RUBÉN A. HIDALGO Universidad Técnica Federico Santa María - Chile

Transcription

1 Proyecciones Vol 19 N o pp August 000 Universidad Católica del Norte Antofagasta - Chile FIXED POINT PARAMETERS FOR MÖBIUS GROUPS RUBÉN A HIDALGO Universidad Técnica Federico Santa María - Chile Abstract Let Γ n 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 Γ n respectively Γ HomΓ n P GL C respectively HomΓ P GL 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 UTFSM 9913 and a Presidential Chair on geometry

2 158 Rubén A Hidalgo 1 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 Ĉ az+b In particular Bz = where a b c d are cz+d complex numbers satisfying ad bc = 0 There is a natural isomorphism between the group of Möbius transformations and the projective linear group P GL C given by a Bz = az+b cz+d c b d 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 ab rb F B as follows 1 If B is Loxodromic then ab and rb are the attracting and repelling fixed points of B If B is parabolic then ab = rb is its unique fixed point

3 Fixed Point Parameters If B is elliptic then ab and rb 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 HaB = HrB = 0 The space of infinitely generated marked groups G A 1 A n can be identified to the set HomΓ P GL C where Γ is a free group of infinite rank This is a infinite dimensional complex manifold isomorphic to P GL C N Similarly the space of finitely generated marked groups G A 1 A n can be identified the set HomΓ n P GL C where Γ n is a free group of rank n This is a 3n-dimensional complex manifold isomorphic to P GL C n Two marked groups G 1 A 1 A n and G B 1 B n are said equivalent if and only if there is a Möbius transformation H P GL C satisfying H A i H 1 = B i for i = 1 n The respective spaces of equivalence classes of marked groups are the algebraic deformation spaces DefΓ P GL C an infinitely complex dimensional space and DefΓ n P GL C a complex analytic space of dimension 3n 1 The sets F DefΓ P GL C and F n DefΓ n P GL C consist of equivalence classes of marked groups [G A 1 A n ] of Möbius transformations non necessarily discrete ones satisfying the following 1 A i = I for all i 1; A j A 1 I for all j ; and 3 F A 1 F A j = for all j 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 A i I for all i 1; A j A j 1 A A 1 I for all j ; and

4 160 Rubén A Hidalgo 3 F A 1 F A j = for all j 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 A n ] in either F n or G n respectively [G A 1 A n ] in either F or G there exists a unique representative G A 1 A n respectively G A 1 A n normalized by aa 1 = aa = 0 and aa A 1 = 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 defined by: Φ n : F n Ĉ C3n 4 Ψ n : G n Ĉ C3n 4 Φ : F Ĉ CN Ψ : G Ĉ CN Φ n [G A 1 A n ] = ra 1 ra n aa 3 aa n ra A 1 ra n A 1 Ψ n [G A 1 A n ] = ra 1 ra n aa 3 aa n ra A 1 ra n A n 1 A A 1 Φ [G A 1 A n ] = ra 1 ra n aa 3 aa n ra A 1 ra n A 1 Ψ [G A 1 A n ] = ra 1 ra n aa 3 aa n ra A 1 ra n A n 1 A A 1 Theorem 1 The functions Φ n Ψ n φ and Ψ are one-to-one complex analytic map 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 C

5 Fixed Point Parameters 161 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 A Couple of Applications 1 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 HomG P GL C of representations of G into P GL C Two representations are said equivalents if they are conjugate by some Möbius transformation The set of equivalence classes DefG P GL C is the algebraic deformation space of G Another deformation space associated to G is the quasiconformal deformation space A quasiconformal homeomorphism w : Ĉ Ĉ is called a deformation of G if w G w 1 is again a group necessarily Kleinian of Möbius transformations Two deformations of G say w 1 and w are equivalent if there exists a Möbius transformation A so that w g w 1 = A w 1 g w1 1 A 1 for all g G 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 DefG P GL C proved In [8] the following is Theorem Kra Maskit If G is a finitely generated Kleinian group then T G is biholomorphically equivalent to a domain in C n 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

6 16 Rubén A Hidalgo which can be generated by Möbius transformations A 1 A n so that the following hold: 1 A i I A j A 1 I for all i 1 and all j F A 1 F A j = for all j 3 aa 1 = aa = 0 and aa A 1 = 1 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 and 3 as above then the map Φ : T G Ĉ C3n 4 defined by Φ[w] = w n r 1 w n r n w n a 3 w n a n w n s w n s n 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 and 3 as above then the map Φ : T G Ĉ CN defined by Φ[w] = w n r 1 w n r n w n a 3 w n a n w n s 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 + R be a finitely generated Fuchsian group acting on the hyperbolic plane H A Fuchsian representation of F is a monomorphism θ : F P GL + R such that there is a quasiconformal homeomorphism φ : H H satisfying θγ = φ γ φ 1 for all γ F

7 Fixed Point Parameters 163 Two Fuchsian representations θ 1 and θ are said Fuchsian equivalent if and only if there exists a Möbius transformation A P GL R such that θ γ = A θ 1 γ A 1 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 + k + 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: F = A 1 B1 A g Bg P1 Pk L 1 L l : lm=1 L kj=1 l m+1 Pk j+1 g i=1[a g i+1 Bg 1+1] = I where [A j Bj ] denotes the commutator between the hyperbolic transformations A j and Bj the transformations L j are also hyperbolic and the transformations Pj are parabolic We also may assume: 1 if g 1 then aa 1 = ab 1 = 0 and ab 1A 1 = 1; if g = 0 and k then ap 1 = ap = 0 and ap P 1 = 1; 3 if g = 0 k = 1 then ap 1 = al 1 = 0 and al 1P 1 = 1; 4 if g = 0 k = 0 then al 1 = al = 0 and al L 1 = 1 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 ; there is an isomorphism ψ : F G so that ψa j = A j ψb j = B j ψp i = P i and ψl r = L r ; 3 if g 1 then aa 1 = ab 1 = 0 and ab 1 A 1 = 1; 4 if g = 0 and k then ap 1 = ap = 0 and ap P 1 = 1; 5 if g = 0 k = 1 then ap 1 = al 1 = 0 and al 1 P 1 = 1; 6 if g = 0 k = 0 then al 1 = al = 0 and al L 1 = 1

8 164 Rubén A Hidalgo If we use the function Ψ n with n = g + 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 := aa j b j := ra j c j := ab j d j := rb j p j := ap j x j := al j y j := rl j e j := ra j A 1 f j := rb j A 1 r j := rp j A 1 s j := rl j A 1 q j := rp j P j 1 P 1 w j := rl j L j 1 L 1 P k P k 1 P 1 and t j := rl j L j 1 L 1 3 Signature g00 g In this case the map Q : T F W R 6g 3 defined by QG A 1 B 1 A g B g = a b c d e f where a = a a g b = b 1 b g c = c c g d = d 1 d g e = e e g f = f 1 f g 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 E 11 = 1 E 1 = 0 and E = 1 where g j=1[a j B j ] is represented by the matrix A 11 A 1 A 1 A 4 Signature gkl g 1 l 1 The map Q : T F R 6g 6+k+3l defined by QG A 1 B 1 A g B g P 1 P k L 1 L l

9 Fixed Point Parameters 165 = a b c d e f p r s x y where a = a a g b = b 1 b g c = c c g d = d 1 d g e = e e g f = f 1 f g p = p 1 p k r = r 1 r k s = s 1 s l 1 x = x 1 x l 1 y = y 1 y l 1 turns out the to be a one-to-one real analytic map 5 Signature gk0 g 1 k 1 The map Q : T F W R 6g 5+k defined by QG A 1 B 1 A g B g P 1 P k = a b c d e f q r where a = a a g b = b 1 b g c = c c g d = d 1 d g e = e e g f = f 1 f g q = q 1 q k 1 r = r 1 r k 1 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

10 166 Rubén A Hidalgo 6 Signature 0k0 k 4 The map Q : T F W R k 5 defined by QG P 1 P k = p q where p = p 3 p k q = q q k 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 7 Signature 0kl k l 1 The map Q : T F R k+3l 6 defined by QG P 1 P k L 1 L l = p q w x y where p = p 3 p k q = q q k w = w 1 w l x = x 1 x l 1 y = y 1 y l turns out the to be a one-to-one real analytic map 8 Signature 01l l The map Q : T F R 3l 4 defined by QG P 1 L 1 L l = w x y where w = w 1 w l x = x x l 1 y = y 1 y l turns out the to be a one-to-one real analytic map 9 Signature 00l l 3 The map Q : T F R 3l 6 defined by QG L 1 L l = t x y where t = t t l x = x 3 x l y = y 1 y l 1

11 Fixed Point Parameters 167 turns out the to be a one-to-one real analytic map Remarks In the last section we do more explicit computations for types and 0We 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 [] For infinitely generated Fuchsian groups we may also use the results in this note to construct models of the respective Teichmüller spaces 3 Proof of Theorem 1 In this section we prove theorem 1 For this we need the following lemmas Lemma 1 Φ : F Ĉ C is one-to-one Lemma Let A and A be two Möbius transformations such that F A F B = If A ab rb and rb A are known then the transformation B is uniquely determined Proof of Theorem 1 The proof is a direct consequence of lemmas 1 and as follows 1 Lemma 1 implies that A 1 and A are uniquely determined Now apply Lemma 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 C

12 168 Rubén A Hidalgo Apply Lemma 1 to obtain uniquely the elements A 1 and A Next apply Lemma to the pair A = A A 1 and B = A 3 to obtain A 3 uniquely We continue inductively applying Lemma to the pairs A = A j 1 A 1 and B = A j to obtain A j uniquely Proof of Lemma 1 We decompose the set F as the disjoint union of four subsets say F = 4 i=1l i where L 1 = {[G A 1 A ] F ; A 1 and A are not parabolics}; L = {[G A 1 A ] F ; A 1 is parabolic and A is not parabolic}; L 3 = {[G A 1 A ] F ; A is parabolic and A 1 is not parabolic}; L 4 = {[G A 1 A ] F ; A 1 and A are both parabolic} It is easy to see that the images under Φ 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 at = rt The injectivity of Φ then follows from the injectivity of Φ restricted to each L i i = In what follows we denote by x y and z the points ra 1 ra and ra A 1 respectively I Φ is injective on L 1 The transformations A 1 and A are not parabolics The matrix representation of these transformations is given by k A 1 = 1 x1 k1 0 1 y 0 A = 1 k yk

13 Fixed Point Parameters 169 where either k j > 1 or k j = e πiθ j θ j 0 1/ for j = 1 In this case the product A A 1 has the matrix representation A A 1 = yk1 xy1 k1 k11 k x1 k11 k + yk Since 1 is a fixed point of A A 1 we have the equation k k 1 + x1 k 1 y = 1 yk 1 + x1 k 1 The facts y = 1 y 0 and k = 0 imply that k 1 +x1 k 1 y 0 In particular k = 1 yk 1 +x1 k 1 k 1 +x1 k 1 y The fixed points of A A 1 are the roots of the quadratic polynomial in w: k 11 k w + [x1 k 11 k + yk yk 1]w xy1 k 1 = 0 In particular z = xy1 k 1 k 1 1 k and since z 0 k = xy1 k 1 +zk 1 k 1 z The equality of the RHS of and gives us the following equation to be satisfied by k 1: k 4 1 k 1 xzy 1+xy1 x+x yz xy y1 xz x + xx y 1 xz x = 0 The solutions to this equations are by 1 1 k 1 and k 1 From that one obtain: k 1 = xx y and 1 xz x k = xy 1y z zx y II Φ is injective on L In this case A 1 and A have the following representation

14 170 Rubén A Hidalgo A = A 1 = 1 a 0 1 y 0 1 k yk where either k > 1 or k = e πiθ θ 0 1/ and a 0 In this case the product A A 1 has the following matrix representation A A 1 = y ay 1 k a1 k + yk The fact that 1 is a fixed point of A A 1 gives us the equation 1 + ay 1 = k y 1 a Since k 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 k = 1+ay 1 y 1 a The fixed points of the transformation A A 1 are the roots of the quadratic equation in w: w + a yw It follows that z = ay and 1 k ay 1 k = 0 k = z+ay z The equality of the RHS of and implies the following equation to be satisfied by a: a y + ayz y + 1 = 0 Since a 0 and y 0 we obtain a = y z 1 and k = y 1y z z

15 Fixed Point Parameters 171 III Φ is injective on L 3 In this case A 1 and A have the following matrix representation 1 0 A = b 1 k A 1 = 1 x1 k1 0 1 where either k1 > 1 or k1 = e πiθ θ 0 1/ and b 0 The product A A 1 has the following matrix representation A A 1 = k 1 x1 k1 bk1 bx1 k1 + 1 The fact that 1 is a fixed point of A A 1 implies the following equation 1 + xb 1 = k 11 + xb 1 b Since k 1 = 0 we must have that 1 + xb 1 = 0 if and only if 1+xb 1 b = 0 in which case b = 0 a contradiction In particular we get k 1 = 1+xb 1 1+xb 1 b The fixed points of the transformation A A 1 are the roots of the quadratic equation in w: bk 1w + bx + 11 k 1w x1 k 1 = 0 It follows that z = x1 k 1 and bk1 k 1 = x x zb The equality of the RHS of and gives the following equation to be satisfied by b: zxb + bz1 x x = 0

16 17 Rubén A Hidalgo Since b 0 x = 0 and z = 0 we obtain b = x+zx 1 xz and k 1 = x x 1x z IV Φ is injective on L 4 In this case A 1 and A have the following matrix representation 1 0 A = b 1 1 a A 1 = 0 1 where ab 0 In this case the product A A 1 has the following matrix representation 1 a A A 1 = b 1 + ab Since 1 is a fixed point of A A 1 we have the following equation a = b1 + a The fact that a = 0 implies a 1 and we obtain the equation b = a 1+a The fixed points of the transformation A A 1 are the roots of the quadratic equation in w: It follows that z = a b and w + aw a b = 0 b = a z The equality of the RHS of and implies a = 1 z and b = 1+z z

17 Fixed Point Parameters 173 Proof of Lemma We normalize so that aa = Case 1 Assume A and B to be parabolic elements In this case A and B have the following matrix representation B = A = 1 a px px p 1 px 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 rb A In this case the product B A has the following matrix representation B A = 1 + px a1 + px px p 1 + ap px Denote by z the point rb A The fact that z is fixed point of B A gives us the equation px zz + a x = a Since a 0 we obtain p = a x zz+a x Case Assume A to be parabolic and B to be non-parabolic In this case A and B have the following matrix representation B = A = 1 a 0 1 x yk xyk 1 1 k xk y

18 174 Rubén A Hidalgo where a 0 rb = x and ab = y are the fixed point of B and either k > 1 or k = e πiθ θ 0 1/ We want to obtain a unique value of k in function of a x y and rb A In this case the product B A has the following matrix representation x yk ax yk B A = + xyk 1 1 k a1 k + xk y Denote by z the point rb A The fact that z is fixed point of B A gives us the equation k y zx z a = x zy z a 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 k = x zy z a y zx z a Case 3 Assume A non-parabolic and B to be parabolic Let ra = r ab = rb = x and rb A = z The transformations A and B have the following matrix representation k r1 k A = px px B = p 1 px where p 0 In this case we want to determine the value of p uniquely The transformation B A has the matrix representation B A = k 1 + px 1 + pxr1 k px pk pr1 k + 1 px The condition that z is a fixed point of B A gives us the equation pk + 1xz + r1 k x z x k z = 1 k z r

19 Fixed Point Parameters 175 Since p = 0 k 1 and z = r both sides of the above equation are necessarily different from zero In particular p = 1 k z r k +1xz+r1 k x z x k z Case 4 Assume A and B to be non-parabolic elements In this case A and B have the following matrix representation B = A = k 1 x1 k u tk utk 1 1 k uk t where ra = x rb = u ab = t and either k j > 1 or k j = e πiθ j θ j 0 1/ We want to obtain a unique value of k in function of x u t k 1 and rb A In this case the product B A has the following matrix representation k B A = 1 u tk x1 k1u tk + utk 1 k11 k x1 k11 k + uk t Denote by z the point rb A The fact that z is fixed point of B A gives us the equation k z tk 1z+x1 k 1 u = z uk 1z+ x1 k 1 t Since t z and u z we have that k 1z+x1 k 1 u = 0 if and only if k 1z + x1 k 1 t = 0; in which case u = t a contradiction In particular k = z uk 1 z+x1 k 1 t z tk 1 z+x1 k 1 u

20 176 Rubén A Hidalgo 4 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 C representing the transformations A 1 A n as follows I If r 1 then A 1 = k 1 r 1 1 k where k 1 = r 1r 1 r 1 r 1 s r 1 II If r 1 = then A 1 = 1 r s III If r = 0 then A = r 0 1 k r k where k = r 1r 1r s s r 1 r IV If r = 0 then A = 1 0 r 1 +s r 1 1 r 1 s 1 V If r j = a j j = 3 n and r 1 = then 1 + pj r A j = j p j rj p j 1 p j r j

21 Fixed Point Parameters 177 where p j = 1+s r r j s j s j r j +r s 1 VI If r j = a j j = 3 n and r 1 then 1 + pj r A j = j p j rj p j 1 p j r j where p j = 1 k 1 s j r 1 k 1 +1r js j +r 1 1 k 1 r j s j r j k 1 s j VII If r j a j j = 3 n and r 1 = then rj a A j = j kj r j a j kj 1 1 kj r j kj a j where k j = r j s j a j s j r +s +1 a j s j r j s j r +s +1 VIII If r j a j j = 3 n and r 1 then rj a A j = j kj r j a j kj 1 1 kj r j kj a j where k j = r j s j a j k 1 s j r 1 1 k 1 a j s j r j k 1 s j r 1 1 k 1 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 C representing the transformations A 1 A n as follows A 1 = k 1 r 1 1 k where k 1 = r 1r 1 r 1 r 1 t r 1 A = r 0 1 k r k where k = r 1r 1r t t r 1 r

22 178 Rubén A Hidalgo A j = rj a j k j a j r j k j 1 1 k j r j k j a j where k j = r j t j m j 1 a j t j 1 +t j 1 µ j t jµ j a jt j +t j 1 t j a j t j 1 µ j +a jt j µ j a j t j m j 1 r j t j 1 +t j 1 µ j t jµ j r jt j +t j 1 t j r j t j 1 µ j +r jt j µ j and m j 1 is the attracting fixed point of the transformation A j 1 A j A A 1 with multiplier µ j where µ j > 1 The above values of m j and µ j are obtained in an inductive way where m = 1 and µ is the multiplier of A A 1 5 Computing Models for Some Teichmüller Spaces 51 Teichmüller Spaces of Riemann Surfaces of Type 04 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 + 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 and α 3 be simple loops on S through the point z as shown in figure 1 Figure 1

23 Fixed Point Parameters 179 The fundamental group of S at the point z S has a presentation Π 1 S z =< α 1 α α 3 > = Γ 3 In this way we have that Γ is a free group of rank 3 generated by A 1 A and A 3 so that the transformations A i are hyperbolic and the axis of these transformations A 1 A A 3 A A 1 and A 3 A 1 are shown in figure Figure 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 B 3 satisfying the following 1 G is discrete subset of P GL + R P GL C ab 1 = ab = 0 and ab B 1 = 1 3 There exists a quasiconformal homeomorphism F : H H such that F A i F 1 = B i for i = 1 3 As a consequence of theorem 1 we have one-to-one real analytic map

24 180 Rubén A Hidalgo Φ 3 : T Γ = V 3 R 6 defined by Φ 3 G B 1 B B 3 = r 1 r r 3 a 3 s 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 r 3 a 3 s and s 3 In fact u = r 1 r 1 a 3 r 1 r 3 r 1 1r 1 s r 1 r 1 r r 1 s 3 The axis of the transformations B 1 B B 3 B B 1 and B 3 B 1 have the same topological configuration as the axis of the transformations A 1 A A 3 A A 1 and A 3 A 1 respectively In particular the fixed points of the above transformations satisfy the following inequalities E1 r < 0 E 1 < s < r 1 < s 3 < u < a 3 < r 3 Let us consider the parameter space R as the open subset of R 6 consisting of the tuples r 1 r r 3 a 3 s s 3 satisfying the inequalities given by E1 and E In particular Φ 3 T Γ is contained R Corollary Φ 3 T Γ =R Proof We have to show that for any point p = r 1 r r 3 a 3 s s 3 contained in the region R there is a normalized marked group G B 1 B B 3 in V 3 so that Φ 3 G B 1 B B 3 = p For p as above we can construct Möbius transformations k B 1 = 1 r 1 1 k1 0 1 B = r 0 1 k r k

25 Fixed Point Parameters 181 B 3 = r3 a 3 k 3 r 3 a 3 k k 3 r 3 k 3 a 3 where k 1 = r 1 r 1 r 1 r 1 s r 1 ; k = r 1r 1r s s r 1 r ; k 3 = r 3 s 3 a 3 s 3 r +s +1 a 3 s 3 r 3 s 3 r +s +1 The inequalities E1 and E ensure that the transformations B 1 B B 3 B B 1 and B 3 B 1 are hyperbolic We also have that ab 1 = rb 1 = r 1 ab = 0 rb = r ab 3 = a 3 rb 3 = r 3 ab B 1 = 1 rb B 1 = s ab 3 B 1 = r 1 r 1 a 3 r 1 r 3 r 1 1r 1 s r 1 r 1 r r 1 s 3 and rb 3 B 1 = s 3 We denote the axis of the transformations B 1 B B 3 B B 1 and B 3 B 1 by N 1 N N 3 N 1 and N 31 respectively as shown in figure 3 Construct the geodesics: Figure 3 L 1 common perpendicular of N 1 and N

26 18 Rubén A Hidalgo L common perpendicular between N and N 1 L 3 common perpendicular between N 1 and N 1 M 1 common perpendicular between N 1 and N 3 and M common perpendicular between N 3 and N 31 M 3 common perpendicular between N 1 and N 31 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 M 1 and M 3 d = log r 1 r 1 r r 1 s r 1 1 In particular we have B 1 = R 1 R 3 = S 1 S 3 B = R R 1 and B 3 = S S 1 We consider the groups G 1 =< B 1 B > and G =< B 1 B 3 > 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 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 Figure 4

27 Fixed Point Parameters 183 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 Remark The angle involve in gluing the pants corresponding to the groups G 1 and G in the above proof is given by θ = θr 1 r r 3 a 3 s s 3 = π r log 1 a 3 r 1 r 3 r log 1 r 1 r r 1 s r 1 1 r 1 r 1 r We can either make r = 0 or s = 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 = 0 s = 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 1r 3 1 r 1 = r 3 r 3 r 1 r 1 The explicit model is M = {r 1 r 3 R ; 1 < r 1 < r 3 } In figure 5 we draw a fundamental domain for the group G =< B 1 B B 3 > such that Φ 3 G =< B 1 B B 3 > = r 1 0 r 3 r 3 1 u

28 184 Rubén A Hidalgo Figure 5 π In this case the angle θ = θr 1 r 3 is θ = θr 1 r 3 = log r 3 r 1 log r 1 r 1 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 defined by Lr 1 r 3 = log r 1 θr r 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 5 Teichmüller Spaces of Riemann Surfaces of Type 11 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

29 Fixed Point Parameters 185 A hyperbolic Riemann surface S of type 1 1 can be constructed from a pant P where two of its boundaries are bounded by closed simple geodesics of the same length by identifying these two boundaries This can be seen as follows Start with a Fuchsian group Γ uniformizing a surface S of type 1 1 Let l be a non-dividing simple closed geodesic on S and denote by R = S l Fix a lifting T of the region R in the hyperbolic plane and consider G 1 the subgroup of G fixing T The group G 1 uniformizes a pant P with two boundaries bounded by simple closed geodesics of the same length We may assume up to conjugation that G 1 =< A 1 A > is normalized by aa 1 = aa = 0 and aa A 1 = 1 and the axis of A 1 and A are projected to the two geodesics of the same length If r 1 = ra 1 r = ra and s = ra A 1 then the axis of the transformations A 1 A and A A 1 denoted as N 1 N and N 1 respectively are as shown in figure 6 A fundamental domain for G 1 is also shown in figure 6 Figure 6

30 186 Rubén A Hidalgo As a consequence of Theorem 1 we have that the multiplier of the transformations A 1 and A are given by k 1 = r 1r 1 r r 1 1r 1 s and k = r 11 r s r sr 1 r respectively Our assumption on the equality of the geodesics lengths implies that s = r 1 r 1 1 r Since s > 1 we have the inequality r r 1 < 1 Observe that r < 0 and r 1 > 1 imply that r 1 r 1 1 r < r 1 On the other hand both geodesics N 1 and N project onto l on S It implies that there is a transformation A 3 in Γ such that A 3 A A 1 3 = A 1 1 We denote by D and D 1 the discs bounded by N and N 1 respectively where the boundary of D 1 contains r and the boundary of D contains r We have that A 3 D 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 =< A 3 > In particular Γ is the HNN-extension of G 1 by A 3 see figure 7 for a fundamental domain of Γ Figure 7

31 Fixed Point Parameters 187 Denote by a and r the attracting and repelling fixed points of A 3 respectively In this case r < r < 0 and r 1 < a If k > 1 denotes the multiplier of A 3 then A 3 = r ak ark 1 1 k rk a Since necessarily A 3 0 = r 1 and A 3 r = we obtain a = r 1r r and k = r r 1 r The inequality r > r rr r implies the inequality r 1 < a In this way we obtain that the only variables are given by r 1 r and r satisfying the inequalities F1 r r 1 < 1; and F r < r < 0 < 1 < r 1 The Teichmüller space of Γ is identified with the set W of marked groups G B 1 B B 3 satisfying: 1 G is discrete subset of P GL + R; ab 1 = ab = 0 and ab B 1 = 1; and 3 There is a quasiconformal homeomorphism F : H H such that F A i F 1 = B i for i = 1 3 In particular for G B 1 B B 3 in W the only relation is given by B 3 B B3 1 = B1 1 Let us consider the region H in R 3 consisting of the points r 1 r r satisfying the inequalities F1 and F As a consequence of theorem 1 we have a one-to-one real analytic map Θ : W H defined by ΘG B 1 B B 3 = rb 1 rb rb 3 Corollary 3 ΘW =H

32 188 Rubén A Hidalgo Proof Start with a point r 1 r r in H Set s = r 1 r 1 1 r and a = r 1r Inequality F1 ensure that 1 < s < r r 1 and F ensure that r 1 < a Using the values r 1 r and s we can construct a pant group G 1 =< B 1 B > where ab 1 = rb 1 = r 1 ab = 0 rb = r ab B 1 = 1 Direct computations shows that rb B 1 = s We also construct B 3 satisfying rb 3 = r ab 3 = a and k = a r r r Inequality F ensures that B 3 maps the disc D bounded by the geodesic joining r and 0 and containing r in the boundary onto the disc D 1 bounded by the geodesic joining and r 1 and containing r in the boundary One can see that the conditions of the second combination theorem of Maskit holds for G 1 and G =< B 3 > As a consequence of the same theorem we have that the group G generated by G 1 and G has the presentation G =< G 1 G >=< B 1 B B 3 ; B 3 B B 1 3 = B 1 1 > and G B 1 B B 3 belongs to H As a consequence of Theorem 4 we have that H is an explicit model for the Teichmüller space of hyperbolic surfaces of type 1 1 This model only uses the fixed points of some elements Compare to Maskit s model in [5] in which one of the parameters is a multiplier In the boundary of H given by r r 1 = 1 we have an explicit model for the Teichmüller space of parabolic Riemann surfaces of type 1 1 In this case the model is given by N = {r 1 r; 1 r 1 < r < 0 < 1 < r 1 } Figure 8 shows a fundamental domain for a marked group G B 1 B B 3 obtained from a point r 1 r in N Figure 8

33 Fixed Point Parameters 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 β 1 and β as shown in figure 9 Figure 9 The fundamental group of S with base point at z has a presentation of the form Π 1 S z =< γ α 1 α β 1 β ; α 1 1 β 1 α 1 = γβ 1 α 1 β α = γβ > Let F be a Fuchsian group acting on the hyperbolic plane H = {z C; lmz > 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 πx = 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 [η]

34 190 Rubén A Hidalgo then f ρ = f η We define λ[η] = f η We set A 1 = λ[γ] A = λ[β 1 ] A 3 = λ[β ] F 1 = λ[α 1 ] and F = λ[α ] In particular a presentation of F is given by F =< A 1 A A 3 F 1 F ; F 1 A F 1 1 = A A 1 F A 3 F 1 = A 3 A 1 > Denote by γ α 1 α β1 and β the projections on S under π of the axis of the transformations A 1 F 1 F A and A 3 respectively as shown in figure 10 Figure 10 We have oriented the axe AxH 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 F in P GL R such that aa 1 = aa = 0 and aa A 1 = 1 The choice made for the transformations A 1 A A 3 F 1 and F imply

35 Fixed Point Parameters 191 that the axis of these transformations are as shown in figure 11 Figure 11 For simplicity we denote aa 3 = a 3 ra i = r i i = 1 3 aa 3 A 1 = u ra j A 1 = s j j = 3 rf k = x k af k = y k k = 1 In particular we have that these fixed points satisfy the following inequalities: r < x 1 < 0 < 1 < y 1 < s < r 1 < s 3 < y < u < a 3 < x < r 3 From the above we observe that the group G 1 = A 1 A 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 β 1 and πaxa A 1 and the loops β and πaxa 3 A 1 respectively We denote by D D D 3 and D 3 the disjoint discs bounded by the axis of the transformations A A A 1 A 3 and A 3 A 1 respectively Since F 1 A F1 1 = A A 1 and F A 3 F 1 = A 3 A 1 we have that F 1 D D = and F 1 D 3 D 3 = 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 =< F 1 > The resulting group G 3 =< G 1 G >= G 1 F1 uniformizes a surface of genus one with two boundary components of the same length Now we apply again the second combination theorem of Maskit

36 19 Rubén A Hidalgo to the pair G 3 and G 4 =< F > to obtain the group F =< G 3 G 4 >= G 3 F uniformizing the surface S For the group F in we have some equalities We denote by k H > 1 the multiplier of the hyperbolic transformation H Then we have the following: 1 k A = k A A 1 ; k A 3 = k A 3 A 1 ; 3 F 1 0 = 1 F 1 r = s ; 4 F a 3 = u F r 3 = s 3 ; 5 k A 1 = r 1r 1 r 1 r 1 s r 1 ; 6 k A = r 1r 1r s s r 1 r ; 7 k A 3 = r 3 s 3 a 3 k A 1 s 3 r 1 1 k A 1 a 3 s 3 r 3 k A 1 s 3 r 1 1 k A 1 Direct computations imply that u = r 1 1 k A 1 +r 1 k A 1 s 3 a 3 r 3 +a 3 r 3 k A 1 s 3 r 1 Equality 1 implies s = r 1r 1 r r 1 Equality and the fact that r 1 < s 3 imply s 3 = a 31 r 1 +r 1 r 1 r r 1 As a consequence we obtain that u = r 11 r +r 3 r 3 r 1 r The transformation F k is uniquely determined by x k y k and k F k Equality 3 implies y 1 = r 11 x 1 r +x 1 r x 1 1 r 1 +r +r 1 r and k F 1 = x 1 1r 1 x 1 1+r x 1 r x 1 r 1 1r x 1 Equality 4 implies y = P Q and k F = a 3+r 1 +a 3 r 1 r 1 x +r x r 1 r r 1 r 1 x +r x r 1 r r 3 +r 1 r 3 a 3 x r 1 1r 1 r x r 3 where P = a 3 r 1 a 3 r 1 x r 1 a 3 r 1 +a 3 r 1x +a 3 r x a 3 r 1 r r 1 r x a 3 r 1 r x + r 1r +a 3 r 1r r 1r x +r 1 r x r 1r a 3 r 3 +r 1 r 3 +3a 3 r 1 r 3 r 1 r 3 x r 1r 3 a 3 r 3 1r 3 +r 1r 3 x a 3 r r 3 +r r 3 x r 1 r r 3 +a 3 r 1 r r 3 r 1 r r 3 x + r 1r r 3 and Q = r r 1 a 3 +x +r 1 +a 3 r 1 r 1 x +r x r 1 r r 3 +r 1 r 3 We observe from the above that the group F is uniquely determined by the fixed points r 1 r r 3 a 3 x 1 and x We consider the open connected region in R 6 F defined as:

37 Fixed Point Parameters 193 F = {r 1 r r 3 a 3 x 1 x ; r < x 1 < 0 < 1 < y 1 < s < r 1 < s 3 < y < u < a 3 < x < r 3 } where the values y 1 s s 3 y and u are given by the formulae above The Teichmüller space T F can be identified with the space subspace of the suitable deformation space consisting of marked groups G B 1 B B 3 H 1 H satisfying the following properties: 1 G is a discrete subgroup of P GL + R; ab 1 = ab = 0 ab B 1 = 1; 3 There exists a quasiconformal homeomorphism K : H H such that K A i K 1 = B i for i = 1 3 and K F j K 1 = H j for j = 1 In particular for G B 1 B B 3 in T F the only relations are given by H 1 B H1 1 = B B 1 and H B 3 H 1 = B 3 B 1 > We have as a consequence of the previous sections an one-to-one real analytic map φ : T F F defined as φg B 1 B B 3 H 1 H = rb 1 rb rb 3 ab 3 rh 1 rh Corollary 4 φt F =F 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 r 3 a 3 x 1 x in F we can construct the values s s 3 u y 1 and y We can also construct values k H 1 and k H Using these values we obtain unique transformations B 1 B B 3 H 1 and H 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 B 3 B B 1 B 3 B 1 H 1 and H is the same as for the axis of the transformations A 1 A A 3 A A 1 A 3 A 1 F 1 and F as shown in figure 11 The group

38 194 Rubén A Hidalgo T 1 =< B 1 B > is a free group of rank two uniformizing a pant The group T =< 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 >= T 1 <B1 >T uniformizes a surface of type 0 4 It is easy to check that H k B k+1 Hk 1 = B k+1 B 1 for k = 1 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 H1 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 > to obtain that the marked group T 7 =< T 5 T 6 >= T 5 H belongs to T F 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 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 AxA 1 AxA and AxA 3 we have associated the multiplier of the transformations A 1 A 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 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 to both AxA 1 and AxA respectively AxA 1 and AxA 3

39 Fixed Point Parameters 195 These two geodesics determine an arc in AxA 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 AxA 1 and AxA A 1 We proceed to see the hyperbolic arc in AxA A 1 determined by the intersection point of L 3 with AxA A 1 and the image by F 1 of the intersection point of L 3 with AxA Similarly for looking at the angle at α The explicit computations are similar to the case 0 4 and are left to the interested reader 4 We can apply Theorem 1 to construct explicit models for the Teichmüller space of arbitrary topologically finite Riemann surfaces This will be done elsewhere References [1] W Abikoff The real analytic theory of Teichmüller space Lecture Notes in Mathematics 80 Springer 1980 [] RA Hidalgo The noded Schottky space I Proc London Math Soc 73 pp [3] R A Hidalgo and G Labbe Fixed point parameters for Teichmüller space of closed Riemann surfaces Revista Proyecciones 19 pp [4] B Maskit Parameters for Fuchsian groups I: Signature 04 Holomorphic Functions and Moduli II pp Math Sci Res Inst Pub 11 Springer -Verlag New York 1988 [5] B Maskit Parameters for Fuchsian groups II: topological type 11 Annal Acad Sci Fenn Ser AI 14 pp [6] B Maskit New parameters for fuchsian groups of genus two Preprints

40 196 Rubén A Hidalgo [7] B Maskit Kleinian Groups Grundlehren der Mathematischen Wissenschaften Vol 87Springer - Verlag Berlin Heildelberg New York 1988 [8] I Kra and B Maskit The deformation space of a Kleinian group Amer J Math 103 pp [9] C Min New parameters of Teichmüller spaces of Riemann surfaces of genus two Ann Acad Scie Fenn Series AI Mathematica Received : March 000 Rubén A Hidalgo Departamento de Matemática Universidad Técnica Federico Santa María Casilla 110-V Valparaíso Chile