EMBEDDING OF THE TEICHMÜLLER SPACE INTO THE GOLDMAN SPACE. Hong Chan Kim. 1. Introduction

Transcription

1 J. Korean Math. Soc , No. 6, pp EMBEDDING OF THE TEICHMÜLLER SPACE INTO THE GOLDMAN SPACE Hong Chan Kim Abstract. In this paper we shall explicitly calculate the formula of the algebraic presentation of an embedding of the Teichmüller space TM into the Goldman space GM. From this algebraic presentation, we shall show that the Goldman s length parameter on GM is an isometric extension of the Fenchel-Nielsen s length parameter on TM. 1. Introduction A convex real projective structure on a smooth surface M is a representation of M as a quotient space Ω/Γ of a convex domain Ω RP by a discrete group Γ PGL3, R acting properly and freely. If χm < 0, then the equivalence classes of convex real projective structures on M form a deformation space GM called the Goldman space. The study of RP structures has been quite active. Ehresmann, Kuiper, Benzécri, Kobayashi, and Thurston have done important work. Recently Goldman and S. Choi lead this field. The deformation space of hyperbolic structures on M is called the Teichmüller space and denoted by TM. Choi and Goldman [] proved GM is a component of the deformation space RP M of real projective structure on M and GM contains the Teichmüller space TM. Goldman [5] defined the length parameters l, m on GM. They seems to be an extension of Fenchel-Nielsen s length parameter l on TM. But during the calculation of the translation length of Goldman and Fenchel-Nielsen s parameters, author realized that they do not fit. Thus we define the modified Goldman s length parameters on GM. Received June 1, 005. Revised November 10, Mathematics Subject Classification: 58D7, 53D30. Key words and phrases: convex real projective structure, Hilbert metric, Teichmüller space, Goldman space, length parameter. The author gratefully acknowledges the support from a Korea University Grant.

2 13 Hong Chan Kim The purpose of this paper is to formulate the explicit algebraic presentation of an embedding of TM into GM which isometrically extends Fenchel-Nielsen s length parameter on TM to the modified Goldman s length parameters on GM. In Section, we recall some preliminary definitions about G, X- structures on a smooth manifold M. In Section 3, we describe the relation between the deformation space DM of G, X-structures on M and the orbit space Homπ, G /G. In Section 4, we recall the positive hyperbolic elements of SL3, R. In Section 5, we give some knowledge about the Hilbert metric. In Section 6, we compare the relations between the Poincaré metric and the Hilbert metric on the unit disc. In Section 7, we define an embedding formula of hyperbolic structures into convex real projective structures. To realize a hyperbolic structure on M as a convex real projective structure, we define an isometry H Ω and an embedding PSL, R SL3, R, where Ω is a strictly convex subset of RP with the conic boundary. In Section 8, we show the modified Goldman s length parameters l, m on GM isometrically extend Fenchel-Nielsen s length parameter l.. Preliminaries The followings are from Kim s paper [7]. For more detail see Kim [7]. Let X be a smooth manifold and G a connected algebraic Lie group. We assume that G acts on X strongly effectively; that is, if g 1, g G agree on a nonempty open set of X, then g 1 = g. We start this section with examples of strongly effective action. Example.1. Let H = {z C Imz > 0} be the upper half plane. Then SL, R acts on H by a b.1 A z = z = az + b c d cz + d. Since we have A z = A z for any A SL, R and z H, the Lie group PSL, R = SL, R/±I acts strongly effectively on H. Example.. Let RP be the space of all lines through the origin in R 3. For a nonzero vector v in R 3, the corresponding point in RP will be denoted by [v]. Then GL3, R acts on RP by. B [v] = [Bv]. Since the scalar matrices R GL3, R acts trivially on RP, the Lie group PGL3, R = GL3, R/R acts strongly effectively on RP.

3 Embedding of Teichmüller space into Goldman space 133 Let Ω be an open subset of X. A map φ : Ω X is called locally- G, X if for each component W Ω, there exists g G such that φ W = g W. Since G acts strongly effectively on X, above element g is unique for each component. Clearly a locally-g, X map is a local diffeomorphism. A G, X-structure on a connected smooth n-manifold M is a maximal collection of coordinate charts {U α, ψ α } such that 1. G acts strongly effectively on X.. {U α } is an open covering of M. 3. For each α, ψ α : U α X is a diffeomorphism onto its image. 4. If two coordinate charts U α U β, then the transition function ψ β ψα 1 : ψ α U α U β ψ β U α U β is locally-g, X. The PSL, R, H -structures and PGL3, R, RP -structures on a smooth surface M are called the hyperbolic structures and real projective structures on M respectively. A smooth manifold equipped with a G, X-structure is called a G, X-manifold. If N is a G, X-manifold and f : M N is a local diffeomorphism, then we can give the induced G, X-structure on M via f. In particular every covering space of a G, X-manifold has the canonically induced G, X-structure. A smooth map f : M N of G, X-manifolds is called a G, X- map if for each coordinate chart U, ψ U on M and V, ψ V on N, the composition ψ V f ψ 1 U : ψ Uf 1 V U ψ V fu V is locally- G, X. If f : M N is a diffeomorphism such that f and f 1 are G, X-maps, then f is called a G, X-diffeomorphism. The following Development Theorem is the fundamental fact about G, X-structures. See Thurston s book [13] for details. Theorem.3. Let p : M M denote a universal covering map of a G, X-manifold M, and π the corresponding group of covering transformations. Then 1. There exist a G, X-map dev : M X called the developing map and homomorphism h : π G called the holonomy homomorphism such that for each γ π the following diagram commutes: γ M dev X M dev hγ X

4 134 Hong Chan Kim. Suppose dev, h is another pair satisfying above conditions. Then there exists g G such that dev = g dev and h = ι g h, where ι g : G G denotes the inner automorphism defined by g. By Theorem.3, the developing pair dev, h is unique up to the G-action by composition and conjugation respectively. 3. Deformation space of G, X-structures For a smooth manifold M, consider a pair f, N where N is a G, X- manifold and f : M N is a diffeomorphism. Then M admits the induced G, X-structure via f. The set of all such pairs f, N is denoted by AM. Then AM is the space of all G, X-structures on M. We say two pairs f, N and f, N in AM are equivalent if there exists a G, X-diffeomorphism g : N N such that g f is isotopic to f. The set of equivalence classes AM/ will be denoted by DM and called the deformation space of G, X-structures on M. The deformation space DM has the natural topology. Let DiffM 0 be the space of all diffeomorphisms of M which are isotopic to the identity map I M. Then we may think the deformation space DM consists of diffeomorphisms f : M N to a G, X-manifold N modulo the action of DiffM 0 given by g : f f g where g DiffM 0. Give DM the quotient topology induced from the C -topology on the space of diffeomorphisms f : M N. Definition 3.1. Let M be a connected smooth surface. The deformation space of hyperbolic structures on M is called the Teichmüller space and denoted by TM. The deformation space of real projective structures on M is denoted by RP M. The deformation space DM is closely related to Homπ, G/G the orbit space of homomorphisms φ : π G. The group G acts on Homπ, G by conjugation; that is, for g G and φ Homπ, G, the action g φ is defined by g φγ = g φγ g 1 where γ π. If G is an algebraic Lie group, then Homπ, G is an algebraic variety. But generally Homπ, G is not smooth. Suppose φ Homπ, G and Zφ is the centralizer of φπ in G. Goldman [3] showed φ is a nonsingular point of Homπ, G if and only if dim Zφ/ZG = 0, where ZG denotes the center of G. Let Homπ, G be the set of nonsingular points of Homπ, G. Then G acts freely on the smooth Zariski open subset Homπ, G. But unfortunately Homπ, G /G is generally not Hausdorff. Let Homπ, G be the subset of Homπ, G consisting

5 Embedding of Teichmüller space into Goldman space 135 of homomorphisms whose image does not lie in a parabolic subgroup of G. Then Homπ, G is a Zariski open subset of Homπ, G, and Homπ, G /G is a Hausdorff smooth manifold of dimension dim G χm. For more detail see Goldman s paper [3]. Suppose M is a closed surface. Taking the holonomy homomorphism of a G, X-structure defines a map hol : DM Homπ, G /G which is a local homeomorphism. See Goldman [4] and Johnson-Millson [6] for details. For a hyperbolic structures on M, hol : TM Homπ, PSL, R /PSL, R is an embedding onto a real analytic manifold of dimension 3 χm = 6g 6. Thus the Teichmüller space TM is diffeomorphic to R 6g 6 and an element of TM will be identified with a conjugacy class of Homπ, PSL, R. Furthermore the developing map dev is a diffeomorphism from M onto a convex domain Ω = dev M H. In this case the holonomy homomorphism h is an isomorphism from π onto a discrete subgroup Γ = hπ PSL, R which acts properly and freely on Ω. Thus M is diffeomorphic to the quotient space Ω/Γ. But for a real projective structures on a closed surface M, hol : RP M Homπ, PGL3, R /PGL3, R is just a local homeomorphism. And the developing map is just a local diffeomorphism and the developing image may be not convex. We can find such examples in Sullivan and Thurston s paper [1]. We consider the convex real projective structures on M. A domain Ω RP is called convex if there exist a projective line l RP such that Ω RP l and Ω is a convex subset of the affine plane RP l. A real projective structure on M is called convex if the developing map dev : M RP is a diffeomorphism onto a convex domain in RP. The following fundamental theorem is due to Goldman [5]. Theorem 3.. Let M be a closed real projective surface. Then the following statements are equivalent. 1. M has a convex real projective structure.. M is projectively diffeomorphic to a quotient space Ω/Γ, where Ω RP is a convex domain and Γ PGL3, R is a discrete group acting properly and freely on Ω. Definition 3.3. The Goldman space GM is the subset of RP M consisting of the equivalence classes of convex real projective structures.

6 136 Hong Chan Kim Choi and Goldman [], [5] proved that if M is a closed real projective surface, then GM is a component of RP M and the restriction hol : GM Homπ, PGL3, R /PGL3, R is an embedding onto a real analytic manifold of dimension 8 χm = 16g 16. Thus the Goldman space GM is diffeomorphic to R 16g 16. The Goldman space GM is an analogue of the Teichmüller space TM. Classically known that TM embeds in GM. That means every hyperbolic structure on M defines a convex real projective structure on M. We shall explicitly calculate the formula of the algebraic presentation of an embedding of TM into GM. 4. Positive hyperbolic elements An element A of SL, R is said to be hyperbolic if A has two distinct real eigenvalues. Since the characteristic polynomial of A is fλ = λ tλ + 1, where t = tra, A is hyperbolic if and only if tra > 4. Let A be an element of PSL, R. Since the absolute value of trace is still defined, A PSL, R is said to be hyperbolic if tra >. A hyperbolic element A in PSL, R can be expressed by the diagonal matrix 4.1 [ α α ] let α 1 0 = ± 0 α via an SL, R-conjugation where α > 1. The homomorphism GL3, R SL3, R defined by B detb 1/3 B induces an isomorphism PGL3, R = GL3, R/R SL3, R. Thus from now on we shall identify the groups PGL3, R and SL3, R. An element B SL3, R is called positive hyperbolic if it has three distinct positive real eigenvalues. If B is positive hyperbolic, then it can be represented by the diagonal matrix 4. λ µ ν via an SL3, R-conjugation where λ µ ν = 1 and 0 < λ < µ < ν. The following theorem is one of the analogues between hyperbolic structures and convex real projective structures proved by Kuiper [9]. Theorem 4.1. Let M be a closed oriented surface with χm < 0.

7 Embedding of Teichmüller space into Goldman space If M is a hyperbolic surface, then every nontrivial element of holonomy group Γ PSL, R is hyperbolic.. If M is a convex real projective surface, then every nontrivial element of holonomy group Γ SL3, R is positive hyperbolic. 3. Either the boundary of the developing image is a conic in RP or is not C 1+ε for any ε > 0. It is known that the boundary Ω is a conic if and only if the convex real projective structure on M arises from a hyperbolic structure on M. Let Ω be the domain in RP defined by 4.3 Ω = {[x 1, x, x 3 ] RP x 1 + x x 3 < 0}. Then Ω has a conic boundary Ω. Let M be a surface with a hyperbolic structure. Composing the developing map M H with an isometry H Ω RP and the holonomy homomorphism π PSL, R with an embedding PSL, R SL3, R realizes M as a convex real projective surface. Thus we shall define an isometry H Ω and an embedding PSL, R SL3, R. 5. The Hilbert metric To define an isometry H Ω, we need a knowledge about the Hilbert metric. Hilbert discovered a metric on a bounded convex domain Ω in R or equivalently in C. This metric is related to the hyperbolic metric but geodesics are still Euclidean segments. We recall some basic definitions and properties. For more detail, see Kobayashi s paper [8]. Definition 5.1. Let z 1, z, z 3, z 4 be four points in the extended complex numbers Ĉ = C { } such that at least three points are distinct. The cross-ratio of z 1, z, z 3, z 4 is defined by 5.1 [z 1, z, z 3, z 4 ] = z 1 z 3 z z 4 z 1 z z 3 z 4. There are six different methods to define the cross-ratio on Ĉ. We adopt this presentation since it is an easy way to understand the Hilbert metric. The properties of this presentation are: 1. [z 1, z, z 3, z 4 ] = [z 4, z 3, z, z 1 ].. [z 1, z, z 3, z 4 ] > 1 if distinct four points lie in a line segment such that z is between z 1 and z 3 and z 3 is between z and z 4.

8 138 Hong Chan Kim Definition 5.. Let Ω be a bounded convex domain in C. For a distinct points z, w in Ω, the Hilbert distance between z and w is defined by 5. d H z, w = log [z, z, w, w ], where z, w are the boundary points in Ω which lie on the straight line joining z and w such that z lies between z and w. If we add d H z, z = 0, then the Hilbert distance d H defines a complete metric on Ω. A strictly convex domain is a convex domain whose boundary Ω does not contain any line segment. Since a strictly convex domain Ω does not contain any full straight line, the Hilbert distance d H still defines a complete metric on Ω and has the following properties: 1. There is a unique geodesic between two points in Ω.. The geodesics are straight lines in Euclidean sense. Kuiper [9] showed that the developing images Ω RP of convex real projective structures are strictly convex domains. This yields that every developing image of a convex real projective structure of a closed surface has the complete Hilbert metric. Let a 1, a, a 3, a 4 be collinear distinct four points in RP. Then there exist corresponding four nonzero vectors v 1, v, v 3, v 4 of R 3 which are contained in a plane P R 3 ; that is, a k = [v k ] = [x k, y k, s k ] for each k. Remark 5.3. Since we use the z for the complex variable z = x + iy, the s will be used for the third Euclidean coordinate in R 3. If P is the xy-plane, then the cross-ratio is defined by [ x1 5.3 [a 1, a, a 3, a 4 ] :=, x, x 3, x ] 4. y 1 y y 3 y 4 If P is not the xy-plane, then the cross-ratio is defined by 5.4 [a 1, a, a 3, a 4 ] := [ x1 + i y 1 s 1, x + i y s, x 3 + i y 3 s 3, x 4 + i y 4 s 4 Proposition 5.4. Suppose z 1, z, z 3, z 4 are distinct four points which are contained in a straight line in C. Let z k = x k + i y k for each k. Then { [x1, x 5.5 [z 1, z, z 3, z 4 ] =, x 3, x 4 ] if x 1, x, x 3, x 4 are distinct [y 1, y, y 3, y 4 ] if y 1, y, y 3, y 4 are distinct. By virtue of Proposition 5.4, the cross ratio of collinear distinct four points a 1, a, a 3, a 4 of RP should be one of the followings: [ x1 [a 1, a, a 3, a 4 ] =, x, x 3, x ] [ 4 x1 or, x, x 3, x ] [ 4 y1 or, y, y 3, y ] 4 y 1 y y 3 y 4 s 1 s s 3 s 4 s 1 s s 3 s 4 ].

9 Embedding of Teichmüller space into Goldman space 139 when the cross ratios of the right hand side are well-defined. 6. Relation between the Poincaré and the Hilbert metric Let H P = H, d P be the upper half plane H = {z C Imz > 0} with the Poincaré metric d P ; i.e., the lines in H P are the semi-circles centered at the x-axis and the rays orthogonal to the x-axis. The Poincaré metric on H P is defined by 6.1 d P z, w = log [z, z, w, w ], where z, w are the boundary points in the extended x-axis ˆR = R { } which lie on the line joining z and w such that z is between z and w. The elements of PSL, R act on H P as the linear fractional transformations in.1. Since the linear fractional transformations on Ĉ preserve the cross-ratio, we have the following theorem. Theorem 6.1. The group I 1 of orientation preserving isometries of the upper half plane H P is {[ ] } a b 6. I 1 = PSL, R = PGL, R c d ad bc = 1. Let D P = D, d P be the unit disc D = {z C : z < 1} with the Poincaré metric d P ; i.e., the lines in D P are the arcs of circles which are orthogonal the boundary of D and the segments through the origin. The Poincaré metric on D P is defined similarly as in 6.1. The linear fractional transformation G 1 : Ĉ Ĉ defined by 6.3 w = G 1 z = z i iz + 1 maps { 1, 0, 1,, i} to { 1, i, 1, i, 0} respectively. Thus the restriction of G 1 to H P is an orientation preserving isometry onto D P with the inverse G 1 let 1 = F 1 such that 6.4 z = F 1 w = w + i iw + 1. The following Theorem 6. is well-known fact. We can find a similar result in Matsuzaki and Taniguchi s book [10] or Ratcliffe s book [11]. Theorem 6.. The group I of orientation preserving isometries of the Poincaré disk D P is {[ ] } α β 6.5 I = PGL, C β ᾱ α β = 1.

10 140 Hong Chan Kim Proof. The linear [ fractional ] transformations [ ] G 1 and F 1 correspond to 1 the matrices 1 i and i i in SL, C respectively. i 1 Thus the isometry of D P is the compositions of G 1 f 1 F 1, where f 1 is an isometry of H P in 6.. In the matrix representation, [ ] [ ] [ ] 1/ i/ G 1 f 1 F 1 = i/ 1/ a b 1/ i/ c d i/ 1/ [ a+d b c = + i b+c a d + i ] [ ] let α β b+c a d i a+d b c i = β ᾱ and α β = ad bc = 1. Conversely, if α = α 1 + i α, β = β 1 + i β for α 1, α, β 1, β R are the complex numbers in the group I, then the corresponding element in I 1 is [ ] α1 + β 6.6 β 1 + α β 1 α α 1 β and α 1 + β α 1 β β 1 + α β 1 α = α β = 1. This completes the proof. Let D H = D, d H be the unit disc D with the Hilbert metric d H ; i.e., the lines in D H are the Euclidean line segments. Figure 1. The Poincaré and the Hilbert metric on D Let Σ = {x, y, s R 3 x + y + s = 1 } be the unit sphere in R 3 with the north pole n = 0, 0, 1. Consider the stereographic projection

11 Embedding of Teichmüller space into Goldman space 141 P : Σ {n} R defined by P x, y, s = x 1 s, y. 1 s Then P is a conformal diffeomorphism with the inverse P 1 such that P 1 u u, v = 1 + u + v, v 1 + u + v, 1 + u + v 1 + u + v. Let Σ = {x, y, s Σ s < 0 } be the lower hemisphere of Σ. Then the restriction of P 1 : R Σ {n} to the unit disk D is diffeomorphic to the lower hemisphere Σ. See Figure. Figure. Inverse of the stereographic projection P 1 We define a mapping G : D P D H by G = p xy P 1, where p xy : R 3 R is the projection to the xy-plain; i.e., u 6.7 G u, v = 1 + u + v, v 1 + u + v. Then G is a diffeomorphism with the inverse G 1 = F : D H D P x 6.8 F x, y = x y, y x y In the complex variables w D P, z D H, the mappings G, F are represented by 6.9 z = G w = w 1 + w, w = F z = let z z. The result in Proposition 6.3 is also found in Thurston s book [13].

12 14 Hong Chan Kim Proposition 6.3. The mappings G : D P D H and F : D H D P preserve the lines in D P and D H. Proof. First we will show that F carries the chords in D H to the arcs in D P which are orthogonal to the boundary D P. Using the rotations on D H, we may assume the cord is {x = a} for 0 a < 1. If a = 0, then the image of {x = 0} is itself {u = 0}. Suppose 0 < a < 1 and u, v = F a, y D P. Then a 6.10 u = a y, v = y a y. Since a 0, u is also non-vanishing. From 6.10, we get the relation y = v a u. Plug in y = v a u to the left equation of 6.10, we obtain the following equation: 6.11 u 1 a + v = 1 a a. Therefore the image of the cord {x = a} is the part of the circle centered at C = 1 a, 0 with the radius 1 a a. Let O be the origin of R and A, B the points in D P which intersect with the circle Then A = a, 1 a and B = a, 1 a. We can easily compute OA + AC = a + 1 a 1 a + = 1 a = OC. By the Pythagorean theorem, OAC = π/. Similarly we can show OBC = π/. Therefore the image of the cords in D H are the arcs in D P which are orthogonal to the boundary D P. Conversely, let l P be an arc in D P which is orthogonal to the boundary D P. Then l P is the F -image of the chord l H joining the boundary points of the arc l P. Since G : D P D H is the inverse mapping of F, G l P = G F l H = l H. Thus G maps the lines in D P to the lines in D H. Unfortunately the mapping F : D H D P is not an isometry. Through a little modification of the Hilbert metric, we can show that F : D H D P is an isometry. Let D H = D, d H be the unit disc D with the modified Hilbert metric d H defined by 6.1 d H z, w = 1 d Hz, w. The following Theorem 6.4 is also found in Thurston s book [13]. I give another proof of it. a

13 Embedding of Teichmüller space into Goldman space 143 Theorem 6.4. The mapping F : D H D P is an isometry. Proof. For z 1, z D H, let w j = F z j D P. Since d P w 1, w = log [w 1, w 1, w, w ] and d H z 1, z = 1 log [z 1, z 1, z, z ], it is equivalent to show that 6.13 [w 1, w 1, w, w ] = [z1, z 1, z, z] 1. Figure 3. The image of lines through F : D H D P Without loss of generality we assume that z 1, z lie on the chord {x = a} where 0 a < 1. Let z 1 = a+b i, z = a+c i for b < c, then the boundary points of line segment in D H are z1 = a 1 a i, z = a a i. From the proof of Proposition 6.3, we know w 1 = z 1 and w = z. Therefore the cross-ratio [w 1, w 1, w, w [ ] is a 1 a a + b i i, a b, a + c i a c, a + ] 1 a i a 1 a i = a 1 a i a+c i 1+ 1 a c a+b i 1+ 1 a b a+b i 1+ 1 a b a+c i 1+ 1 a c a + 1 a i a + 1 a i let = B C A D = A B C D. Since z 1 = a + b i is a point in the unit disc D H, we have 1 a > b. Thus 1 a b > 0, 1 a + b > 0. It follows a b = 1 a b 1 1 a + b 1. Hence 1+ 1 a b A = 1+ 1 a b a+ 1 a i a+b i = 1 a b i + 1 a b a + 1 a i = 1 a b 1 1 a b 1 i + 1 a + b 1 a + 1 a i

14 144 Hong Chan Kim let = 1 a b 1 α. Through the similar computations, we have the followings: where a b A = 1 a b 1 α, a c B = 1 a + c 1 β, a b C = 1 a + b 1 γ, a c D = 1 a c 1 δ, α = 1 a b 1 i + 1 a + b 1 a + 1 a i, β = 1 a + c 1 i 1 a c 1 a + 1 a i, γ = 1 a + b 1 i 1 a b 1 a + 1 a i, δ = 1 a c 1 i + 1 a + c 1 a + 1 a i. By simple calculation we can get the relation α β = γ δ. Therefore the theorem is proved as follow: [ w 1, w 1, w, w ] A = C B D = a b A a b C a c B a c D = 1 a b 1 α 1 a + b 1 γ 1 a + c 1 β 1 a c 1 δ = 1 a b 1 1 a + c 1 1 a + b 1 1 a c 1 1 a = b 1 1 a + c 1 a + b 1 a c [ = a 1 a i, a + b i, a + c i, a + 1 a i = [z 1, z 1, z, z ] 1. Theorem 6.5. The group I 3 of orientation preserving isometries of the Hilbert disk D H is { α z + β z } + αβ 6.15 I 3 = Reα βz + α + β α β = 1. ] 1

15 Embedding of Teichmüller space into Goldman space 145 Proof. Since G : D P D H is an isometry with the inverse F, the isometry of D H has the expression such that G f F, where f is an isometry of D P in 6.5. Therefore the isometries of D H are G f F z = G f = G = = z z αz + β1 + 1 z βz + ᾱ1 + 1 z α z + β z + αβ1 + 1 z let A = G = A B B B B + AĀ α βz + ᾱβ z + α + β z α z + β z + αβ Reα βz + α + β, where α, β C such that α β = 1. Let Ω H = Ω, d H be the convex domain Ω RP defined by 6.16 Ω = {[x, y, s] RP x + y s < 0} with the Hilbert metric d H. Then Ω H is a strictly convex domain with the conic boundary. Consider the mapping G 3 : D H Ω H defined by 6.17 G 3 z = [ Rez, Imz, 1 ] = [ x, y, 1 ] for z = x + i y. Then G 3 is a diffeomorphism with the inverse G 1 let 3 = F 3 such that 6.18 F 3 [x, y, s] = x s + i y s. Theorem 6.6. The mapping G 3 : D H Ω H is an isometry. Proof. For two distinct points z 1 = x 1 +i y 1, z = x +i y D H with the boundary points z 1, z, we denote w j = G 3 z j and w j = G 3z j for each j. Then w 1, w 1, w, w are four distinct collinear points in RP since the corresponding four lines in R 3 are contained in the plane P spanned by two linearly independent vectors x 1, y 1, 1 and x, y, 1 in R 3. Since the plane P is not the xy-plane, by Definition 5.4 of the cross-ratio, [w 1, w 1, w, w ] = [[x 1, y 1, 1], [x 1, y 1, 1], [x, y, 1], [x, y, 1]] = [x 1 + i y 1, x 1 + i y 1, x + i y, x + i y ] = [z 1, z 1, z, z ].

16 146 Hong Chan Kim This completes the proof because d H w 1, w = log [w 1, w 1, w, w ] and d H z 1, z = log [z 1, z 1, z, z ]. 7. Embedding of PSL, R into SL3, R The goal of this section is to define an isometry H Ω and an embedding of PSL, R into SL3, R. Since H P and D H are isometric and D H and Ω H are isometric, to define an isometry between H and Ω, we need to change the metric on Ω from d H to d H as in 6.1. Let Ω H = Ω, d H be the strictly convex domain Ω in RP with the modified Hilbert metric d H. Then the mapping G 3 : D H Ω H is also an isometry. Theorem 7.1. The mapping G : H P Ω H defined by 7.1 Gz = [ x, x + y 1, x + y + 1 ] is an isometry with the inverse G 1 let 7. F [x, y, s] = x s y + i = F such that s s y 1 x s y s. Proof. Since G 1 : H P D P, G : D P D H, G 3 : D H Ω H are isometries, clearly their composition G = G 3 G G 1 is an isometry from H P to Ω H. With the identification z = x+i y H P, the isometry G can be calculated as in 7.1. The expression of the inverse G 1 = F is 7.. Clearly F : Ω H H P is the isometry calculated by the composition F = F 1 F F 3. To realizes a hyperbolic structure on M as a convex real projective structure, consider the final goal of this section, which is to define an embedding PSL, R SL3, R. Theorem 7.. The mapping ϕ : PSL, R SL3, R defined by 7.3 ad + bc ac bd ac + bd [ ] ϕa = ab cd a b c +d a +b c d a b for A = c d ab + cd a b +c d a +b +c +d is an embedding of PSL, R into SL3, R. Proof. Since the mapping G : H P Ω H is an isometry with the inverse F, we can define a mapping ϕ : PSL, R SL3, R such that

17 Embedding of Teichmüller space into Goldman space 147 the following diagram commutes. H P A H P G Ω H ϕa G Ω H Let [x, y, s] = [z, s] be a point in Ω H RP and A PSL, R the matrix representation of an isometry f 1 of H P. By Theorems 6.1, 6., and 6.5, G f 1 F [ x, y, s ] = G 3 f 3 F 3 [ x, y, s ] = G 3 f 3 z s α z + β z + αβs let = G 3 Reα βz + α s + β = G 3 w [ w ] s t = t, 1 = [w, t]. To describe the matrix representation of G f 1 F, plug in the standard basis [1, 0, 0], [0, 1, 0], [0, 0, 1] to the linear transformation G f 1 F. Then we obtain G f 1 F [1, 0, 0] = G f 1 F [ 1, 0 ] = [ α + β, Reα β ], G f 1 F [0, 1, 0] = G f 1 F [ i, 0 ] = [ α i β i, Reα βi ], G f 1 F [0, 0, 1] = G f 1 F [ 0, 1 ] = [ αβ, α + β ]. Recall α = a+d b c b+c a d + i and β = + i from the proof of Theorem 6.. After some calculations, we have the following equations. α + β = ad + bc + iab cd, Reα β = ab + cd, α i β i = ac bd + i a b c + d, Reα βi = a b + c d, αβ = ac + bd + i a + b c d, α + β = a + b + c + d. Thus we have the matrix representation ϕa as in 7.3. Suppose ϕa 1 = ϕa for A 1, A SL, R. Then A 1 = A or A 1 = A. We can also compute ϕa 1 A = ϕa 1 ϕa. Thus ϕ : PSL, R

18 148 Hong Chan Kim SL3, R is an injective homomorphism. Therefore ϕ is an embedding of PSL, R into SL3, R. After some the Mathematica computations, we have the following properties. Proposition 7.3. Let B = ϕa SL3, R for A PSL, R. Then, 1. DetB = DetA 3 = ad bc 3 = 1.. TrB = TrA 1 = a + d B PSO, 1 SL3, R. 4. Suppose ±{α, α 1 } is the eigenvalues of A, respectively. Then {α, 1, α } is the eigenvalues of B. Therefore if A PSL, R is a hyperbolic element, then B = ϕa SL3, R is a positive hyperbolic element. We can derive that the hyperbolic structures embeds into convex real projective structures through the identification of the conjugacy classes of [PSL, R] [SL3, R] 7.4 [ α α ] α +α 0 α α α α α +α = α α. 8. The Goldman s length parameters The set of positive hyperbolic elements of SL3, R is denoted by Hyp +. Goldman [5] defined the length parameters l, m on Hyp + as lb = log ν λ, mb = 3 logµ, where B is a positive hyperbolic element represented by the diagonal matrix 4.. In this paper we will modify Goldman s length parameters l, m in order to maintain the consistency with the Fenchel-Nielsen s length parameter l. The modified Goldman s length parameters l, m are 8.1 lb = 1 ν log, mb = 3 λ 4 logµ with λ µ ν = 1 and 0 < λ < µ < ν. For a hyperbolic manifold M, let Ω be the developing image in H and A an element of the holonomy group Γ PSL, R. The translation length la of A is defined by la = inf z Ω d P z, Az,

19 Embedding of Teichmüller space into Goldman space 149 where d P is the Poincaré metric on Ω. Then the translation length la of A is achieved if and only if z lies on the principal line of A which is the line joining the repelling and attracting fixed point of A. From Beardon s book [1], we get the relation 8. tra la = cosh between the translation length and trace of A. Since cosh 1 t = logt + t 1 and tra = α + α 1 for α > 1, Equation 8. becomes la α + α = cosh 1 1 α + α 1 α = log α 1 4 α + α 1 = log + α α 1 α + α 1 = log + α α 1 = logα. Therefore the Fenchel-Nielsen s length parameter l can be defined as 8.3 la = logα for a hyperbolic element A PSL, R represented by the diagonal matrix 4.1 with α > 1. Theorem 8.1. The modified Goldman s length parameter l is an isometric extension of the Fenchel-Nielsen s length parameter l. Proof. Let B = ϕa. Since the length parameter l is invariant under the conjugation, consider the identifications λ = α, µ = 1 and ν = α in 7.4. Then we have lb = 1 ν log = 1 α λ log α = 1 logα4 = logα = la. Therefore the modified Goldman s length parameter lb is exactly the same parameter to the Fenchel-Nielsen s length parameter la. We can extend the concept of translation length of hyperbolic structures to that of convex real projective structures. For any B Hyp + there exist three non-collinear fixed points and a B-invariant line in RP. We shall refer that the repelling, saddle, attracting fixed points

20 150 Hong Chan Kim Fix B, Fix 0 B, Fix + B are the fixed points corresponding to the eigenvectors in R 3 for the smallest eigenvalue λ, middle eigenvalue µ, the largest eigenvalue ν, respectively. The principal line σb is the line joining the repelling and attracting fixed points of B. Each positive hyperbolic element B can be uniquely decomposed as HV up to SL3, R-conjugation, where 8.4 H = λ µ ν µ and V = 1/ µ µ / µ We call H the horizontal factor and V the vertical factor of B Hyp +. H will be also called the pure hyperbolic factor of B. Consider the horizontal factor H of B. For any point a = [x, 0, s] t in the principal line σb such that x 0, s 0, the modified Hilbert distance d H between a and Ha is d H a, Ha = 1 log [Fix B, a, Ha, Fix + B] = 1 log 1 0, x 0, λ µ x 0 0 s ν, 0 0 µ s 1 = 1 λ x x log ν s s 0 x λ x s ν s 0 = 1 ν λ log = lb. We call lb = d H a, Ha the horizontal translation length and it is the length of the boundary component represented by B.. Figure 4. The horizontal and vertical translation lengths

21 Embedding of Teichmüller space into Goldman space 151 Consider the vertical factor V of B. Then the stationary set is the principal line σb and the saddle fixed point Fix 0 B. Without loss of generality we assume µ > 1. Then for any b = [x, y, s] t in the segment joining [x, 0, s] t and [0, 1, 0] t, the point V b goes toward [0, 1, 0] t since µ > 1. Then the modified Hilbert distance d H between b and V b is d H b, V b = 1 log [ [x, 0, s] t, b, V a, Fix 0 B ] = 1 log x 0, s [ = 1 log x 0, x y, x y s x µ yµ, 0 1, ] x µ yµ x µ, [ ] = 1 log 1 0, 1 y, 1, 0 yµ 3 1 y = 1 y 1 µ log y 1 y 1 µ 3 0 = 1 µ log 3 = 3 log µ = mb. 4 We call mb = d H b, V b the vertical translation length. Therefore B = HV SL3, R moves a point in Ω vertically by 3 4 logµ and horizontally by 1 log ν λ. We can easily compute the following relations. lh = 1 ν µ log λ = 1 ν µ log = lb, λ mh = 3 log1 = 0. 4 The above equations imply the horizontal translation lengths of B and H are the same. If a positive hyperbolic element B SL3, R is derived from a hyperbolic element, then mb = 0. Therefore the length parameter m measures the deviation of convex real projective structures from hyperbolic structures. Acknowledgements. I want to thank W. Goldman for introducing me to this subject and Jin-Hwan Cho for an advice to convert metapost graphics to eps files. I also thank the referee who points out some errors and mistakes.

22 15 Hong Chan Kim References [1] A. F. Beardon, The Geometry of Discrete Groups, Graduate Texts in Mathematics 91, Springer-Verlag, [] S. Choi and W. M. Goldman, Convex real projective structures on closed surfaces are closed, Proc. Amer. Math. Soc , no., [3] W. M. Goldman, The symplectic nature of fundamental groups of surfaces, Adv. in Math , no., [4], Geometric structures on manifolds and varieties of representations, Geometry of group representations Boulder, CO, 1987, , Contemp. Math., 74, Amer. Math. Soc., Providence, RI, [5], Convex real projective structures on compact surfaces, J. Differential Geom , no. 3, [6] D. Johnson and J. J. Millson, Deformation spaces associated to compact hyperbolic manifolds, Discrete groups in geometry and analysis New Haven, Conn., 1984, , Progr. Math., 67, Birkhäuser Boston, Boston, MA, [7] H. C. Kim, Matrix presentations of the Teichmüller space of a pair of pants, J. Korean Math. Soc , no. 3, [8] S. Kobayashi, Invariant distances for projective structures, Symposia Mathematica, Vol. XXVI Rome, 1980, pp , Academic Press, London-New York, 198. [9] N. Kuiper, On convex locally projective spaces, Convegno Int. Geometria Diff., Italy, 1954, 00 13, Edizioni Cremonese, Roma, [10] K. Matsuzaki and M. Taniguchi, Hyperbolic manifolds and Kleinian groups, Oxford Science Publications, Oxford University Press, New York, [11] J. Ratcliffe, Foundations of hyperbolic manifolds, Graduate Texts in Mathematics, 149. Springer-Verlag, New York, [1] D. Sullivan and W. Thurston, Manifolds with canonical coordinates charts: some examples, Enseign. Math , no. 1-, [13] W. Thurston, Three-dimensional geometry and topology. Vol. 1, Princeton Mathematical Series, 35. Princeton University Press, Department of Mathematics Education Korea University Seoul , Korea hongchan@korea.ac.kr