Generalization of a formula of Wolpert for balanced geodesic graphs on closed hyperbolic surfaces

A well-known theorem of Wolpert shows that the Weil-Petersson symplectic form on Teichm\"uller space, computed on two infinitesimal twists along simple closed geodesics on a fixed hyperbolic surface, equals the sum of the cosines of the intersection angles. We define an infinitesimal deformation starting from a more general object, namely a balanced geodesic graph, by which any tangent vector to Teichm\"uller space can be represented. We then prove a generalization of Wolpert's formula for these deformations. In the case of simple closed curves, we recover the theorem of Wolpert.


Introduction
Given a closed oriented surface S of genus g ≥ 2, Teichmüller space T (S) is classically defined as the space of complex structures on S, up to biholomorphisms isotopic to the identity. By the uniformization theorem, it is naturally identified to what is sometimes called Fricke space F (S), namely the space of hyperbolic metrics on S up to isometries isotopic to the identity.
The Weil-Petersson metric is a Kähler metric on T (S), whose definition only involves the point of view of T (S) as the space of complex structures on S. However, it turns out to be an extremely interesting object from the hyperbolic geometric point of view. An example of the bridge between the two viewpoints is in fact provided by Wolpert's theorem ( [Wol81,Wol83]), which we now briefly recall. Given two simple closed geodesics c, c ′ on a fixed closed hyperbolic surface (S, h), let us denote by t c and t ′ c the infinitesimal twists along c and c ′ , namely where E w·c l (h) is the new hyperbolic surface obtained from h by a left earthquake along c -that is, by cutting along c and re-glueing after a left twist of length w. Then Wolpert showed that, if ω WP denotes the symplectic form of the Weil-Petersson Kähler metric, then where, for every point of intersection p between c and c ′ , θ p denotes the angle of intersection at p.
Balanced geodesic graphs. In this paper, we will define a more general type of infinitesimal deformations on T (S), which generalize twists along simple closed geodesics. These will be associated to a balanced geodesic graph, namely a weighted graph (G, w) on (S, h) whose edges are geodesic segments and whose weights satisfy a balance condition at every vertex p, namely: where the sum is over all edges e which are adjacent to the vertex p, w e denotes the weight of e and v e is the tangent vector to e in T p S.
Clearly, every simple closed geodesic can be regarded as a balanced geodesic graph, just by adding a number of vertices so as to make sure that every edge is a segment, and choosing the same weight w for all edges. The balance condition is then trivially satisfied. In fact, the infinitesimal deformation t (G,w) we define reduces to the infinitesimal twist in this case.
In order to define t (G,w) , it is easier to use the identification of T (S) to the space of discrete and faithful representations of π 1 (S) into Isom(H 2 ) (i.e. the group of orientation-preserving isometries of the hyperbolic plane), up to conjugacy. The identification is obtained by associating to a hyperbolic metric h on S its holonomy representation ρ. The tangent space to the space of representations up to conjugacy is then known to be identified (see [Gol84]) to the group cohomology H 1 Adρ (π 1 (S), isom(H 2 )). The tangent vector t (G,w) is then defined by lifting to the universal cover of S a generic closed loop representing γ ∈ π 1 (S), and taking the weighted sum of the infinitesimal twists along the lifts of the edges of G met by the lift of the closed loop. The balance condition ensures that t (G,w) is well-defined. If (G, w) = (c, 1) is a simple closed geodesic with weight 1, t (G,w) actually coincides with the variation of the holonomy of the twisted metrics E w·c l (h), and we thus recover the infinitesimal twist t c .
Motivations. Besides the main result explained below, we would like to highlight two main motivations behind the study of such objects. The first motivation is directly related to hyperbolic geometry, in the spirit of the study of earthquakes of hyperbolic surfaces, see for instance [Thu86,Bon92,McM98]. In fact, in [FS12] left/right flippable tilings were introduced. These are (non-continuous) transformations between hyperbolic surfaces, associated to certain geodesic graphs on (S, h) such that the faces of the graph are divided into black and white faces, and the transformation is obtained by flipping the black faces. A balanced geodesic graph (G, w) can be interpreted as a tangent vector to a path of flippable tilings on hyperbolic surfaces, where the black faces are collapsing to the vertices of G and the weights w are the derivatives of the lengths of such black faces. This is thus a generalization of the deformations by earthquakes along simple closed curves, which have an infinitesimal twist as tangent vector. The second motivation comes from the deep relation between Teichmüller theory and geometric structures on three-dimensional manifolds. In the case of hyperbolic structures in dimension three, this is an important phenomenon, which goes back to Bers' simultaneous uniformization theorem for quasi-Fuchsian manifolds [Ber60], and has been widely developed [Bro03,Bon86,Uhl83,Tau04,Sep16]. Analogous of quasi-Fuchsian manifolds in Lorentzian geometry are maximal globally hyperbolic manifolds, and their relation with Teichmüller theory has been initiated in [Mes07]. Here we are mostly interested in flat maximal globally hyperbolic manifolds, as studied in [Bon05,Bar05,BS16b]. The relevant point here is that a balanced geodesic graph with positive weights, and such that S \ G is a disjoint union of convex polygons, is naturally the dual to convex polyhedral surface in a flat globally hyperbolic manifold M of dimension three homeomorphic to S × R. Moreover, as observed by Mess, the isomorphism between Minkowski space R 2,1 and the Lie algebra isom(H 2 ) ∼ = so(2, 1) induces a correspondence between the tangent space of T (S), in the model H 1 Adρ (π 1 (S), isom(H 2 )) of the representation variety, and the translation part of the holonomies of manifolds M as above.
This enables us to show that the map which associated to a balanced geodesic graph (G, w) the deformation t (G,w) in T [h] T (S) is surjective. In other words, any tangent vector to T (S) can be represented as the deformation associated to some balanced geodesic graph. This is not true if one only considers (weighted) simple closed geodesics. Hence in this sense our main result below, which extends Wolpert's formula, is quite more general since it can be used to represent the Weil-Petersson form applied to any two tangent vectors to T (S).
Main result. Let us now come to the statement of the main result. Recall that ω WP denotes the symplectic form of the Weil-Petersson metric on T (S).
Theorem. Let (G, w) and (G ′ , w ′ ) be two balanced geodesic graphs on the hyperbolic surface (S, h). Then where e and e ′ are intersecting edges of G and G ′ and θ e,e ′ is the angle of intersection between e and e ′ according to the orientation of S.
Therefore, when (G, w) = (c, 1) is a simple closed geodesic with weight 1, we recover Wolpert's formula (1). Let us remark again that our construction permits to represent any tangent vector in T [h] T (S) as t (G,w) for some balanced geodesic graph (G, w), and thus our result is in a greater generality than Wolpert's.
There is a small caveat in interpreting (3). If the two geodesic graphs G and G ′ do not have vertices in common and intersect transversely, it is clear what the intersection points and angles are. On the other hand, if G and G ′ have some non-transverse intersection or share some vertex, the points of intersection must be counted by perturbing one of the two graphs by a small isotopy, so as to make the intersection transverse. See Figures 10 and 11. This is just a caveat for counting points of intersections (it turns out that the result not depend on the chosen perturbation, as a consequence of the balance condition (2)), while of course the angles cos θ e,e ′ are the angles between the original geodesic edges e, e ′ .
To conclude the introduction, let us mention that the proof of our main result relies on two main tools. The first tool is a theorem of Goldman ( [Gol84]) which shows that the Weil-Petersson symplectic form on T (S) equals (under the holonomy map) a form defined only in terms of the group cohomology H 1 Adρ (π 1 (S), isom(H 2 )). The second tool is de Rham cohomology with values in certain flat vector bundles over S of rank 3. In this setup, our proof becomes rather elementary, and we thus also provide a simple proof of Wolpert's formula for simple closed geodesics as a particular case.
On the other hand, the cotangent space T * [X] T (S) is naturally identified to the space of holomorphic quadratic differentials QD(X) = H 0 (S, K 2 X ). The identification is given by the pairing on QD(X) × BD(X) defined by In fact, if in local complex coordinates φ = φ(z)dz 2 and µ = µ(z)dz/dz, then φµ = φ(z)µ(z)dz ∧ dz is a quantity which can be naturally integrated over S. The fundamental property is then the fact that for every φ ∈ QD(X), S φµ = 0 for every µ ∈ BD(X) ⇔ µ ∈ BD 0 (X) .
The Weil-Petersson product is then easily defined on the cotangent space, by where h X is the unique hyperbolic metric (i.e. Riemannian metric of constant curvature −1) compatible with the complex structure X, provided by the Uniformization Theorem ( [Koe09]). By a similar argument as above, the quantity φφ ′ /h X is indeed of the correct type to be integrated on S.
Hyperbolic metrics and Fuchsian representations. We will be using other two important models of T (S).
(1) By the aforementioned Uniformization Theorem, given any complex structure X on S, there is a unique hyperbolic metric h X on S compatible with X. Let us define the Fricke space of S as: where Diff 0 (S) clearly acts by pull-back. It is easy to check that the map X → h X is equivariant for the actions of Diff 0 (S), and therefore induces a diffeomorphism The inverse of U is simply the map which associates to a hyperbolic metric h the complex structure induced by h, which is obtained by choosing local isothermal coordinates.
(2) Given a hyperbolic metric h on S, let π : S → S be the universal cover of S. Then π * h is a hyperbolic metric, which is complete since S is compact, on the simply connected surface S. Hence ( S, π * h) is isometric to the hyperbolic plane H 2 by [Rat06, Theorem 8.6.2]. Such an isometry (chosen to be orientation-preserving) is unique up to post-compositions with elements in group Isom(H 2 ) of orientationpreserving isometries of H 2 , and is called developing map. Let us denote it by dev : ( S, π * h) → H 2 . It turns out to be equivariant for some representation ρ : π 1 (S) → Isom(H 2 ), called the holonomy map. That is: for every γ ∈ π 1 (S). Since dev is well-defined up to post-composition, ρ is welldefined up to conjugacy by elements of Isom(H 2 ). This provides a map where χ(π 1 (S), Isom(H 2 )) is the character variety Hom(π 1 (S), Isom(H 2 ))/ /Isom(H 2 ) .
Group cohomology. The model of Teichmüller space T (S) as χ fd (π 1 (S), Isom(H 2 )) enables to give a simple description of the tangent space. In fact, using the differentials of the maps U and Hol, we can identify . In [Gol84], the tangent space to the space of representations is described as the group cohomology with values in the Lie algebra isom(H 2 ): where: with respect to the adjoint action of ρ, that is, functions with values in the Lie algebra isom(H 2 ) satisfying: (6) This is essentially the condition of being a representation of π 1 (S) into Isom(H 2 ), at first order.
Adρ (π 1 (S), isom(H 2 )) is the space of coboundaries, namely cocycles of the form for some τ 0 ∈ isom(H 2 ). This is the first-order condition for a deformation of being trivial in χ(π 1 (S), Isom(H 2 )), that is, of being tangent to a path of representations ρ t obtained from ρ by conjugation.
Goldman symplectic form. In the fundamental paper [Gol84], Goldman introduced a symplectic form ω G on the space χ(π 1 (S), isom(H 2 )) (actually, the construction holds when replacing isom(H 2 ) by a more general Lie group G) and showed that it coincides (up to a factor) with the symplectic form ω WP of the Weil-Petersson Hermitian metric. The Goldman form is defined as follows. Recall from the previous section that the tangent space T [ρ] χ(π 1 (S), Isom(H 2 )) is identified to the group cohomology H 1 Adρ (π 1 (S), isom(H 2 )). Then one can define a pairing The first arrow is obtained by the cup product in group cohomology, paired by using the Killing form of isom(H 2 ). The identification between H 2 (π 1 (S), R) and R is then given by evaluation on the fundamental top-dimensional class of the closed oriented manifold S.
Recall that the map Hol • U associates to the Teichmüller class of a complex structure X on S the holonomy representation of the uniformizing hyperbolic metric h X . The differential of Hol • U should hence be consider as a vector space isomorphism In [Gol84], Goldman proved: Concerning Equation (8), we remark that in Goldman's original paper [Gol84] there is a different factor appearing. This is due to two reasons: • Goldman uses the trace form in the sl(2, R) model of isom(H 2 ), which is defined as b(X, Y ) = tr(XY ), instead of the Killing form which turns out to be κ = 4b. As a reference for Equation (8), which is the formula we actually need in this paper, see [Lou15, Section 2].
There actually is another description of the Goldman form in terms of de Rham cohomology, which will be introduced in Section 6.

Some properties of the hyperboloid model
It will be useful for this paper to consider the hyperboloid model of H 2 . Namely, let us consider (2+1)-dimensional Minkowski space, which is the vector space R 3 endowed with the standard bilinear form of signature (2, 1): It turns out that the induced bilinear form on the upper connected component of the twosheeted hyperboloid (which is simply connected) gives a complete hyperbolic metric. It is thus isometric to H 2 , again by [Rat06, Theorem 8.6.2]. Hence we will identify Description of the Lie algebra. By means of this identification, we have namely, the group of orientation-preserving isometries of H 2 is the identity component in the group of linear isometries of the Minkowski bilinear form. We then also have the following identification for the Lie algebra: where so(2, 1) are skew-symmetric matrices with respect to the Minkowski metric. A useful description for this Lie algebra is provided by the Minkowski cross product, which is the analogue for R 2,1 of the classical Euclidean cross product. This provides an isomorphism for any y ∈ R 2,1 . More explicitly, Hyperbolic isometries. An example is given by hyperbolic isometries. Every geodesic ℓ of H 2 is of the form ℓ = H 2 ∩ x ⊥ ⊂ R 2,1 , for some x ∈ R 2,1 with x, x = 1, and x ⊥ is the plane orthogonal to the vector x for ·, · . Moreover, the orientation of H 2 and the direction of x determine an orientation of ℓ. Then we define T t (ℓ) the hyperbolic isometry which preserves ℓ setwise and translates every point of ℓ by a length t according to the orientation of ℓ. We will also denote namely, t(ℓ) is the generator of the 1-parameter group T t (ℓ), or in other words the infinitesimal translation along ℓ.
Example 3.1. If we pick v = (1, 0, 0), then ℓ = H 2 ∩ {x 1 = 0} (see also Figure 1) and Hence the infinitesimal isometry is In general, if ℓ is an (oriented) geodesic, intersection of H 2 and x ⊥ endowed with the induced orientation, then the following formula holds: Additional structures on so(2, 1). Finally, two important properties of the isomorphism Λ are the following. See also [BS16a, Section 2] • Λ is equivariant for the actions of SO 0 (2, 1): the standard action on R 2,1 and the adjoint action on so(2, 1). Namely, for every η ∈ SO 0 (2, 1), • Λ is an isometry between the Minkowski metric and the Killing form on so(2, 1), up to a factor: where the Killing form κ for so(2, 1) is: κ(a, a ′ ) = tr(aa ′ ) .

Balanced geodesic graphs on hyperbolic surfaces
In this section we introduce balanced geodesic graphs on a closed hyperbolic surface, we show that there is a vector space structure on the space of such objects, and we construct an element in T [h] F (S) from any balanced geodesic graph on (S, h). For convenience, we will use H 2 in the hyperboloid model defined in the previous section, and hence we will identify Isom(H 2 ) = SO 0 (2, 1) and isom(H 2 ) = so(2, 1).
Balanced geodesic graphs. Let us fix a hyperbolic metric h on the closed oriented surface S.
where E is the set of (unoriented) edges of G and V is the set of vertices; • The assignment w : E → R of weights w(e) = w e for every e ∈ E; satisfying the following conditions: • Every edge e is a geodesic interval between its endpoints; • For every p ∈ V, the following balance condition holds: where p ∈ e denotes that p is an endpoint of an edge e ∈ E, and in this case v e is the unit tangent vector at p to the geodesic edge e.
We provide several classes of examples which should account for the abundance of such objects on any surface (S, h).
Example 4.2. Given a simple closed geodesic c on (S, h) and a weight w, c is the support of a balanced geodesic graph with weight w. In fact, it suffices to declare that the vertex set V consists of n ≥ 1 points on c. Then c is split into n edges, and we declare that each edge has weight w. Clearly the balance condition (12) is satisfied, since there are only two vectors to consider at every vertex, opposed to one another, with the same weight. See the curve in the left of Figure 2. Hence the class of balanced geodesic graphs include weighted simple closed geodesics.
Example 4.3. More generally, given any finite collection c 1 , . . . , c n of (not necessarily simple) closed geodesics (S, h), and any choice of weights w 1 , . . . , w n , the union ∪ i c i can be made into a balanced geodesic graph. In fact, it suffices to choose the vertices of V on the geodesics c 1 , . . . , c n , so that every intersection point between c i and some other geodesic c j (including self-intersections of c i ) is in the vertex set V. Moreover, it is necessary to add vertices to geodesics c i which are disjoint from all the geodesics c j (in particular, c i has no selfintersection) so as to make every edge of the graph an interval, as in Example 4.2. Then we declare that the weight of an edge contained in c i is w i . In fact, in this case the balance condition will be automatically satisfied, since at every vertex p ∈ V, tangent vectors come in opposite pairs with the same weight. (A pair is composed of the two opposite vectors tangent to the same geodesic c i .) See Figure 2. Therefore the balance condition (12) is satisfied regardless of the initial choice of w i . In particuar, weighted multicurves (i.e. collections of disjoint simple closed geodesics endowed with positive weights) are balanced geodesic graphs.
Example 4.4. Colin de Verdière in [CdV91] proved that, given any topological triangulation of (S, h), and any choice of positive weights w e assigned to each edge e, there exists a geodesic triangulation, with the same combinatorics of the original triangulation, which satisfies the balance condition (12) for the prescribed weights w e . Therefore, this geodesic realization of a topological triangulation is a balanced geodesic graph in the sense of Definition 4.1.  Figure 3. New assignements of weights for the refinement of two balanced geodesic graphs.
Vector space structure. In this subsection we will introduce the space of balanced geodesic graphs on a hyperbolic surface, and show that this space has a vector space structure.
Let us consider the space where the equivalence relation ∼ is defined as follows: two balanced geodesic graphs are equivalent if they can be obtained from one another by adding, or deleting: • Points which are endpoints of only two edges (possibly coincident); • Edges of weight zero. The space BG (S,h) defined in this way is naturally endowed with a structure of vector space, defined in the following way. For (G, w) in BG (S,h) and λ ∈ R, we define λ(G, w) = (G, λw). For the addition, we define (G, w)+(G ′ , w ′ ) = (G +G ′ , w+w ′ ), where G +G ′ is the refinement of G and G ′ , and w + w ′ are the weights on the edge set of G + G ′ defined as follows. When the refinement does not create new vertices for G and G ′ , then the same weights are kept on the corresponding edges. When the refinement creates a new vertex for G but not for G ′ , new weights are assigned as in the bottom of Figure 3. When the refinement creates a new vertex for both G and G ′ , then new weights are assigned as in the top of Figure 3. It is easy to check that (12) is satisfied in every case.
Finally, the zero element of the vector space is the class of any geodesic graph whose weights are all zero. Let us observe that, for every (fixed) geodesic graph G 0 , the subset of BG (S,h) composed of classes of balanced geodesic graphs having underlying geodesic graph G 0 is a finite-dimensional vector subspace.
Construction of the map Φ. We are now ready to define the map from the space BG (S,h) to the tangent space of Teichmüller space. We will actually define a deformation of the holonomy representation of h, thus providing a tangent vector to the character variety of π 1 (S). Namely, we will define a map: Adρ (π 1 (S), so(2, 1)) ,

where [ρ] = Hol([h]
). For this purpose, consider the universal cover π : S → S and fix a developing map dev : ( S, π * h) → H 2 which is a global isometry, equivariant for the representation ρ : π 1 (S) → SO 0 (2, 1). Let us fix a balanced geodesic graph (G, w) and lift it to ( G, w) on ( S, π * h). Let us also fix a basepoint x 0 ∈ S, which we assume does not lie in G. We define a cocycle τ : π 1 (S) → so(2, 1) in the following way. We say a path σ : [0, 1] → S is transverse to G if the intersection Imσ ∩ G of the image of σ and G consists of a finite number of points, which are not vertices of G, and for every point p = σ(t 0 ) ∈ Imσ ∩ G there exists ǫ > 0 such that σ| (t0−ǫ) and σ| (t0+ǫ) are contained in two different connected component of U \ G, where U is a small neighborhood of p in S. Now, pick γ ∈ π 1 (S) and let σ : [0, 1] → S be a path such that σ(0) = x 0 and σ(1) = γ ·x 0 , transverse to G. Then we define the following element of so(2, 1): where: • The sum is over all points p of intersection of the image of the path σ with the lift G of the balanced geodesic graph G, see Figure 4; • If ℓ is an oriented (subinterval of) a geodesic of H 2 , recall from Section 3 that we denote t(ℓ) ∈ so(2, 1) the infinitesimal hyperbolic translation along ℓ, using the orientation of ℓ. Namely, where T t (ℓ) ∈ SO 0 (2, 1) is the isometry of H 2 which preserves ℓ and translates every point of ℓ by a length t according to the orientation of ℓ. This is thus applied in Equation (13)  We then define Φ(G, w) = [τ ] ∈ H 1 Adρ (π 1 (S), so(2, 1)) . Under the identification between H 1 Adρ (π 1 (S), so(2, 1)) and T [X] T (S), we thus define Well-definiteness of Φ. There is a number of points to be verified in order to check that the map Φ is well-defined. First, we need to show that the value τ (γ) does not depend on the chosen path σ : [0, 1] → S, as long as σ is transverse to G. (This is the same as choosing a representative of the closed loop in S, based at π(x 0 ), which represents γ and is transverse to G.) In fact, there are three cases to consider. See also Figure 6.
(1) If two representatives σ and σ ′ can be isotoped to one another by a family of paths σ t : [0, 1] → S which is transverse for all t, then the value of τ is the same when computed with respect to σ or σ ′ , since the quantities w e and t(dev( e)) only depend on the edge e, and not on the intersection point of σ with e.
PSfrag replacements x 0 γ · x 0 σ pẽ Figure 4. To define Φ, we lift a generic closed loop representing γ ∈ π 1 (S) to a path σ inS, and consider a sum over the points of intersection with lifted edges e.

PSfrag replacements dev(σ)
ℓ dev(ẽ) (2) Suppose σ and σ ′ agree on the complement of a small neighborhood U p of a vertex p ∈ V, and consider an isotopy for σ and σ ′ which crosses p at some time t 0 ∈ (0, 1) and is constant in S \ U p . Observe that the balance condition (12) is equivalent to the following condition: p∈ e w e t(dev( e)) = 0 .
Indeed, from Equation (12), by lifting to the universal cover and rotating all vectors by π/2, one obtains p∈ e w e y e = 0 , where y e is the unit vector orthogonal to the geodesic containing the edge e, and can thus be interpreted as a unit spacelike vector in R 2,1 . This shows that the result for τ , defined in Equation (13), is the same if computed using σ or σ ′ . (3) Finally, suppose σ and σ ′ only differ in such a way that one of the two paths intersects the same edge e at two consecutive points while the other does not. Then the contributions given by such consecutive intersections cancel out, hence the result is again the same.
Since every two transverse paths connecting x 0 and γ · x 0 can be deformed to one another by a sequence of moves of the three above types, we have shown that the definition of τ (γ) does not depend on the choice of the path σ representing γ.
Second, we need to check τ ∈ Z 1 ρ (π 1 (S), so(2, 1)), that is, τ satisfies the cocycle condition of Equation (6). In fact, let σ 1 be a transverse path connecting x 0 and γ · x 0 as in the definition, and similarly let σ 2 be a transverse path connecting x 0 and η · x 0 . To represent ηγ, we can use the concatenation of σ 1 and η · σ 2 . Let σ be such a concatenation of paths. where we have applied the equivariance of ( G, w) and of dev for the holonomy representation ρ, and the property that the infinitesimal hyperbolic translation along the geodesic ρ(η) · ℓ coincides with the infinitesimal translation along ℓ composed with Adρ(η). This shows that the second term in (16) coincides with Adρ(η) · τ (γ) and thus concludes the claim. It now only remains to show that the definition of Φ does not depend on the choice of the basepoint x 0 . In fact, given another basepoint x ′ 0 , let σ ′ be a path connecting x ′ 0 and γ · x ′ 0 and let τ ′ (γ) be obtained by applying the definition (13) to σ ′ . Then one has where τ 0 is the quantity obtained by a summation, exactly as in (13), along a transverse path which connects x 0 and x 1 . This shows that [τ ] is well-defined in H 1 Adρ (π 1 (S), so(2, 1)).

Geometric description
In this section we will give two types of interpretations of the infinitesimal deformation Φ(G, w) we have produced out of a balanced geodesic graph. The first interpretation should be interpreted as a motivation, since it generalizes infinitesimal twist along simple closed geodesics. The second concerns polyhedral surfaces in Minkowski space, and is then applied to show that Φ is surjective. Some details are omitted, since investigation of these viewpoints is beyond the scope of this paper and is thus left for future work.
Infinitesimal twist along simple closed geodesics. Let us consider a simple closed geodesic c on (S, h). As in Example 4.2, one can turn c into a balanced geodesic graph G c , with constant weight w. If we put w = 1, then Φ(G c , 1) is the infinitesimal left twisting, or infinitesimal left earthquake, along the simple closed geodesic c, see Proposition B.3 in [BS12]. This is exactly the object which appears in Wolpert's formula in the articles [Wol83] and [Wol81].
An infinitesimal left earthquake is easily generalized to the case of weighted multicurves. Weighted multicurves are a particular case of our balanced geodesic graph, see Example 4.3. Another classical generalisation of weighted multicurves are measured laminations. See [SB01], where a generalisation of Wolpert formula to the case of geodesic laminations is given.
Flippable tilings on hyperbolic surfaces. More generally, let us assume G is a geodesic graph, which disconnects S in convex geodesic faces. Then there are (differentiable) deformations h t of the hyperbolic metric h so that h t contains a geodesic graph G t which is a left flippable tiling in the sense of [FS12] (which is the reference to be consulted for more details). Roughly speaking, this means that the faces of G t can be divided into black faces and white faces, and the black faces can be flipped to obtain a new hyperbolic metric h ′ t . The metric h ′ t is also endowed with a geodesic graph, which is a right flippable tiling. As t → 0, the metrics h t and h ′ t converge to the original metric h. The black faces of G t and G ′ t collapse continuously to the vertices of the original graph G, while the white faces converge to the connected components of S \ G. Moreover, the derivatives of the lengths of the edges of the black faces satisfy the balance condition, and thus define (positive) weights w such that (G, w) is a balanced geodesic graph. See Figure 7. A deformation of (G, w) by left and right flippable tilings is not canonical, but by a direct computation one can show that their difference at first order is uniquely determined by (G, w), and coincides with the quantity Φ(G, w) we defined. Namely, where ρ t is the holonomy of h t and ρ ′ t is the holonomy of h ′ t .
Polyhedral surfaces in Minkowski space. Let us now move to the second interpretation of the map Φ. Let ρ : π 1 (S) → SO 0 (2, 1) be the holonomy representation of a hyperbolic surface (S, h) and let [τ ] ∈ H 1 Adρ (π 1 (S), so(2, 1)). Using the isomorphism Λ : R 2,1 → so(2, 1) introduced at the end of Section 3, and the equivariance of Λ for the natural SO 0 (2, 1)actions, we have an isomorphism, which we still denote by Λ, between the spaces of cocycles: Λ : Z 1 ρ (π 1 (S), R 2,1 ) → Z 1 Adρ (π 1 (S), so(2, 1)) . Analogously, it induces an isomorphism between the spaces of coboundaries, and thus we have an isomorphism so(2, 1)) . Recall that the identity component of the isometry group of R 2,1 is isomorphic to SO 0 (2, 1) ⋊ R 2,1 , and that, given a representation ρ : π 1 (S) → SO 0 (2, 1) ⋊ R 2,1 with linear part ρ, the translation part of such representation is a cocycle in Z 1 ρ (π 1 (S), R 2,1 ). Now, let [τ ] ∈ H 1 Adρ (π 1 (S), so(2, 1)) and suppose Σ is a convex polyhedral surface in R 2,1 with spacelike faces, invariant by the action of ρ(π 1 (S)), where the linear part is ρ(π 1 (S)) and the translation part is Λ(τ ). (This corresponds to the lift to R 2,1 of a convex polyhedral surface in a maximal globally hyperbolic flat three-manifold, studied from this point of view in [Mes07].) It then turns out that the Gauss map G : Σ → H 2 is a set-valued map equivariant with respect to the action of ρ on Σ, and of ρ on H 2 . The image of the faces of Σ are points in H 2 , the image of edges of Σ are geodesic edges. Therefore G defines a geodesic graph G on (S, h).
Moreover, we can define a set of weights w on G. Given an edge e of G, let e be any lift of e to H 2 , and let e ′ = G −1 ( e) be the corresponding edge of the polyhedral surface Σ. Then we define w e as the length (for the Minkowski metric) of e ′ , which clearly does not depend on the chosen lift. It can be easily seen that (G, w) is balanced, since the balance condition (12) is exactly equivalent to the condition that the faces of Σ are bounded by a set of edges which "close up" in R 2,1 .
Finally, let us remark that the balanced geodesic graph constructed in this way has positive weights on all edges. Moreover, the connected components of S \ G are convex geodesic polygons.
This construction shows the abundance of examples of balanced geodesic graphs on a closed hyperbolic surface. In fact, it enables to show the following.

De Rham cohomology
In this section we introduce the final tool needed to prove our main theorem, namely de Rham cohomology with values in a certain vector bundle over (S, h). We then conclude the proof of the generalization of Wolpert's formula.
Flat vector bundles. Given a hyperbolic surface (S, h), let ρ : π 1 (S) → SO 0 (2, 1) be its holonomy representation. Let us consider the flat vector bundle F ρ defined in the following way. Let S be the universal cover of S, and let F ρ = ( S × so(2, 1))/π 1 (S) , where π 1 (S) is acting on S × so(2, 1) by Then F ρ is endowed with a flat vector bundle structure with base the original surface (S, h). Hence we can consider de Rham cohomology with values in F ρ , namely where Z 1 dR (S, F ρ ) is the vector space of closed F ρ -valued 1-forms and B 1 dR (S, F ρ ) the subspace of exact 1-forms.
Since α is closed (hence also α) and S is simply connected, τ α (γ) does not depend on the choice of the path σ with fixed endpoints. This authorizes us to use the notation γ·x0 x0 α. By an argument analogous to that of Section 4, changing the basepoint x 0 results in changing τ α by a coboundary.
Moreover, if α = df is exact, where f is a global section of F ρ , then α = d f and f satisfies Hence in this case, by the fundamental theorem of calculus, is a coboundary as in (7). This shows that the map α → τ α induces a well-defined map from H 1 dR (S, F ρ ) to H 1 Adρ (π 1 (S), so(2, 1)). As a warm-up, and for convenience of the reader, we also show that Ψ is injective.
Proof. Suppose the image of α is trivial in the cohomology H 1 Adρ (π 1 (S), so(2, 1)). Since S is simply connected, there exists a function g : S → so(2, 1) such that α = dg. Since [τ α ] = 0, there exists ξ ∈ so(2, 1) such that Now choose a section f 0 ∈ H 0 (S, F ρ ) such that its lift f 0 : S → so(2, 1) at the point x 0 takes the value f 0 (x 0 ) = ξ. By this choice, τ df (γ) = Adρ(γ) · ξ − ξ, and thus τ α−df0 = 0. This implies that where γ ∈ π 1 (S) and σ is is any path connecting x 0 and γ · x 0 . The proof will be concluded if we show that α − df 0 is an exact 1-form. In fact, define f 1 : S → so(2, 1) by Clearly α = d f 0 + df 1 . If we show that f 1 induces a section of F ρ , this will imply that df 1 induces an exact F ρ -valued 1-form on S and α = df 0 + df 1 , and we will be done. In fact, we have where we have used Equation (18) and the equivariance of α − d f 0 . Hence f 1 comes from a section in H 0 (S, F ρ ) and this shows the claim.
Reconstructing 1-forms from cocycles. Although surjectivity of the map H 1 dR (S, F ρ ) → H 1 Adρ (π 1 (S), so(2, 1)) follows from injectivity and the fact that both vector spaces have dimension 3|χ(S)|, we will need an explicit construction for the inverse of this map.
Second, for every edge e of G, let e be a lift of e to S, and put: where: , and the derivative f ′ is compactly supported in (−ǫ, ǫ); • t(dev( e)) is the infinitesimal hyperbolic isometry along the geodesic containing dev( e), where dev is the developing map of (S, h) and e is a lift of the edge e. We choose t(dev( e)) to be the infinitesimal translation on the left, as seen from {δ < 0} to {δ > 0}. Observe that the definition of t(dev( e)) depends on the chosen orientation of dev( e), which is implicitely induced by the choice of the sign of the distance function δ. However, the quantity (w e t(dev( e))) d(f • δ) is independent of such choice, since if one replaces δ by −δ, t(dev( e)) takes the opposite sign, and also d(f • δ) changes sign since f (−x) = −f (x).
Moreover, since dev(γ · e) = ρ(γ) · dev( e), we have t(dev(γ · e)) = Adρ(γ) · t(dev( e)), and therefore α is well-defined as a 1-form with values in F ρ (which is only defined on the complement of the balls B(p, 2ǫ) centered around vertices, for the moment). To conclude the definition, it remains to define α on the balls B(p, 2ǫ) centered at vertices of V. Before doing this, we give a remark.
Remark 6.2. For every path σ : [0, 1] → S transverse to G (which we suppose has image in the complement of the 2ǫ-balls centered at vertices, for the moment), if σ(0) and σ(1) lie outside the ǫ-neighborhood N ǫ ( G), then the integral of α over σ is given by the sum of w e t(dev( e)) over all edges e met by σ, and e = π( e).
Hence for every vertex p of G, if γ is any loop which generates the fundamental group of the annulus B(p, 3ǫ) \ B(p, 2ǫ), γ α = 0 as a consequence of the balance condition (15). It follows that α is exact on the annulus, hence it can be smoothly extended to a closed 1-form on the disc, and in turn on S. See Figure 8.

Proof of the generalized Wolpert formula
We are now ready to prove our main result, namely: Theorem. Let (G, w) and (G ′ , w ′ ) be two balanced geodesic graphs on the hyperbolic surface (S, h). Then ω WP (t (G,w) , t (G ′ ,w ′ ) ) = 1 2 p∈e∩e ′ w e w ′ e cos θ e,e ′ , where e and e ′ are intersecting edges of G and G ′ and θ e,e ′ is the angle of intersection between e and e ′ according to the orientation of S.

PSfrag replacements
p N ǫ (G ′ ) N ǫ (G) Q p Figure 9. The regions Q p in the intersection of the tubular neighborhoods of G and G ′ , which are used in the proof.
If G and G ′ have some non-trasverse intersection, the points of intersection must be counted by perturbing one of the two graphs by a small isotopy, so as to make the intersection transverse.
Proof. Let α and α ′ be the closed 1-forms constructed in the previous subsection, so that Ψ([α] dR ) = Φ(G, w) and Ψ([α ′ ] dR ) = Φ(G ′ , w ′ ). By Goldman theorem (recall Equation (8)), the definition in Equation (14) and Equation (20), we have: To compute the integral, first suppose that all points of intersection of G and G ′ are transverse and do not coincide with any of the vertices in V and V ′ . Replace the ǫ in the construction of α and α ′ by a smaller ǫ if necessary, so that N ǫ (G) and N ǫ (G ′ ) intersect only in the complement of the vertices of G and G ′ . Since α vanishes in the complement of N ǫ (G), and α ′ vanishes in the complement of N ǫ (G ′ ), we have ω WP (t (G,w) , t (G ′ ,w ′ ) ) = 1 4 Nǫ(G)∩Nǫ(G ′ ) κ(α ∧ α ′ ) .
Now, for every point of intersection p of edges e of G and e ′ of G ′ , let Q p be the connected component of N ǫ (G) ∩ N ǫ (G ′ ) containing p. See Figure 9.

Hence we have
To show the claim, by Stokes Theorem we have: Figure 11. Perturbation by a small isotopy, when G and G ′ share an edge.
As we said, if G and G ′ are not transverse, or intersect on vertices, the only caveat in applying the formula (3) is to slightly perturb (topologically) one of the two graphs and use the perturbed graph to count intersection points. It follows from the Theorem that the result will not change if one chooses two different isotopies to perturb G, which therefore result in different edge intersections. Let us remark that this is also a consequence of the balance condition (12).