FIXED POINT PARAMETERS FOR MÖBIUS GROUPS

Let Γn (respectively, Γ∞) be a free group of rank n (respectively, a free group of countable infinite rank). We consider the space of algebraic representations of the group Γn (respectively, Γ∞) Hom(Γn, PGL(2,C)) (respectively, Hom(Γ∞, PGL(2,C))). Inside each of these spaces we consider a couple of open and dense subsets. These subsets contain non-discrete groups of Möbius transformations. We proceed to find complex analytic parameters for these spaces given by fixed points. ∗Partially supported by projects Fondecyt 1000715, UTFSM 991223 and a Presidential Chair on geometry 158 Rubén A. Hidalgo


Introduction
In [3] we have considered a real parameterization of the Teichmüller space of a closed Riemann surface.This parameterization is real analytic and given by a collection of fixed points of a particular set of generators of a Fuchsian group acting on the upper-half plane H.One of the main ideas used in that paper is a geometric configuration of axis for a particular set of generators (inequalities of real numbers).In this note, we produce parameterizations of the deformation space of finitely generated groups of Möbius transformations by a collection of fixed points of a particular set of generators.We do not use axis configurations and it is important to note that, in this general situation, we work with groups which may not be discrete ones nor Kleinian groups nor Fuchsian groups (this, including the complex nature of the parameters, is the main difference with the above work).This parameterization can be used in particular for describing (fixed points) complex analytic parameters for the deformation space of a Kleinian group.We also compute explicit (real analytic) models of some fuchsian groups (including an example of genus two).To describe this parameterization we start with some basic definitions.
A Möbius transformation B is a conformal automorphism of the Riemann sphere C. In particular, B(z) = az+b cz+d , where a, b, c, d are complex numbers satisfying ad − bc = 0.There is a natural isomorphism between the group of Möbius transformations and the projective linear group P GL (2, C) given by For each Möbius transformation B, we denote its set of fixed points by F (B).If B is neither the identity nor elliptic of order two, we can define the values a(B), r(B) ∈ F (B) as follows.
(1) If B is Loxodromic, then a(B) and r(B) are the attracting and repelling fixed points of B.
(2) If B is parabolic, then a(B) = r(B) is its unique fixed point.
(3) If B is elliptic, then a(B) and r(B) are its fixed points for which there exists a Möbius transformation H such that H • B • H −1 (z) = λz, where λ = 1, imaginary part of λ is positive and H(a(B)) = ∞, H(r(B)) = 0.
The space of infinitely generated marked groups (G, A 1 , ..., A n , ...)) can be identified to the set Hom(Γ ∞ , P GL(2, C)), where Γ ∞ is a free group of infinite rank.This is a infinite dimensional complex manifold (isomorphic to P GL(2, C) N ).Similarly, the space of finitely generated marked groups (G, A 1 , ..., A n )) can be identified the set Hom(Γ n , P GL(2, C)), where Γ n is a free group of rank n.This is a 3n-dimensional complex manifold (isomorphic to P GL(2, C) n ).
Analogously, we define the subsets G ∞ and G n consisting of the equivalence classes of marked (non necessarily discrete ones) groups [(G, (A 1 , ..., A n , ...))] of Möbius transformations satisfying the following. (1) for all j ≥ 2; and (3) The sets F ∞ , G ∞ , F n and G n are open dense subsets of the respective deformation spaces.In particular, they are complex analytic spaces of the same respective dimensions.
In each class [(G, (A 1 , ..., In this way, we may think of the elements of F n and G n (respectively, F ∞ and G ∞ ) as normalized marked groups satisfying the above respective properties.
Using the unique normalized representative, we may construct functions For each normalized marked group (G, (A 1 , ..., A n , ...)) in either F n , G n , F ∞ and G ∞ , we write down explicit matrices in P GL(2, C) representing all the transformations A i .The entries of such matrices are rational functions in the corresponding fixed point coordinates given by the above theorem.

Deformation Spaces of Möbius groups
Let G be a group of Möbius transformations, maybe infinitely generated.The algebraic deformations of G are defined in similar fashion as it was done for Γ n and Γ ∞ .More precisely, we consider the space Hom(G, P GL(2, C)) of representations of G into P GL (2, C).Two representations are said equivalents if they are conjugate by some Möbius transformation.The set of equivalence classes Def (G, P GL(2, C)) is the algebraic deformation space of G. Another deformation space associated to G is the quasiconformal deformation space.A quasiconformal homeomorphism w : again a group (necessarily Kleinian) of Möbius transformations.Two deformations of G, say w 1 and w 2 , are equivalent if there exists a Möbius transformation A so that The set of equivalence classes of deformations of G is called the deformation space of G and denoted by T (G).In each class there is a unique representative deformation w n satisfying w n (x) = x, for x ∈ {∞, 0, 1}.In the case that G is a geometrically finite Kleinian group, then the above two deformation spaces are the same.In general, we have T (G) ⊂ Def (G, P GL(2, C)).In [8] the following is proved.
The proof of such a theorem is a consequence of the existence of certain points called stratification points.Our coordinates are a kind of stratification points for Möbius groups (non necessarily discrete ones and maybe infinitely generated).In particular, we have the following concerning the above Kra-Maskit's result.Let G be a Möbius group which can be generated by Möbius transformations A 1 ,..., A n ,...so that the following hold: (1) A 2 i = I, (A j • A 1 ) 2 = I, for all i ≥ 1 and all j ≥ 2.
(2) F (A 1 ) ∩ F (A j ) = ∅, for all j ≥ 2. ( We denote by a i the attracting fixed point of A i , r i the repelling fixed point of A i , and s k the repelling fixed point of A k • A 1 .Then theorem 1 implies the following: Corollary 1.If G is a Möbius group finitely generated by Möbius transformations A 1 ,..., A n , so that they satisfy conditions (1), ( 2) and (3) as above, then the map Φ : turns out to be a one-to-one holomorphic map.Similarly, if G is a Möbius group infinitely generated by Möbius transformations A 1 ,..., A n ,..., so that they satisfy conditions (1), ( 2) and (3) as above, then the map Φ : T (G) → C×C N , defined by Φ([w]) = (w n (r 1 ), ..., w n (r n ), ..., w n (a 3 ), ..., w n (a n ), ..., w n (s 2 ), ..., w n (s n ), ...) is a one-to-one holomorphic map.
The above is in really true at the level of the algebraic deformation spaces by theorem 1.

Models of Teichmüller spaces
Let F < P GL + (2, R) be a finitely generated Fuchsian group acting on the hyperbolic plane H.A Fuchsian representation of F is a monomorphism θ : Two Fuchsian representations θ 1 and θ 2 are said Fuchsian equivalent if and only if there exists a Möbius transformation for all γ ∈ F .The set of equivalence classes of Fuchsian representations of the Fuchsian group F is called the Teichmüller space of F and denoted as T (F ).This set is a simply connected real analytic manifold of dimension 6g − 6 + 2k + 3l, where H/F is a Riemann surface of genus g with k punctures and l holes (see [1]).We say that F has signature or type (g, k, l).
The Fuchsian group F has a presentation of the form: where [A * j , B * j ] denotes the commutator between the hyperbolic transformations A * j and B * j , the transformations L * j are also hyperbolic and the transformations P * j are parabolic.We also may assume: (2) if g = 0 and k ≥ 2, then a(P * 1 ) = ∞, a(P * 2 ) = 0 and a(P We can identify the Teichmüller space of F with the set of marked groups (G, (A 1 , B 1 , ..., A g , B g , P 1 , ..., P k , L 1 , ..., L l )) for which: (1) G is a fuchsian group acting on H of same type as F ; (2) there is an isomorphism ψ : (4) if g = 0 and k ≥ 2, then a(P 1 ) = ∞, a(P 2 ) = 0 and a(P 2 P 1 ) = 1; (5) if g = 0, k = 1, then a(P 1 ) = ∞, a(L 1 ) = 0 and a(L If we use the function Ψ n , with n = 2g + k + l, then theorem 1 implies it is a one-to-one real analytic map into R 3n−3 .The image is contained inside an algebraic variety defined by (3 + k) real polynomials.Let us set the following notation a j := a(A j ), b j := r(A j ), c j := a(B j ), d j := r(B j ), p j := a(P j ), x j := a(L j ), y j := r(L j ), e j := r(A j A 1 ), f j := r(B j A 1 ), r j := r(P j A 1 ), s j := r(L j A 1 ), q j := r(P j P j−1 • • • P 1 ), turns out the to be a one-to-one real analytic map into the real affine variety W defined by three polynomials.These three polynomials are b, c, d, e, f, p, r, s, x, y), where turns out the to be a one-to-one real analytic map.
turns out the to be a one-to-one real analytic map into the real affine variety W defined by a polynomial E = 0, where E is defined by the following observation.The above data determines uniquely the transformations P 1 ,..., P k−1 .Since the transformation P k is the inverse of the compositions of these transformations and it is parabolic, then the polynomial corresponds to have square of the trace of P k equal to 4.
turns out the to be a one-to-one real analytic map into the affine real variety defined by one polynomial obtained in the same way as in the case above.
turns out the to be a one-to-one real analytic map.
2.8.Signature (0,1,l), l ≥ 2 The map Q : turns out the to be a one-to-one real analytic map.
2.9.Signature (0,0,l), l ≥ 3 The map turns out the to be a one-to-one real analytic map.
Remarks.In the last section we do more explicit computations for types (0, 4), (1, 1) and (2, 0).We obtain parameter spaces related to the ones given by Maskit in [4], [5], [6] and Min in [9].We must remark that Min's parameters use multipliers and ours (also Maskit's ones) only use fixed points.Unfortunately, our parameters look more difficult to Maskit's ones.Application to the Schottky space and noded Riemann surfaces can be found in [2].
For infinitely generated Fuchsian groups, we may also use the results in this note to construct models of the respective Teichmüller spaces.

Proof of Theorem 1
In this section we prove theorem 1.For this, we need the following lemmas.Proof of Theorem 1.The proof is a direct consequence of lemmas 1 and 2 as follows.
(1) Lemma 1 implies that A 1 and A 2 are uniquely determined.Now, apply Lemma 2 to the pair A = A 1 and B = A j to obtain A j uniquely, for every j ≥ 3. The complex analyticity of the map Φ n and Φ ∞ follows easily from the explicit matrix description of the generators in P GL(2, C).
(2) Apply Lemma 1 to obtain uniquely the elements A 1 and A 2 .
Next, apply Lemma 2 to the pair A = A 2 • A 1 and B = A 3 to obtain A 3 uniquely.We continue inductively applying Lemma 2 to the pairs Proof of Lemma 1.We decompose the set F 2 as the disjoint union of four subsets, say where ))] ∈ F 2 ; A 1 and A 2 are not parabolics}; It is easy to see that the images under Φ 2 of L i and L j are disjoint if i = j.This is a consequence of the fact that, T is parabolic if and only if a(T ) = r(T ).The injectivity of Φ 2 then follows from the injectivity of Φ 2 restricted to each L i , i = 1, 2, 3, 4. In what follows, we denote by x, y and z the points r(A 1 ), r(A 2 ) and r(A 2 • A 1 ), respectively.
(I) Φ 2 is injective on L 1 .The transformations A 1 and A 2 are not parabolics.The matrix representation of these transformations is given by where either k 2 j > 1 or k 2 j = e 2πiθ j , θ j ∈ (0, 1/2), for j = 1, 2. In this case, the product A 2 • A 1 has the matrix representation Since 1 is a fixed point of A 2 • A 1 , we have the equation ).
The facts y = 1, y = 0 and The fixed points of A 2 •A 1 are the roots of the quadratic polynomial in w: .
The equality of the RHS of ( * ) and ( * * ) gives us the following equation to be satisfied by k 2 1 : The solutions to this equations are by 1, −1, k 1 and − k 1 .¿From that one obtain: .
(II) Φ 2 is injective on L 2 .In this case A 1 and A 2 have the following representation where either k 2 2 > 1 or k 2 2 = e 2πiθ , θ ∈ (0, 1/2) and a = 0.In this case the product A 2 • A 1 has the following matrix representation The fact that 1 is a fixed point of A 2 • A 1 gives us the equation Since k 2 = 0 and y = 1, we must have that 1 + a = 0 if and only if y = 1 + a; in which case y = 0, a contradiction.In particular, we get The fixed points of the transformation A 2 • A 1 are the roots of the quadratic equation in w: (III) Φ 2 is injective on L 3 .In this case, A 1 and A 2 have the following matrix representation where either k 2 1 > 1 or k 2 1 = e 2πiθ , θ ∈ (0, 1/2) and b = 0.The product A 2 • A 1 has the following matrix representation The fact that 1 is a fixed point of Since k 1 = 0, we must have that 1 + x(b − 1) = 0 if and only if 1 + x(b − 1) − b = 0, in which case b = 0 a contradiction.In particular, we get The fixed points of the transformation A 2 • A 1 are the roots of the quadratic equation in w: bk 2  1 , and The equality of the RHS of ( * ) and ( * * ) gives the following equation to be satisfied by b: (IV) Φ 2 is injective on L 4 .In this case, A 1 and A 2 have the following matrix representation where ab = 0.In this case the product A 2 • A 1 has the following matrix representation The fact that a = 0 implies a = −1, and we obtain the equation The fixed points of the transformation A 2 • A 1 are the roots of the quadratic equation in w: It follows that z = − a b , and The equality of the RHS of ( * ) and ( * * ) implies Proof of Lemma 2.
We normalize so that a(A) = ∞.
Case 1. Assume A and B to be parabolic elements.In this case, A and B have the following matrix representation where ap = 0 and x is the fixed point of B. We want to obtain a unique value of p in function of a, x and r(B • A).
In this case, the product B • A has the following matrix representation Case 2. Assume A to be parabolic and B to be non-parabolic.In this case, A and B have the following matrix representation where a = 0, r(B) = x and a(B) = y are the fixed point of B and either k > 1 or k 2 = e 2πiθ , θ ∈ (0, 1/2).We want to obtain a unique value of k 2 in function of a, x, y and r(B • A).
In this case, the product B • A has the following matrix representation Since x = z and y = z, we have that x − z − a = 0 if and only if y − z − a = 0 in which case x = y a contradiction.In particular, where p = 0.In this case we want to determine the value of p uniquely.The transformation B • A has the matrix representation The condition that z is a fixed point of B • A gives us the equation Since p = 0, k 2 = 1 and z = r, both sides of the above equation are necessarily different from zero.In particular, Case 4. Assume A and B to be non-parabolic elements.In this case, A and B have the following matrix representation where r(A) = x, r(B) = u, a(B) = t, and either k j > 1 or k 2 j = e 2πiθ j , θ j ∈ (0, 1/2).We want to obtain a unique value of k 2 2 in function of x, u, t, k 2  1 and r(B • A).

Explicit Matrix Representation
For the case of normalized marked groups in F n or F ∞ , as a consequence of Theorem 1, we can write matrices in P GL(2, C) representing the transformations A 1 ,..., A n , ..., as follows.
(VIII) If r j = a j (j = 3, ..., n) and r 1 = ∞, then where For the case of normalized marked groups in V n or V ∞ , as a consequence of Theorem 1, we can write matrices in P GL(2, C) representing the transformations A 1 ,..., A n , ..., as follows. where . where . where , and m j−1 is the attracting fixed point of the transformation The above values of m j and µ 2 j are obtained in an inductive way, where m 2 = 1 and µ 2  2 is the multiplier of A 2 • A 1 .

Computing Models for Some Teichmüller Spaces
5.1.Teichmüller Spaces of Riemann Surfaces of Type (0,4) A Riemann surface S is said to be of type (0, 4) if it is a Riemann surface of genus zero with exactly 4 boundary components.If some of the boundaries is a puncture, then S is called a parabolic Riemann surface of type (0, 4); otherwise, it is called a hyperbolic Riemann surface of type (0, 4).Let Γ < P GL + (2, R) be a Fuchsian group acting on the hyperbolic plane such that S = H/Γ is a hyperbolic Riemann surface of type (0, 4).Let α 1 , α 2 and α 3 be simple loops on S (through the point z) as shown in figure 1.The fundamental group of S, at the point z ∈ S, has a presentation In this way, we have that Γ is a free group of rank 3 generated by A 1 , A 2 and A 3 , so that the transformations A i are hyperbolic and the axis of these transformations The Teichmüller space of Γ (or S) T (Γ) is in this case given by the subset V (3) consisting of those marked groups (G, (B 1 , B 2 , B 3 )) satisfying the following.
(3) There exists a quasiconformal homeomorphism F : As a consequence of theorem 1, we have one-to-one real analytic map r 3 , a 3 , s 2 , s 3 ), where r i , a 3 and s k are the repelling fixed point of B i , the attracting fixed point of B 3 and the repelling fixed point of B k • B 1 , respectively.
If we denote by u the attracting fixed point of the transformation B 3 • B 1 , then we have that the value u is a real analytic function on r 1 , r 2 , r 3 , a 3 , s 2 and s 3 .In fact, .
The axis of the transformations B 1 , B 2 , B 3 , B 2 •B 1 and B 3 •B 1 have the same topological configuration as the axis of the transformations In particular, the fixed points of the above transformations satisfy the following inequalities.
Let us consider the parameter space R as the open subset of R 6 consisting of the tuples (r 1 , r 2 , r 3 , a 3 , s 2 , s 3 ) satisfying the inequalities given by (E1) and (E2).In particular, Φ 3 (T (Γ)) is contained R.
Proof.We have to show that for any point p = (r 1 , r 2 , r 3 , a 3 , s 2 , s 3 ) contained in the region R there is a normalized marked group (G, (B 1 , B 2 , B 3 )) in V (3) so that Φ 3 ((G, (B 1 , B 2 , B 3 ))) = p.For p as above we can construct Möbius transformations where ; The inequalities (E1) and (E2) ensure that the transformations and r(B 3 • B 1 ) = s 3 .We denote the axis of the transformations
Denote by R i and S i the reflection on L i and M i , respectively (see figure 3).Direct computations show that the hyperbolic distance d between L 1 and L 3 is the same as the hyperbolic distance between )).In particular, we have The group G i uniformizes a pant P i , both of them having a boundary (given by the axe N 1 ) of the same length.It is easy to see that we can apply the first combination theorem of Maskit [7] to these groups with common subgroup J =< B 1 > (the discs used in such a theorem are the discs bounded by the axe N 1 ).As a consequence, the group G generated by B 1 , B 2 and B 3 is a free group of rank three and H/G is a hyperbolic Riemann surface of type (0, 4) (see figure 4 for a fundamental domain of G).The construction of a quasiconformal homeomorphism of the hyperbolic plane as required is standard using the fundamental domains for Γ and G which are topologically the same. 2 Remark.The angle involve in gluing the pants corresponding to the groups G 1 and G 2 in the above proof is given by ).
We can either make r 2 = 0 or s 2 = 1 or u = s 3 or a 3 = r 3 to obtain explicit models for the Teichmüller space (as boundaries of the above model) of parabolic Riemann surfaces of type (0, 4).We must remark that if we make r 1 = ∞, then we get only a model of the Teichmüller space of pants.In the particular case, r 2 = 0, s 2 = 1, u = s 3 and a 3 = r 3 , we obtain Maskit's model for the Teichmüller space of marked surfaces of genus zero with four punctures (see [4]).In this case, the above formula for u gives us u = r 1 + (r 1 −1)(r 3 −1) 5 we draw a fundamental domain for the group In this case the angle θ = θ(r 1 , r 3 ) is θ = θ(r 1 , r 3 ) = π log( r 1 r 1 −1 ) log( r 3 −r 1 r 1 ).We have an explicit one-to-one real analytic diffeomorphism between M and the Fricke space (angle, length coordinates) L : M → R 2 defined by L(r 1 , r 3 ) = (log( r 1 r 1 −1 ), θ(r 1 , r 3 )).Similarly, one can use the above parameters to find explicit models for the Teichmüller spaces of surfaces of type (0, m), where m ≥ 5.

Teichmüller Spaces of Riemann Surfaces of Type (1,1)
A Riemann surface of type (1, 1) is topologically equivalent to a surface of genus one with a deleted point.If the boundary is a puncture, then we call it a parabolic Riemann surface of type (1,1).; otherwise, we call it a hyperbolic Riemann surface of type (1,1).
As a consequence of Theorem 1, we have that the multiplier of the transformations A 1 and A 2 are given by k , respectively.Our assumption on the equality of the geodesics lengths implies that s = r 2 (1−r 1 ) 1−r 2 .Since s > 1 we have the inequality r 2 (2 − r 1 ) < 1. Observe that r 2 < 0 and r 1 > 1 imply that 1−r 2 < r 1 .On the other hand, both geodesics N 1 and N 2 project onto l on S. It implies that there is a transformation We denote by D 2 and D 1 the discs bounded by N 2 and N 1 , respectively, where the boundary of D 1 contains r 1 + 1 and the boundary of D 2 contains r 2 2 .We have that A 3 (D 2 ) is equal to the complement of D 1 ∪ N 1 .In this way the conditions of the Maskit's second combination theorem ( [7]) are satisfied for G 1 and G 2 =< A 3 >.In particular, Γ is the HNN-extension of G 1 by A 3 (see figure 7 for a fundamental domain of Γ).Denote by a and r the attracting and repelling fixed points of A 3 , respectively.In this case, r 2 < r < 0 and r 1 < a.If k 2 > 1 denotes the multiplier of A 3 , then Since necessarily A 3 (0) = r 1 and A 3 (r 2 ) = ∞, we obtain a = r 1 r 2 r and k 2 = r 2 (r 1 −r) r(r−r 2 ) .The inequality r > r 2 implies the inequality r 1 < a.In this way, we obtain that the only variables are given by r 1 , r 2 and r satisfying the inequalities (F1) r 2 (2 − r 1 ) < 1; and The Teichmüller space of Γ is identified with the set W of marked groups (G, (B 1 , B 2 , B 3 )) satisfying: (1) G is discrete subset of P GL + (2, R); (2) a(B 1 ) = ∞, a(B 2 ) = 0 and a(B 2 • B 1 ) = 1; and (3) There is a quasiconformal homeomorphism F : In particular, for (G, (B 1 , B 2 , B 3 )) in W , the only relation is given by Let us consider the region H in R 3 consisting of the points (r 1 , r 2 , r) satisfying the inequalities (F1) and (F2).As a consequence of theorem 1, we have a one-to-one real analytic map Θ : W → H defined by Θ((G, (B 1 , B 2 , B 3 ))) = (r(B 1 ), r(B 2 ), r(B 3 )).

Teichmüller Spaces of Closed Riemann Surfaces of Genus Two
Let S be a closed Riemann surface of genus two.On S we consider a set of oriented simple loops (through a point z) γ, α 1 , α 2 , β 1 and β 2 as shown in figure 9.
The fundamental group of S, with base point at z, has a presentation of the form Let F be a Fuchsian group (acting on the hyperbolic plane H = {z ∈ C; lm(z) > 0}) uniformizing the surface S, that is, there is a holomorphic covering π : H → S with F as covering group.Choose a point x in H such that π(x) = z.We have a natural isomorphism λ : Π 1 (S, z) → F as follows.For a class [η] ∈ Π 1 (S, z) we consider a representative η.Now we lift η under π at the point x.The end point of such a lifting is of the form f η (x) for a unique element f η ∈ F .Basic covering theory asserts that if ρ is another representative of ).In particular, a presentation of F is given by Denote by γ, α1 , α2 , β1 and β2 the projections on S under π of the axis of the transformations A 1 , F 1 , F 2 , A 2 and A 3 , respectively, as shown in figure 10.We have oriented the axe Ax(H) of a (hyperbolic) transformation H in such a way that the attracting fixed point of H is the end point.The orientations of the projections of the above axis carry the natural orientation induced from the one given to the axis.We can normalize The choice made for the transformations A 1 , A 2 , A 3 , F 1 and F 2 imply that the axis of these transformations are as shown in figure 11.For simplicity, we denote a( In particular, we have that these fixed points satisfy the following inequalities: From the above, we observe that the group G 1 = A 1 , A 2 , A 3 uniformizes a hyperbolic surface of type (0, 4) with two pairs of holes of the same length.These two pairs of holes are the ones bounded by the loops β1 and π(Ax(A 2 •A 1 )), and the loops β2 and π(Ax(A 3 •A 1 )), respectively.We denote by D 2 , D 2 , D 3 and D 3 the disjoint discs bounded by the axis of the transformations A where the bar represents the Euclidean closure.One can apply the second combination theorem of Maskit [7] to the pair of groups G 1 and G a surface of genus one with two boundary components of the same length.Now we apply again the second combination theorem of Maskit where the values y 1 , s 2 , s 3 , y 2 and u are given by the formulae above.

Proof.
The map φ is a surjective map.This is a consequence of the combination theorems of Maskit in [7].We sketch the idea of the proof of this assertion.Given a point p = (r 1 , r 2 , r 3 , a 3 , x 1 , x 2 ) in F, we can construct the values s 2 , s 3 , u, y 1 and y 2 .We can also construct values k 2 (H 1 ) and k 2 (H 2 ).Using these values we obtain unique transformations B 1 , B 2 , B 3 , H 1 and H 2 .The inequalities defining the set F asserts that the above transformations are all hyperbolic.The configuration of the axis of the transformations B 1 , B 2 , B 3 , B 2 • B 1 , B 3 • B 1 , H 1 and H 2 is the same as for the axis of the transformations A 1 , A 2 , A 3 , A 2 • A 1 , A 3 • A 1 , F 1 and F 2 as shown in figure 11.The group T 1 =< B 1 , B 2 > is a free group of rank two uniformizing a pant.The group T 2 =< B 1 , B 3 > also uniformizes a pant.One can check that the conditions of the first combination theorem of Maskit are satisfied and one obtain that T 3 =< T 1 , T 2 >= T 1 * <B 1 > T 2 uniformizes a surface of type (0, 4).It is easy to check that H k • B k+1 • H −1 k = B k+1 • B 1 , for k = 1, 2. The conditions of the second combination theorem holds for the group T 3 and T 4 =< T 1 >.The group T 5 =< T 3 , T 4 >= T 3 * H 1 uniformizes a surface of genus one with two boundary components.Again apply the second combination theorem to the groups T 5 and T 6 =< H 2 > to obtain that the marked group T 7 =< T 5 , T 6 >= T 5 * H 2 belongs to T (F ). 2 Remarks.
(1) The main difference between the above parameters and the ones in [9] is the fact that we only use fixed points.The parameters given by Maskit also only contain fixed points and are more easy than the ones obtained here.
(2) Maskit's model uses the fact that any Riemann surface of genus two is constructed from two isometric pants.Min model uses the fact that any Riemann surface of genus two is constructed from two surfaces of type (1, 1).Our construction uses the fact that any Riemann surface of genus two is constructed from a surface of type (0, 4) with two pairs of boundaries of the same length.
(3) We can relate our fixed point parameters to other parameters of Teichmüller space, for instance to Fenchel-Nielsen Parameters in the same way as done for the case of signature (0, 4).The way to do this is the following.At each each axis Ax(A 1 ), Ax(A 2 ) and Ax(A 3 ) we have associated the multiplier of the transformations A 1 , A 2 and A 3 , respectively.These multipliers are in function of the fixed point parameters and, in particular, the hyperbolic lengths of the geodesics α 1 , α 2 and γ are in function of these fixed points parameters.To obtain the angle at γ, we look at the common orthogonal geodesic L 1 (respectively, L 2 ) to both Ax(A 1 ) and Ax(A 2 ) (respectively, Ax(A 1 ) and Ax(A 3 )).
These two geodesics determine an arc in Ax(A 1 ), whose hyperbolic length determine the angle (also in function of the fixed point parameters).To determine an angle at α 1 we look the common orthogonal geodesic L 3 of Ax(A 1 ) and Ax(A 2 A 1 ).We proceed to see the hyperbolic arc in Ax(A 2 A 1 ) determined by the intersection point of L 3 with Ax(A 2 A 1 ) and the image by F 1 of the intersection point of L 3 with Ax(A 2 ).Similarly for looking at the angle at α 2 .The explicit computations are similar to the case (0, 4) and are left to the interested reader.

Lemma 2 .
Let A and A be two Möbius transformations such that, F (A) ∩ F (B) = ∅.If A, a(B), r(B) and r(B • A) are known, then the transformation B is uniquely determined.

Denote by z
the point r(B • A).The fact that z is fixed point of B • A gives us the equation p(x − z)(z + a − x) = −a.Since a = 0, we obtain p = −a (x−z)(z+a−x) .

Case 3 .
Assume A non-parabolic and B to be parabolic.Let r(A) = r, a(B) = r(B) = x and r(B • A) = z.The transformations A and B have the following matrix representation