sinh(ν − σ_{3}) & e^{σ_{1}} sinh σ_{3} \\ −e^{−σ_{1}} sinh σ_{3} & sinh(ν + σ_{3}) \end{matrix} ) .

Step 5. In order to write down the matrices for the handle closing geodesics, we need to note that λ_{1} = λ_{2} = σ_{1} , and we need matrices for the untwisted elementary conjugators, e_{1,2}^{0R}, e_{2,2}^{0L}, e_{1,3}^{0R} and e_{2,3}^{0L} . We find from equations (16) and (18):

ẽ_{1,2}^{0R} = \frac{1}{\sqrt{2 sinh μ}} ( \begin{matrix}exp(\frac{1}{2}μ) & exp(−\frac{1}{2}μ) \\ exp(−\frac{1}{2}μ) & exp(\frac{1}{2}μ) \end{matrix} ) ,
ẽ_{2,2}^{0L} = \frac{1}{\sqrt{2 sinh μ}} ( \begin{matrix}exp(\frac{1}{2}μ) & − exp(−\frac{1}{2}μ) \\ − exp(−\frac{1}{2}μ) & exp(\frac{1}{2}μ) \end{matrix} ) .

For the first handle closer, equation (20) becomes

(21)  d̃_{1} = c̃_{2}^{L}ẽ_{2,2}^{0L}f̃_{τ_{2}}^{R}(c̃_{1}^{R}ẽ_{1,2}^{0R})^{−1}.

Since c_{1}^{R} = 1, we obtain

ã_{4} = d̃_{1} = \frac{1}{sinh μ} ( \begin{matrix}e^{τ_{1}} cosh(μ − τ_{2}) & −e^{τ_{1}} cosh τ_{2} \\ −e^{−τ_{1}} cosh τ_{2} & e^{−τ_{1}} cosh(μ + τ_{2}) \end{matrix} ) .

For the second handle-closer, we need the untwisted elementary conjugators, e_{1,3}^{0R} and e_{2,3}^{0L}; we write

ẽ_{1,3}^{0R} = ( \begin{matrix}exp(\frac{1}{2}(ν + σ_{1})) & exp(\frac{1}{2}(−ν + σ_{1})) \\ exp(\frac{1}{2}(−ν − σ_{1})) & exp(\frac{1}{2}(ν − σ_{1})) \end{matrix} ) ,  and
ẽ_{2,3}^{0L} = ( \begin{matrix}exp(\frac{1}{2}(ν + σ_{1})) & − exp(\frac{1}{2}(−ν + σ_{1})) \\ − exp(\frac{1}{2}(−ν − σ_{1})) & exp(\frac{1}{2}(ν − σ_{1})) \end{matrix} ) .

We obtain

ã_{5} = c̃_{2}^{L}ẽ_{2,3}^{0L}f̃_{τ_{3}}^{R}(ẽ_{1,3}^{0R})^{−1} = \frac{1}{sinh ν} ( \begin{matrix}e^{τ_{1}} cosh(ν − τ_{3}) & −e^{(τ_{1}+σ_{1})} cosh τ_{3} \\ −e^{(−τ_{1}−σ_{1})} cosh τ_{3} & e^{−τ_{1}} cosh(ν + τ_{3}) \end{matrix} ) .

14. The second variation

In the second variation, which can start with either the original algorithm or with the first variation, we change the definitions of the conjugators, and of the handle closing generators.

The first two conjugators, c_{1} = 1 and c_{2} , remain unchanged. Assume that we have constructed J_{k} representing Q′_{k}, k < q. Then there is some j so that A_{k+j}^{k} projects onto L_{k+1} . We redefine the untwisted conjugator c_{k+1}^{0} , so that it maps the left half-plane onto the action half-plane of a_{k+j}^{k}, while preserving the common orthogonal between the imaginary axis and A_{k+j}^{k}. The twisted conjugator c_{k+j} is then the composition of a twist by 2τ_{k} in the positive direction along the imaginary axis, followed by the untwisted conjugator.

We follow the above procedure, until we reach the point where K^{0} is defined. We assume that we have K^{0} , and we now redefine the base points and matrices for the handle closers. It suffices to describe this new procedure for the first handle-closer; the others are treated analogously.

The handle closer d_{1} conjugates a = a_{2p+1} onto the inverse of a′ = a_{2p+2} . From the preceding step, we already have the matrices ã and ã′. Using formula (3), we can write down the matrices for the reflections r_{A} and r_{A′}, in the axes A and A′, respectively. Then h = r_{A′}r_{A} is a hyperbolic element of PSL(2, R) whose axis is the common orthogonal between A and A′, where the translation length of h is exactly twice the distance between these axes. It follows that the square root of h (i.e., the unique hyperbolic isometry whose square is h) maps A onto A′, while preserving the common orthogonal between A and A′. We define the new untwisted handle-closer, d_{1}^{0} , to be this square root. Then the twisted handle-closer d_{1} is given as: First twist along A by 2τ_{p−q+1} , then apply d_{1}^{0} . Since we can compute the fixed points of a and a′, we can find a matrix for h, and then construct a matrix for the square root of h. Likewise, once we have the fixed points of a, and the trace, 2 cosh(τ_{p−q+1}), we can construct the matrix for the corresponding transformation. We then choose d̃_{1} to have positive trace.

We remark that, while the entries in d̃_{1} can be easily computed in each case, and are well defined algebraic functions of the entries in ã and ã′, there are no easy formulae for these entries.

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304 Bernard Maskit

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Received 15 September 1999