arXiv:1609.05891v1 [math.GT] 19 Sep 2016
A NON-CANCELLATION LEMMA FOR GOLDMAN BRACKET ARPAN KABIRAJ Abstract. Given an oriented surface, we study the properties of two terms of the Goldman bracket between two free homotopy classes of oriented closed curves with different signs, when one of the free homotopy classes contains a simple representative. We prove that, in the above situation, the geometric intersection number and the number of terms (counted with multiplicity) in the Goldman bracket are the same.
1. Introduction Let F be an oriented surface and x, y be two oriented closed curves in F intersecting transversally in double points. The Goldman bracket of x and y is defined as X (p)hx ∗p yi [x, y] = p∈x∩y
where x ∗p y denotes the loop product of x and y based at p, x ∩ y denotes the set of all intersection points between them, (p) denotes the sign of the intersection between x and y at p (in the positive direction of both axes) and hxi denotes the free homotopy class of the curve hxi. Goldman [Gol86] showed that this bracket satisfy the following remarkable properties: • If x is freely homotopic to x1 and y is freely homotopic to y1 such that x1 and y1 intersect transversally in double points then [x, y] = [x1 , y1 ]. • [x, y] = −[y, x]. • If x, y and z are three oriented closed curves intersecting pairwise transversally in double points then [[x, y], z] + [[y, z], x] + [[z, x], y] = 0. We denote the collection of all free homotopy classes of oriented closed curves on F by C and the free module generated by C by Z(C). Let hxi denote the free homotopy class of a closed curve x. For any two elements hxi and hyi in C, we consider two representatives x and y (of hxi and hyi respectively), intersecting transversally in double points 1
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and define the Goldman bracket [hxi, hyi] between hxi and hyi to be [x, y]. By the first property, the Goldman bracket is well defined on C. We extend the Goldman bracket linearly on Z(C). By the above properties, the Goldman bracket is a Lie bracket on Z(C). The pair (Z(C), [, ]), denoted by L(F ), consisting of Z(C) and the Goldman bracket is called the Goldman Lie algebra. The structure of the Goldman Lie Algebra for surfaces of non negative Euler Characteristic is either trivial or well understood (see [Cha10b, Lemma 7.6] for the torus case). Therefore throughout the paper we assume F to be of negative Euler Characteristic. By [Far11, Theorem 1.2], we can endow F with a hyperbolic metric. By a hyperbolic surface we mean an orientable surface of finite type with negative Euler characteristic and with a given hyperbolic metric. Let hxi and hyi be elements in C. Denote by i(x, y) the minimal possible number of intersection points (counted with multiplicity) between curves representing hxi and hyi which intersect transversally. The number i(x, y), known as the geometric intersection number between hxi and hyi, plays a crucial role in low dimensional topology and geometric group theory. In his paper [Gol86, Section 5.17], Goldman proved the following relation between [hxi, hyi] and i(x, y) when hxi contains a simple representative (a closed curve without self intersections). Theorem 1.1. Let hxi and hyi be two free homotopy classes of closed oriented curves in F . If hxi contains a simple representative and [hxi, hyi] = 0 then i(x, y) = 0. Goldman used the convexity properties of Teichm¨ uller space to prove the above theorem. In this paper we prove the following stronger version of the above theorem using basic techniques from hyperbolic geometry. Main Theorem. Let hxi and hyi be elements in C. If x contains a simple representative then i(x, y) is same as the number of terms in [hxi, hyi] counted with multiplicity. Remark 1. In [Cha10b], Chas gave a proof of the above theorem using combinatorial topology and group theory. She used HNN extensions and amalgamated products of the fundamental group of a surface to prove the theorem. Goldman discovered this bracket while studying the Weil-Petersson symplectic form on Teichm¨ uller space. The Teichm¨ uller space of F , denoted by T (F ), is the space of all marked hyperbolic surfaces upto isotopy. The Weil-Petersson symplectic form induces a Poisson bracket
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(denoted by {, }) on the space of smooth functions on Teichm¨ uller space. Each free homotopy class of oriented closed curve hxi on the surface F induces a smooth map l(x) on the Teichm¨ uller space, namely the length of the unique geodesic in the free homotopy class of x. Let x and y be two oriented closed curves intersecting transversally in double points. The Goldman bracket [x, y] between them and the Poisson bracket {l(x), l(y)} of their length functions satisfy the following relation ([Gol86, Theorem 3.15]) 1 {l(x), l(y)} = l[x, y] − 2
X
! (p)
l(x)l(y)
p∈x∩y
where the length is extended linearly on the right hand side. Goldman used this relation and convexity results of geodesic length functions [Wol83] to prove Theorem 1.1. Later the combinatorial structure of L(F ) has been studied. Chas and Krongold [CK10] proved that for a compact surface with nonempty boundary, [x, x3 ] determines the self-intersection number of x. Using hyperbolic geometry, Chas and Gadgil [CG14] proved that there exists a positive integer m0 such that for all m ≥ m0 , the geometric intersection number between x and y is same as the number of terms in [xm , y] divided by m. Kawazumi and Kuno [KK13] used Goldman algebra to study the Torelli-Johnson-Morita theory. There is a Lie cobracket defined by Turaev [Tur91] on Z(C) which is the dual object of the Goldman bracket. This structure has been studied in [Cha10a], [Cha04]. Idea of the proof: We use techniques from hyperbolic geometry to prove the theorem. Let X be any point in T (F ). Let x1 be an oriented simple closed curve and y1 be an oriented closed curve. The free homotopy class of x1 (respectively y1 ) contains a unique geodesic, which we denote by x (respectively y). We have, [x1 , y1 ] = [x, y]. Given two elements of the Goldman bracket between x and y with different signs, we consider the lifts of the terms to the upper half plane H. If two terms corresponding to two intersection points are freely homotopic, then the length of their geodesic representatives with respect to X is the same. We use the geometry of the product of geodesics in the upper half plane H to show that the angle of intersection (in the positive direction of both axes) at both intersection points must be the same. This should be true for any X in T (F ). We use Fenchel-Nielsen twist deformation to construct a new point Y in T (F ) where the angle of intersection is
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different which leads to a contradiction. The main ingredient to show that the angles are different is [Ker83, Proposition 3.5]. Organization of the paper: In Section 2 we recall some basic facts from hyperbolic geometry and Teichm¨ uller space. In Section 3 we recall various definitions of Goldman bracket and their equivalences. In Section 4 we prove that it is enough to consider the problem for closed surface. In Section 5 we describe the lifts of two terms of Goldman bracket which are equal and have different signs. In Section 6 we give the proof of the main theorem. Acknowledgements: The author would like to thank Siddhartha Gadgil for his encouragement and enlightening conversations. The author is supported by the Department of Science & Technology (DST); INSPIRE faculty. 2. preliminaries In this section we recall some basic facts about hyperbolic geometry, hyperbolic surfaces and Teichm¨ uller space. References for the results mentioned in this section are [Bea83], [Kat92], [Rat06], [Far11], [Ker83]. Let F be a hyperbolic surface of finite type, i.e. F is a surface of genus g with b boundary components and n punctures such that, 2−2g−b−n < 0. Let π1 (F ) be the fundamental group of F . We identify π1 (F ) with a discrete subgroup of P SL2 (R), the group of orientation preserving isometries of the upper half plane H. The action of π1 (F ) on H is properly discontinuous and does not fix any point. Therefore the quotient space is isometric to F. Henceforth by an isometry of H, we mean an orientation preserving isometry and by a closed curve we mean an oriented close curve. A homotopically non-trivial closed curve in F is called essential if it is not homotopic to a puncture. By a lift of a closed curve γ to H, we mean the image of a lift R → H of the map γ ◦ π where π : R → S 1 is the usual covering map. Every isometry f of H belongs to one of the following classes: 1) Elliptic: f has a unique fixed point in H. 2) Parabolic: f has a unique fixed point in ∂H. 3) Hyperbolic: f has exactly two fixed points in ∂H. The geodesic joining these two fixed points is called the axis of f and is denoted by Af . The isometry f acts on Af by translation by a fixed positive number, called the translation number of f which we denote by τf .
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Since π1 (F ) acts on H without any fixed point, π1 (F ) does not contain any elliptic element. Essential closed curves in G correspond to hyperbolic isometries and closed curves homotopic to punctures correspond to parabolic isometries. Let x, y ∈ π1 (F ) be two hyperbolic elements whose axes intersect at a point P (Figure 1). By [Bea83, Theorem 7.38.6], xy is also hyperbolic. Let Q be the point in Ax at a distance τx /2 from P in the positive direction of Ax and R be the point in Ay at a distance τy /2 from P in the negative direction of Ay . Then the unique geodesic joining R and Q with orientation from R to Q is the axis of xy and the distance between Q and R is τxy /2.
2
P
2
2
Figure 1. Axis of xy. There is a bijective correspondence between the set of all conjugacy classes in π1 (F ) and the set of all free homotopy classes of oriented closed curves in F. Given an oriented closed curve x in F , we denote both its free homotopy class and the corresponding conjugacy class in π1 (F ) by hxi. Every free homotopy class of essential closed curve contains a unique closed geodesic whose length is same as the translation length of any element of the corresponding conjugacy class. Given an oriented surface F of negative Euler Characteristic, the Teichm¨ uller space of F , denoted by T (F ), is defined as follows. Consider a pair (X, φ) where X is a finite area hyperbolic surface with totally geodesic boundary and φ : F → X is a diffeomorphism. We call (X, φ) a marked hyperbolic surfaces. We say (X1 , φ1 ) and (X2 , φ2 ) are equivalent if there exists an isometry I : X1 → X2 such that I ◦ φ1 is
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homotopic to φ2 . The Teichm¨ uller space T (F ) of F is the space of all equivalence classes of marked hyperbolic surfaces. Abusing notation we denote the point (X, φ) in T (F ) by X. Given a point X in T (F ) and a simple closed geodesic x in X, we define the Fenchel-Nielsen left twist deformation of X at time s along x as follows. Cut the surface along x and form a new hyperbolic surface Xs by gluing the two boundary component obtained from x with a left twist of distance s. We call the Fenchel-Nielsen left twist deformation just left twist deformation. To consider Xs as a point in T (F ), we have to construct a homotopy class of diffeomorphism from F to Xs . There are many ways to do this, one of which is the following. Consider an small annular neighbourhood N of x. Define a diffeomorphism from N to N which is homotopic to the twist deformation and then extend it by identity to the whole surface. Let X be a point in T (F ) and x be a simple closed geodesic in X. Let y be any closed geodesic in X intersecting x. Let p be an intersection point between x and y and φp (X) ∈ [0, π) be the angle of intersection between x and y at p, where the angle is considered anticlockwise from y to x. From [Ker83, Proposition 3.5] and [Ker83, Remark on page 254] we have the following theorem. Theorem 2.1. For every intersection point p between x and y, The function φp (Xs ) is a strictly decreasing function of s. For our purpose, we do not need the full strength of [Ker83, Proposition 3.5] which is proved for any geodesic lamination x. The idea of the proof for simple closed geodesic is the following. Consider the unit disc model and suppose x is the horizontal diameter. Now observe that after left twist deformation, the endpoints of the geodesic representative of y are strictly to the left (when viewed from a lift of the intersection point) of the endpoints of y ([Ker83, Lemma 3.6]). 3. Goldman bracket Consider two free homotopy classes of oriented closed curves in a hyperbolic surface F . The Goldman bracket is well defined in the set of free homotopy classes of oriented closed curves C. The geodesic representative of each free homotopy class is unique and two geodesics always intersect transversally. Hence we consider the geodesic representatives from the corresponding classes to define the Goldman bracket. The only problem is geodesic representatives may not intersect in double points. But we can modify the definition to avoid this problem. The following algebraic definition is from [CG14, Section 3]. For the equivalence of these two definitions see [CG14, Section 3].
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Given two hyperbolic transformation x and y with axes Ax and Ay respectively, that intersect in a point P , let I(x, y) denote the segment of Ax of length τx starting from P in the positive direction of Ax , containing P but not containing xP . Definition 3.1. Let hxi and hyi be two non-trivial conjugacy classes in π1 (F ). If either x or y is parabolic, define [hxi, hyi] = 0. If both x and y are hyperbolic, let Y be the cyclic subgroup generated by y and define J(x, y) = {gY ∈ π1 (F )/Y : I(x, y) ∩ gAy 6= ∅}. Then X [hxi, hyi] = ι(x, y g )hxy g i gY ∈J(x,y)
where ι(x, y) denotes the sign of intersection between the axis of x and y if they intersects and 0 otherwise. Remark 2. The geometric intersection number between a boundary curve and any other closed curve is zero. Therefore from the above definition, there is no difference between a compact surface with boundary and a punctured surface with the same fundamental group as far as the Goldman bracket is concerned. Hence we consider only compact surfaces. 4. Closed surface versus Surface with boundary Let F be a hyperbolic surface with geodesic boundary. To each boundary component δ, we attach a hyperbolic surface of genus one with one boundary component of length l(δ) by isometry. We call the new surface F¯ . Lemma 4.1. The inclusion of F into F¯ induces an injection from π1 (F ) to π1 (F¯ ). Proof. Let C denote the collection of boundary geodesics of F in F¯ . Consider a hyperbolic metric on F¯ such that length of each element of C is less than the Margulis constant. If a geodesic in F intersects any element of C in F¯ then its length is non-zero and hence it is nontrivial. Now suppose g1 and g2 are two elements of π1 (F ) and are conjugate to each other. Then by Lemma 4.1, their images in π1 (F¯ ) are also conjugate. Suppose two terms of Goldman bracket are same, i.e. conjugate to each other and they have different signs. If we prove that in F¯ they are not conjugate to each other then by the above reasoning they are not
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conjugate in F . Hence it is enough to prove the main result for closed surface. 5. Lifts of a term in Goldman bracket with different sign Let X be a point in T (F ). Let x be a simple closed geodesic and y be any closed geodesic. Suppose p and q are two intersection points between them such that (p) = −(q) and hx ∗p yi = hx ∗q yi. Given any two oriented curves x and y, we always consider the angle at any intersection point to be the angle which is in the positive direction of both axes. For any point X ∈ T (F ) we denote θp (X) and θq (X) to be the angle of intersection between x and y, at p and q, in X. Lift of a term x ∗p y in H, passing through a given lift of the point p is a bi-infinite polygonal path [CG14, Section 7]. This polygonal path is a quasi-geodesic [CG14, Lemma 7.1] and therefore follows a unique geodesic. We define this geodesic to be the axis of the lift. Consider lifts x ∗p y and x ∗q y to the upper half plane H. As they are freely homotopic, they are conjugate to each other. Therefore (taking conjugate if necessary) we assume that they have a common axis and the axis is Y -axis.
~ Q2
~ P2
b3
a3
q a2
Q1
P1
p b2
~ Q1
b1
a1 Q2
~ P1
P2
Figure 2. Lift of x ∗p y (left hand side where (Q1 ) = −1) and lift of x ∗q y (right hand side where (P1 ) = 1) We consider the anticlockwise orientation in H. If P1 and Q1 are lifts of p and q such that the common axis of the lifts of x ∗p y and x ∗q y is
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Y -axis then the lifts look like a bi-infinite polygonal path as Figure 2, where lX (a2 Q1 ) = lX (b2 P1 ) = τy /2 and lX (a3 Q1 ) = lX (b3 P1 ) = τx /2. Also from the geometry of product isometries, we have lX (x ∗q y) lX (x ∗p y) = lX (a2 a3 ) = lX (b2 b3 ) = . 2 2 Therefore by symmetries of the triangles, we have (1)
θq (X) = θp (X).
As X ∈ T (F ) is arbitrary, the above equation holds for all X ∈ T (F ). 6. Proof of the Main Theorem We prove the following lemma from which the Main Theorem follows. Lemma 6.1. Let F be an oriented surface of finite type with negative Euler Characteristic and x, y be two oriented closed geodesics intersecting each other. Suppose x is a simple geodesic. If p, q ∈ x ∩ y such that p = −q then hx ∗p yi = 6 hx ∗q yi.
q
Ax Q
p
q Ay g
p P
Ay
Figure 3. Two lifts P and Q of the points p and q to H. If we consider the anticlockwise orientation of H then P = p = 1 and Q = q = −1 Proof. We prove by contradiction. Suppose p = −q and hx ∗p yi = hx ∗q yi. As they are freely homotopic to each other, their lifts are conjugate to each other by an element of π1 (F ).
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Let X be any point in T (F ). Then from equation 1, we have θp (X) = θq (X) f or all X ∈ T (F ). Let Xs be the point in T (F ) obtained by a left twist deformation after time s from X along x. By Figure 3 and Theorem 2.1, φp (Xs ) = π − θp (Xs ) and φp (Xs ) = θq (Xs ) and both the functions are strictly decreasing. Therefore θp (Xs ) is an strictly increasing function and θq (Xs ) is a strictly decreasing function. As θp (X0 ) = θp (X) = θq (X) = θq (X0 ), we have θp (Xs ) 6= θq (Xs ) for all small s ∈ (0, π), a contradiction. References [Bea83]
A. F. Beardon, The geometry of discrete groups, Springer-Verlag, New York, 1983. [CG14] M. Chas and S. Gadgil, The Goldman bracket determines intersection numbers for surfaces and orbifolds, To appear in Algebaric & Geometric Topology (2014). [Cha04] M. Chas, Combinatorial Lie bialgebras of curves on surfaces, Topology 43 (2004), 543–568. [Cha10a] F. Chas, M.and Krongold, Algebraic characterization of simple closed curves via Turaev’s cobracket, To appear in the Journal of topology (2010). [Cha10b] M. Chas, Minimal intersection of curves on surfaces, Geometriae Dedicata 144 (1) (2010), 25–60. [CK10] M. Chas and F. Krongold, An algebraic characterization of simple closed curves on surfaces with boundary, Journal of Topology and Analysis 2 (2010), 395–417. [Far11] D. Farb, B.and Margalit, A primer on mapping class groups, Princeton University Press (PMS-49), Princeton and Oxford, 2011. [Gol86] W. M. Goldman, Invariant functions on Lie groups and Hamiltonian flows of surface group representations, Invent. Math. 85, No. 2 (1986), 263–302. [Kat92] S. Katok, Fuchsian groups, Chicago Lectures in Mathematics, University of Chicago Press, Chicago, IL, 1992. [Ker83] Steven P Kerckhoff, The nielsen realization problem, Annals of mathematics (1983), 235–265. [KK13] N. Kawazumi and Y. Kuno, The goldman-Turaev Lie bialgebra and the Johnson homomorphisms, Preprint (2013). [Rat06] J. Ratcliffe, Foundations of hyperbolic manifolds, GTM, Springer, 2006. [Tur91] V. G. Turaev, Skein quantization of Poisson algebras of loops on surfaces, Ann. Sci. Ecole Norm. Sup. 24, No. 6 (1991), 635–704. [Wol83] S. Wolpert, On the symplectic geometry of deformations of a hyperbolic surface, Ann. Math. 117 (1983), 207–234.
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Department of Mathematics, Chennai Mathematical Institute, Chennai 603103, India E-mail address:
[email protected]
