ON FOCK-GONCHAROV COORDINATES OF THE ONCE-PUNCTURED TORUS GROUP

ON FOCK-GONCHAROV COORDINATES OF THE ONCE-PUNCTURED TORUS GROUP YUICHI KABAYA Introduction In their seminal paper [3], Fock and Goncharov defined positive representations of the fundamental group of a surface S into a split semi-simple real Lie group G (eg PSL(n, R)) They showed that the space of positive representations satisfies properties similar to those of the Teichmüller space: a positive representation is faithful, has discrete image in G, and the moduli space of positive representations is diffeomorphic to R χ(s) dim G In fact, when G = PSL(2, R), the space of positive representations coincides with the Teichmüller space They showed that the space of positive representations coincides with the Hitchin component [9] in the representation space of π (S) into G It should be mentioned here that Labourie introduced in [] the notion of Anosov representations, whose moduli space coincides with the Hitchin component and the space of positive representations [], [8] When the Lie group is PSL(n, R) and an ideal triangulation of S is fixed, Fock and Goncharov defined two types of invariants for positive representations: vertex functions and edge functions A vertex function is also called a triple ratio, which we will use in this note They showed that these invariants give a set of coordinates of positive representations (Their coordinates are also defined for more general representations into PSL(n, C)) The Fock-Goncharov coordinates are extensively studied: there are generalizations to 3-manifolds groups [], [6], [5]; the McShane identities are studied in [2]; Fenchel-Nielsen type coordinates for the Hitchin component in [2] In [0], I and Xin Nie give a parametrization of PGL(n, C)-representations of a surface group as an analogue of the Fenchel-Nielsen coordinates In this note, I will explain Fock-Goncharov coordinates and give an explicit construction of matrix generators for once-punctured torus group, in terms of Fock-Goncharov coordinates 2 Flags Let GL(n, C) be the general linear group of n n complex matrices We define two subgroups B and U by B =, U = O O The center of GL(n, C) is isomorphic to C, the set of diagonal matrices with the same diagonal entries We let PGL(n, C) = GL(n, C)/C We have a short exact sequence Z/nZ SL(n, C) PGL(n, C) Received December 3, 202?

A (complete) flag in C n is a sequence of subspaces {0} = V 0 V V 2 V n = C n We denote the set of all flags by F n GL(n, C) and PGL(n, C) act naturally on F n from the left We represent X GL(n, C) by n column vectors as X = ( x x 2 x n) where x i = t (x i,, x i n) are column vectors By setting X i = span C {x,, x i }, we obtain a flag {0} X X n from an element of GL(n, C) Thus we have a map from GL(n, C) to F n Since an upper triangular matrix acts from the right as b b n () X = ( b x b 2 x + b 22 x 2 b n x + + b nn x n), O b nn the map induces a map GL(n, C)/B F n We can easily show that this is bijective and equivariant with respect to the left action of GL(n, C) Thus we can identify F n with GL(n, C)/B We can also identify F n with PGL(n, C)/B where we also denote by B for the quotient in PGL(n, C) by abuse of notation We let AF n = GL(n, C)/U and call an element of AF n an affine flag We have the following short exact sequence: B/U AF n F n GL(n, C)/U GL(n, C)/B Example 2 When n = 2, F n can be identified with the set of lines in C 2 In other words, F 2 is the projective line CP If we regard CP as C { }, PGL(2, C) acts on CP by linear fractional transformations and the stabilizer at is the subgroup B of upper triangular matrices Thus we have F 2 = CP = PGL(2, C)/B 3 Triples of flags We will describe the moduli space of configurations of generic n-tuples of flags Definition 3 Let (X,, X k ) be a k-tuple of flags We fix a matrix representative X i = (x i x n i ) GL(n, C) for each i A k-tuple of flags (X,, X k ) is called generic if (2) det(x x i x 2 x i 2 2 x k x i k k ) 0 for any 0 i,, i k n satisfying i + i 2 + + i k = n We remark that the genericity does not depend on the choice of the matrix representatives Moreover the determinant in (2) is a well-defined complex number if X,, X k AF n (recall ()) We denote the determinant by det(x i X i 2 2 X i k k ) for a k-tuple of affine flags In this note, we only consider generic triples or quadruples of flags Let (X, Y, Z) be a generic triple of F n We fix lifts of X, Y, Z to AF n For a triple (i, j, k) of integers satisfying 0 i, j, k n and i + j + k = n, we denote i,j,k = det(x i Y j Z k ) Consider a big triangle subdivided into n 2 small triangles as in Figure Such a triple 2

(4, 0, 0) (3, 0, ) (2, 0, 2) (, 0, 3) (0, 4, 0) (0, 0, 4) Figure A subdivision into n 2 triangles (n = 4) (i, j, k) corresponds to a vertex of the subdivided triangle For an interior vertex (i, j, k) (in other words i, j, k n and i + j + k = n), the triple ratio is defined by T i,j,k (X, Y, Z) = i+,j,k i,j+,k i,j,k+ i+,j,k i,j+,k i,j,k+ We show a graphical representation of T i,j,k (X, Y, Z) in Figure 2 Each factor of the numerator (resp denominator) corresponds to a vertex colored by black (resp white) in Figure 2 We remark that T i,j,k (X, Y, Z) does not depend on the choice of the matrix representatives By definition, we have (3) (4) (5) T i,j,k (X, Y, Z) = T j,k,i (Y, Z, X) = T k,i,j (Z, X, Y ), T i,j,k (X, Y, Z) = T i,k,j (X, Z, Y ), T i,j,k (X, Y, Z) = T i,j,k (AX, AY, AZ), for any generic triple X, Y, Z F n and A PGL(n, C) X (i +, j, k ) (i +, j, k) (i, j +, k ) (i, j, k) (i, j, k + ) Y Z (i, j +, k) (i, j, k + ) Figure 2 The black (resp white) vertices correspond to the factors of the numerator (resp denominator) of the triple ratio If we denote Conf k (F n ) = GL(n, C)\{(X,, X k ) generic k-tuple of F n }, T i,j,k are functions on Conf 3 (F n ) by (5) Moreover, we have the following theorem Theorem 32 (Fock-Goncharov) A point of Conf 3 (F n ) is completely determined by the (n )(n 2) 2 triple ratios In particular, Conf 3 (F n ) = (C ) (n )(n 2)/2 This theorem follows from the existence of the following normal form of a generic triple of flags 3

Lemma 33 Let (X, Y, Z) be a generic triple of F n Then there exists a unique A GL(n, C) and upper triangular matrices B, B 2, B 3 up to scalar multiplication such that 0 0 O O AXB = O, AY B 2 =, AZB 3 = O O This means that the lower triangular part of AZB 3 gives a set of complete invariants for configurations of generic triples of flags We will later give a brief sketch of the proof of Lemma 33, which gives an explicit construction of the matrix A Combining with the following lemma, we complete the proof of Theorem 32 Lemma 34 Each entry of the lower triangular part of AZB 3 in Lemma 33 is written by a Laurent polynomial of the triple ratios T i,j,k (X, Y, Z) This can be proved by induction Probably the Laurent polynomial might be a polynomial Here are some examples for small n Example 35 When n = 3, let T = T,, (X, Y, Z), then we have the following normal form: (6) X = 0 0 0 0, Y = 0 0 0 0, Z = 0 0 0 0 0 0 0 T + In fact, we have T,, (X, Y, Z) = det det 0 0 0 0 0 0 0 0 0 0 det det 0 0 0 0 0 0 0 0 0 0 det det 0 0 0 T + 0 0 0 T + = T When n = 4, let T ijk = T i,j,k (X, Y, Z), then we have the following normal form: 0 0 0 X = I 4, Y = C 4, Z = 0 0 T 2 + 0, (T 2 + )T 2 + (T 2 + )T 2 + where I 4 is the identity matrix and C 4 is the counter diagonal matrix with all counter diagonal entries Sketch of proof of Lemma 33 First we show that for a generic triple of flags (X, Y, Z), there exists a unique matrix A GL(n, C) such that O AX =, AY =, AZ = O 4

We need to find a matrix A = (a ij ) satisfying a i x j + a i2 x j 2 + + a in x j n = 0, (j < i) a i y j + a i2 y j 2 + + a in y j n = 0, (j < n i + ) a i z + a i2 z 2 + + a in z n = This system of linear equations is equivalent to the matrix equation x x n x i x i n a i (7) y yn = 0 0, i =,, n a in y n i yn n i z zn Since (X, Y, Z) is generic, we can show that the n n-matrix in the above equation is invertible So we have a unique solution A M(n, C) We can show that det A 0 by genericity Multiplying an upper triangular matrix from the right, we can eliminate the upper right (or lower right) triangular part of a matrix This completes the proof of Lemma 33 From the proof of Lemma 33, we have the following proposition Proposition 36 () Let X, Y F n and z CP n be a generic triple, and X, Y F n and z CP n another generic triple Then there exists a unique matrix A PGL(n, C) such that AX = X, AY = Y, Az = z (2) Let X, Y F n and z CP n be a generic triple and T i,j,k be nonzero complex numbers for i, j, k satisfying i, j, k n and i + j + k = n Then there exists a unique flag Z such that Z = z and T i,j,k (X, Y, Z) = T i,j,k 4 Quadruples of flags Let X, Z be affine flags and y, t be non-zero n-dimensional vectors We say that (X, Z, y) is generic if det(x k Z n k y) 0 for k = 0,, n If (X, Z, y) and (X, Z, t) are generic, we define the edge function for i =,, n by (8) δ i (X, y, Z, t) = det(xi yz n i ) det(x i Z n i t) det(x i yz n i ) det(x i Z n i t) We show a graphical representation of δ i (X, y, Z, t) in Figure 3 We can easily check that δ i (X, y, Z, t) is well-defined for X, Z F n and y, t CP n By definition, we have (9) δ i (X, y, Z, t) = δ i (X, t, Z, y), (0) δ i (X, y, Z, t) = δ n i (Z, t, X, y), () δ i (X, y, Z, t) = δ i (AX, Ay, AZ, At), for any A PGL(n, C) For a generic quadruple X, Y, Z, T F n, we simply denote δ i (X, Y, Z, T ) = δ i (X, Y, Z, T ) 5

By (), δ i (X, Y, Z, T ) are functions on Conf 4 (F n ) For a generic quadruple (X, Y, Z, T ), we have (n )(n 2) triple ratios for each (X, Y, Z) and (X, Z, T ) and (n ) edge functions 2 These (n )(n 2) + (n ) = (n ) 2 invariants completely determine a point of Conf 4 (F n ) First we show the following proposition X (i,, n i ) (i, n i, ) y t (i, n i, 0) Z (i,, n i) (i, n i, ) Figure 3 The black (resp white) vertices correspond to the factors of the numerator (resp denominator) of the edge function Proposition 4 Let X, Z F n and y CP n such that the triple (X, Z, y) is generic For any d,, d n C, there exists a unique t CP n such that δ i (X, y, Z, t) = d i, i =,, n In fact, by () and Proposition 36 (), we can assume that the triple (X, Z, y) is of the form O O (2) X =, Z =, y = O O We denote the identity matrix of size i by I i and the counter diagonal matrix of size i with counter diagonal entries by C i We let t = [t : : t n ] CP n, then we have I i O I i O O O O O t i+ O C n i O C n i δ i (X, y, Z, t) = = t i+ I i O I i O t i O O O O t i O C n i O C n i Thus t is uniquely determined by d,, d n Corollary 42 A point (X, Y, Z, T ) of Conf 4 (F n ) is uniquely determined by T i,j,k (X, Y, Z), T i,j,k (X, Z, T ) and δ i (X, Y, Z, T ) In fact, (X, Y, Z) is uniquely determined by T i,j,k (X, Y, Z) by Theorem 32 Then T CP n is determined by δ i (X, Y, Z, T ) by Proposition 4, and then T F n is determined by T i,j,k (X, Z, T ) by Proposition 36 (2) We remark that the quadruple (X, Y, Z, T ) determined by arbitrary given T i,j,k (X, Y, Z), T i,j,k (X, Z, T ) and δ i (X, Y, Z, T ) might not be generic but the triples (X, Y, Z) and (X, Z, T ) are generic (If we further assume positivity of triple ratios and edge functions, then the quadruple must be generic) By a similar argument, we can show that a configuration of generic k flags is uniquely determined by some triple ratios and edge functions 6

Example 43 When n = 2, we observed in Example 2 that F 2 is nothing but CP So we assume that X, Z CP In this identification, the normalization (2) corresponds to X = [ : 0] =, Z = [0 : ] = 0, y = [ : ] = Then we have δ (,, 0, t) = t (See Figure 4) Thus if we define the cross ratio by [x 0 : x : x 2 : x 3 ] = x 3 x 0 x 2 x, x 3 x x 2 x 0 we have δ (X, y, Z, t) = [X : Z : y : t] X = Z = 0 t = δ (X, y, Z, t) y = Figure 4 5 Fock-Goncharov coordinates We will use triple ratios and edge functions to give a parametrization of PGL(n, C)- representations of a surface group Let S be an orientable surface with at least one puncture We assume that S admits a hyperbolic metric An ideal triangle is a triangle with the vertices removed An ideal triangulation of S is a system of disjointly embedded arcs on S which decomposes S into ideal triangles,, N, see Figure 5 (If S is a surface of genus g with p punctures, then N = 4g 4 + 2p) We denote the universal cover of S by S, which can be identified with the hyperbolic plane H 2 The ideal triangulation of S lifts to an ideal triangulation of S Each ideal vertex of an ideal triangle of S defines a point on the ideal boundary H 2 Let S H 2 be the set of these ideal points The fundamental group π (S) acts on the universal cover S by deck transformations It also acts on the ideal triangulation of S and the ideal boundary S γ γ 2 Figure 5 Let ρ : π (S) PGL(n, C) be a representation A map f : S F n is called a developing map for ρ if it is ρ-equivariant ie it satisfies f(γx) = ρ(γ)f(x) for x S and γ π (S) The representation ρ is recovered from the developing map as follows Fix an 7

ideal triangle of S, and denote its ideal vertices by v, v 2, v 3 Since f is ρ-equivariant, we have f(γv i ) = ρ(γ)f(v i ) for any γ π (S) and i =, 2, 3 By Proposition 36 (), ρ(γ) is uniquely determined by these data as an element of PGL(n, C) Let be an ideal triangle of the ideal triangulation of S We take a lift of to S Then the ideal vertices of the triangle are mapped to a triple of flags by f If the triple is generic, we can define the triple ratios for Since f is ρ-equivariant and by (5), the triple ratios do not depend on the choice of the lift We can similarly define the edge functions for each edge of the ideal triangulation Thus, if S is a surface of genus g with p punctures, we have (4g 4+2p) (n )(n 2) triple ratio parameters and (6g 6+3p)(n ) edge functions Altogether we have (n 2 )(2g 2+p) parameters These parameters completely determine f and hence ρ up to conjugacy In fact, we can reconstruct f from these parameters First we choose one ideal triangle in S and denote the ideal vertices by v, v 2, v 3 Then take arbitrary X, X 2 F n and x 3 CP n Define f(v i ) = X i for i =, 2 By Proposition 36 (2), there exists unique f(v 3 ) F n such that f(v 3 ) = x 3 and the triple ratios T i,j,k (f(v ), f(v 2 ), f(v 3 )) are the same as the prescribed ones Let (v, v 2, v 4 ) be the ideal triangle of S adjacent to (v, v 2, v 3 ) By Proposition 4, f(v 4 ) CP n is uniquely determined by the edge functions δ i (f(v ), f(v 3 ), f(v 2 ), f(v 4 ) ) Again by Proposition 36 (2), f(v 4 ) F n is determined by the triple ratios T i,j,k (f(v ), f(v 2 ), f(v 4 )) Iterating these steps, f : S Fn is uniquely determined by these data If we change the first choice of X, X 2 F n and x 3 CP n, then the result differs by a conjugation The conjugating element is explicitly given by Proposition 36 () This system of triple ratio and edge function parameters are called Fock-Goncharov coordinates 6 Once-punctured torus case Let S be a once punctured torus Fix an ideal triangulation of S as in Figure 5 We take a system of generators γ, γ 2 of π (S) as in the right of Figure 5 We give the explicit representation ρ : π (S) PGL(n, C) when n = 3 parametrized by Fock-Goncharov coordinates X 5 X c d X 3 a f w a b z e b X 4 c d X 2 X 6 Figure 6 Figure 6 shows a part of the universal cover S We let z, w be the triple ratios for the two ideal triangles and a, b, c, d, e, f be the edge functions for the three edges as indicated in Figure 6 Each X i in Figure 6 indicates the flag corresponding to the ideal vertex 8

First we fix X = 0 0 0 0, X 2 = 0 0 0 0, X4 = 0 0 0 0 By (4), we have z = T,, (X, X 4, X 2 ) = (T,, (X, X 2, X 4 )) From the normal form (6), we have X 4 = 0 0 0 + /z Next we compute X 5 Put X 5 = [s : s 2 : s 3 ] By the definition (8), we have a = δ 2 (X, X 5, X 4, X 2) = s 0 s 2 0 s 3 0 s 0 s 2 0 0 s 3 0 0 0 0 0 0 0 0 0 0 b = δ (X, X 5, X 4, X 2) = s /z s 2 ( + /z) + s 3 s 2 s 3 = s 2 s 3 s 3, Solving these equations, we have X 5 = [s : s 2 : s 3 ] = [abz + az + a + : a + : ] Similarly we have X 3 = [ : e : ef], X 6 = [cdz : cdz + cz : cdz + cz + c + ] Next we determine X 3 in F n We have X = I 3, X 2 = C 3, X 3 = e, ef where I 3 and C 3 are defined as in Example 35 Since this triple is obtained from the normal form of (X, X 2, X 3 ) by multiplication by a diagonal matrix with diagonal entries (, e, ef), we have X 3 = 0 0 e e 0 ef ef( + w) ef The matrix ρ(γ ) maps the triple (X, X 2, X 3 ) to (X5, X 4, X ) Decompose ρ(γ ) into two matrices as (X 2, X 3, X) A (I 3, C 3, ) B (X 4, X, X5), each of which is calculated explicitly by (7) After some computation, we have ( + a(z + ) + abz)efw f(w + ) + afw(z + ) ρ(γ ) = ( + a)efw f(w + ) + afw efw f(w + ) 9

Similarly, since ρ(γ 2 ) maps (X, X2, X 3 ) to (X 4, X6, X 2 ), we obtain cdefwz cdf(w + )z cdz ρ(γ 2 ) = cdefwz cfz + cdf(w + )z cz + cdz cdefwz cf(z + ) + cdf(w + )z + c(z + ) + cdz We end this note by drawing some pictures of the images of developing maps If we restrict the coordinates to real numbers, we obtain a PGL(3, R)-representation A PGL(3, R) representation preserving a convex set in RP 2 is called a convex projective representation In [4], Fock and Goncharov showed that, when all triple ratios and edge functions are positive, the associated PGL(3, R)-representation is convex projective We remark that Goldman gave a parametrization of convex projective structures in [7] Figures 7, 8 and 9 are drawn in local coordinates of RP 2 given by ( z y [x : y : z] x + z, x y ) x + z In particular, X = [ : 0 : 0] maps to (0, ), X 2 = [0 : 0 : ] to (0, 0) and X 4 = [ : : ] to (0, 0) I only drew triangles developed by the products of ρ(γ ) and ρ(γ 2 ) whose word lengths within 4 by using Sage [3] I remark that these pictures might miss large triangles in the developed images References [] N Bergeron, E Falbel and A Guilloux, Tetrahedra of flags, volume and homology of SL(3), preprint 20, arxiv:742 [2] F Bonahon and G Dreyer, Parametrizing Hitchin components, prepreint 202, arxiv:2093526 [3] V Fock and A Goncharov, Moduli spaces of local systems and higher Teichmüller theory, Publ Math Inst Hautes Études Sci No 03 (2006), 2 [4] V Fock and A Goncharov, Moduli spaces of convex projective structures on surfaces, Adv Math 208 (2007), no, 249 273 [5] S Garoufalidis, M Goerner and C Zickert, Gluing equations for PGL(n,C)-representations of 3- manifolds, preprint 202, arxiv:20767 [6] S Garoufalidis,D Thurston and C Zickert, The complex volume of SL(n,C)-representations of 3- manifolds, pareprint 20, arxiv:2828 [7] W Goldman, Convex real projective structures on compact surfaces, J Differential Geom, 3, 3 (990), 79 845 [8] O Guichard, Composantes de Hitchin et représentations hyperconvexes de groupes de surface, J Differential Geom 80, 3 (2008), 39 43 [9] N Hitchin, Lie groups and Teichmüller space, Topology, 3, 3 (992), 449 473 [0] Y Kabaya and X Nie, Parametrization of PSL(n, C)-representations of surface groups, in preparation [] F Labourie, Anosov flows, surface groups and curves in projective space, Invent Math, 65, (2006), 5 4 [2] F Labourie and G McShane, Cross ratios and identities for higher Teichmüller-Thurston theory, Duke Math J, 49, 2 (2009), 279 345 [3] W Stein et al, Sage Mathematics Software (Version 50), The Sage Development Team, 202, http://wwwsagemathorg Department of Mathematics, Osaka University, JSPS Research Fellow E-mail address: y-kabaya@crmathsciosaka-uacjp 0

- 2 2 2 2-2 - 2 - e = f = 05 - e = f = 2 - e = f = 2 Figure 7 a = b = c = d = 2, z = w = (These correspond to Fuchsian representations, so the developed images are in a round disk) 2 2 05 2 2 z =, w = - - z = 5, w = z = 3, w = Figure 8 a = b = c = d = e = f = 2 2 5 2 05-2 - - 05 5 - - - e = 2, f = e = 2, f = e = 2, f = 5 e = 2, f = 2 Figure 9 a = b = c = d = 2, z = w =