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 Teichmüller 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 Teichmüller 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.
Given a closed oriented surface of genus , Teichmüller space is classically defined as the space of complex structures on , up to biholomorphisms isotopic to the identity. By the uniformization theorem, it is naturally identified to what is sometimes called Fricke space , namely the space of hyperbolic metrics on up to isometries isotopic to the identity.
The Weil–Petersson metric is a Kähler metric on , whose definition only involves the point of view of as the space of complex structures on . 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 on a fixed closed hyperbolic surface , let us denote by and the infinitesimal twists along and , namely
where is the new hyperbolic surface obtained from by a left earthquake along — that is, by cutting along and re-glueing after a left twist of length . Then Wolpert showed that, if denotes the symplectic form of the Weil–Petersson Kähler metric, then
where, for every point of intersection between and , denotes the angle of intersection at .
Balanced geodesic graphs
In this paper, we will define a more general type of infinitesimal deformations on , which generalize twists along simple closed geodesics. These will be associated to a balanced geodesic graph, namely a weighted graph on whose edges are geodesic segments and whose weights satisfy a balance condition at every vertex , namely:
where the sum is over all edges which are adjacent to the vertex , denotes the weight of and is the tangent vector to in .
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 for all edges. The balance condition is then trivially satisfied. In fact, the infinitesimal deformation we define reduces to the infinitesimal twist in this case.
In order to define , it is easier to use the identification of to the space of discrete and faithful representations of into (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 on its holonomy representation . The tangent space to the space of representations up to conjugacy is then known to be identified (see [Gol84]) to the group cohomology . The tangent vector is then defined by lifting to the universal cover of a generic closed loop representing , and taking the weighted sum of the infinitesimal twists along the lifts of the edges of met by the lift of the closed loop. The balance condition ensures that is well-defined. If is a simple closed geodesic with weight 1, actually coincides with the variation of the holonomy of the twisted metrics , and we thus recover the infinitesimal twist .
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 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 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 and the weights 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 is a disjoint union of convex polygons, is naturally the dual to convex polyhedral surface in a flat globally hyperbolic manifold of dimension three homeomorphic to . Moreover, as observed by Mess, the isomorphism between Minkowski space and the Lie algebra induces a correspondence between the tangent space of , in the model of the representation variety, and the translation part of the holonomies of manifolds as above.
This enables us to show that the map which associated to a balanced geodesic graph the deformation in is surjective. In other words, any tangent vector to 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 .
Let us now come to the statement of the main result. Recall that denotes the symplectic form of the Weil–Petersson metric on .
Let and be two balanced geodesic graphs on the hyperbolic surface . Then
where and are intersecting edges of and and is the angle of intersection between and according to the orientation of .
Therefore, when 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 as for some balanced geodesic graph , 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 and do not have vertices in common and intersect transversely, it is clear what the intersection points and angles are. On the other hand, if and 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 are the angles between the original geodesic edges .
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 equals (under the holonomy map) a form defined only in terms of the group cohomology . The second tool is de Rham cohomology with values in certain flat vector bundles over of rank . 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.
We would like to thank Jean-Marc Schlenker for many discussions and encouragements. The second author would like to thank Gabriele Mondello for several discussions and for the help in figuring out the correct factors from the literature.
2. Teichmüller space and Weil–Petersson metric
Let be a closed oriented surface of genus . The Teichmüller space of is defined as:
where is the group of diffeomorphisms of isotopic to the identity, and it acts by pre-composition of a complex atlas. Namely, two complex structures on are equivalent in if and only if there exists a biholomorphism isotopic to the identity. In this section we collect some preliminary results on Teichmüller space, spaces of representations of the fundamental group of , and Weil–Petersson metric.
Teichmüller space is a manifold of real dimension , and is endowed with a structure of complex manifold. Moreover, it is endowed with several metric structures, one of which is the Weil–Petersson metric, which turns out to be a natural Kähler structure on . Let us recall briefly its definition.
Let us fix a complex structure on . It is known that the tangent space of is naturally identified to a quotient of the vector space of Beltrami differentials, namely sections of the vector bundle , where is the canonical bundle of . More precisely, , where is the subspace of Beltrami differentials which induce trivial infinitesimal deformations of .
On the other hand, the cotangent space is naturally identified to the space of holomorphic quadratic differentials . The identification is given by the pairing on defined by
In fact, if in local complex coordinates and , then is a quantity which can be naturally integrated over . The fundamental property is then the fact that for every ,
Hence we have a vector space isomorphism .
The Weil–Petersson product is then easily defined on the cotangent space, by
where is the unique hyperbolic metric (i.e. Riemannian metric of constant curvature ) compatible with the complex structure , provided by the Uniformization Theorem ([Koe09]). By a similar argument as above, the quantity is indeed of the correct type to be integrated on .
Hyperbolic metrics and Fuchsian representations
We will be using other two important models of .
By the aforementioned Uniformization Theorem, given any complex structure on , there is a unique hyperbolic metric on compatible with . Let us define the Fricke space of as:
where clearly acts by pull-back. It is easy to check that the map is equivariant for the actions of , and therefore induces a diffeomorphism
The inverse of is simply the map which associates to a hyperbolic metric the complex structure induced by , which is obtained by choosing local isothermal coordinates.
Given a hyperbolic metric on , let be the universal cover of . Then is a hyperbolic metric, which is complete since is compact, on the simply connected surface . Hence is isometric to the hyperbolic plane by [Rat06, Theorem 8.6.2]. Such an isometry (chosen to be orientation-preserving) is unique up to post-compositions with elements in group of orientation-preserving isometries of , and is called developing map. Let us denote it by . It turns out to be equivariant for some representation , called the holonomy map. That is:
for every . Since dev is well-defined up to post-composition, is well-defined up to conjugacy by elements of . This provides a map
where is the character variety
By a theorem of Goldman [Gol80], this map is a diffeomorphism onto the space of faithful and discrete representations (called Fuchsian representations) up to conjugacy, which is precisely a connected component of :
and denotes the faithful and discrete representations.
The model of Teichmüller space as enables to give a simple description of the tangent space. In fact, using the differentials of the maps and , we can identify
where . In [Gol84], the tangent space to the space of representations is described as the group cohomology with values in the Lie algebra :
The vector space is the quotient
is the space of cocycles with respect to the adjoint action of , that is, functions with values in the Lie algebra satisfying:
This is essentially the condition of being a representation of into , at first order.
is the space of coboundaries, namely cocycles of the form
for some . This is the first-order condition for a deformation of being trivial in , that is, of being tangent to a path of representations obtained from by conjugation.
Goldman symplectic form
In the fundamental paper [Gol84], Goldman introduced a symplectic form on the space (actually, the construction holds when replacing by a more general Lie group ) and showed that it coincides (up to a factor) with the symplectic form of the Weil–Petersson Hermitian metric.
The Goldman form is defined as follows. Recall from the previous section that the tangent space is identified to the group cohomology . 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 . The identification between and is then given by evaluation on the fundamental top-dimensional class of the closed oriented manifold .
Recall that the map associates to the Teichmüller class of a complex structure on the holonomy representation of the uniformizing hyperbolic metric . The differential of should hence be consider as a vector space isomorphism
In [Gol84], Goldman proved:
Goldman uses the trace form in the model of , which is defined as , instead of the Killing form which turns out to be .
The original theorem of Goldman concerns the Weil–Petersson metric on , hence the choice of an identification between and may result in different coefficients for the Weil–Petersson metric.
There actually is another description of the Goldman form in terms of de Rham cohomology, which will be introduced in Section 6.
3. Some properties of the hyperboloid model
It will be useful for this paper to consider the hyperboloid model of . Namely, let us consider (2+1)-dimensional Minkowski space, which is the vector space endowed with the standard bilinear form of signature :
It turns out that the induced bilinear form on the upper connected component of the two-sheeted hyperboloid (which is simply connected) gives a complete hyperbolic metric. It is thus isometric to , 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 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 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 of the classical Euclidean cross product. This provides an isomorphism
for any . More explicitly,
An example is given by hyperbolic isometries. Every geodesic of is of the form , for some with , and is the plane orthogonal to the vector for . Moreover, the orientation of and the direction of determine an orientation of . Then we define the hyperbolic isometry which preserves setwise and translates every point of by a length according to the orientation of . We will also denote
namely, is the generator of the 1-parameter group , or in other words the infinitesimal translation along .
If we pick , then (see also Figure 1) and
Hence the infinitesimal isometry is
In general, if is an (oriented) geodesic, intersection of and endowed with the induced orientation, then the following formula holds:
Additional structures on
Finally, two important properties of the isomorphism are the following. See also [BS16a, Section 2]
is equivariant for the actions of : the standard action on and the adjoint action on . Namely, for every ,
is an isometry between the Minkowski metric and the Killing form on , up to a factor:
where the Killing form for is:
4. 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 from any balanced geodesic graph on . For convenience, we will use in the hyperboloid model defined in the previous section, and hence we will identify and .
Balanced geodesic graphs
Let us fix a hyperbolic metric on the closed oriented surface .
A balanced geodesic graph on is the datum of
A finite embedded graph in , where is the set of (unoriented) edges of and is the set of vertices;
The assignment of weights for every ;
satisfying the following conditions:
Every edge is a geodesic interval between its endpoints;
For every , the following balance condition holds:
where denotes that is an endpoint of an edge , and in this case is the unit tangent vector at to the geodesic edge .
We provide several classes of examples which should account for the abundance of such objects on any surface .
Given a simple closed geodesic on and a weight , is the support of a balanced geodesic graph with weight . In fact, it suffices to declare that the vertex set consists of points on . Then is split into edges, and we declare that each edge has weight . 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.
More generally, given any finite collection of (not necessarily simple) closed geodesics , and any choice of weights , the union can be made into a balanced geodesic graph. In fact, it suffices to choose the vertices of on the geodesics , so that every intersection point between and some other geodesic (including self-intersections of ) is in the vertex set . Moreover, it is necessary to add vertices to geodesics which are disjoint from all the geodesics (in particular, has no self-intersection) 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 is . In fact, in this case the balance condition will be automatically satisfied, since at every vertex , tangent vectors come in opposite pairs with the same weight. (A pair is composed of the two opposite vectors tangent to the same geodesic .) See Figure 2. Therefore the balance condition (12) is satisfied regardless of the initial choice of . In particuar, weighted multicurves (i.e. collections of disjoint simple closed geodesics endowed with positive weights) are balanced geodesic graphs.
Colin de Verdière in [CdV91] proved that, given any topological triangulation of , and any choice of positive weights assigned to each edge , there exists a geodesic triangulation, with the same combinatorics of the original triangulation, which satisfies the balance condition (12) for the prescribed weights . Therefore, this geodesic realization of a topological triangulation is a balanced geodesic graph in the sense of Definition 4.1.
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 defined in this way is naturally endowed with a structure of vector space, defined in the following way. For in and , we define . For the addition, we define , where is the refinement of and , and are the weights on the edge set of defined as follows. When the refinement does not create new vertices for and , then the same weights are kept on the corresponding edges. When the refinement creates a new vertex for but not for , new weights are assigned as in the bottom of Figure 3. When the refinement creates a new vertex for both and , 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 , the subset of composed of classes of balanced geodesic graphs having underlying geodesic graph is a finite-dimensional vector subspace.
Construction of the map
We are now ready to define the map from the space to the tangent space of Teichmüller space. We will actually define a deformation of the holonomy representation of , thus providing a tangent vector to the character variety of . Namely, we will define a map:
where . For this purpose, consider the universal cover and fix a developing map
which is a global isometry, equivariant for the representation .
Let us fix a balanced geodesic graph and lift it to on . Let us also fix a basepoint , which we assume does not lie in . We define a cocycle in the following way. We say a path is transverse to if the intersection of the image of and consists of a finite number of points, which are not vertices of , and for every point there exists such that and are contained in two different connected component of , where is a small neighborhood of in .
Now, pick and let be a path such that and , transverse to . Then we define the following element of :
The sum is over all points of intersection of the image of the path with the lift of the balanced geodesic graph , see Figure 4;
If is an oriented (subinterval of) a geodesic of , recall from Section 3 that we denote the infinitesimal hyperbolic translation along , using the orientation of . Namely,
where is the isometry of which preserves and translates every point of by a length according to the orientation of . This is thus applied in Equation (13) to the geodesic of which contains the image of under the developing map dev. See Figure 5.
The coefficient in the sum equals the weight of the edge .
We then define
Under the identification between and , we thus define
There is a number of points to be verified in order to check that the map is well-defined. First, we need to show that the value does not depend on the chosen path , as long as is transverse to . (This is the same as choosing a representative of the closed loop in , based at , which represents and is transverse to .)
In fact, there are three cases to consider. See also Figure 6.
If two representatives and can be isotoped to one another by a family of paths which is transverse for all , then the value of is the same when computed with respect to or , since the quantities and only depend on the edge , and not on the intersection point of with .
Suppose and agree on the complement of a small neighborhood of a vertex , and consider an isotopy for and which crosses at some time and is constant in . Observe that the balance condition (12) is equivalent to the following condition:
Indeed, from Equation (12), by lifting to the universal cover and rotating all vectors by , one obtains
where is the unit vector orthogonal to the geodesic containing the edge , and can thus be interpreted as a unit spacelike vector in . This shows that the result for , defined in Equation (13), is the same if computed using or .
Finally, suppose and only differ in such a way that one of the two paths intersects the same edge 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 and can be deformed to one another by a sequence of moves of the three above types, we have shown that the definition of does not depend on the choice of the path representing .
Second, we need to check , that is, satisfies the cocycle condition of Equation (6). In fact, let be a transverse path connecting and as in the definition, and similarly let be a transverse path connecting and . To represent , we can use the concatenation of and . Let be such a concatenation of paths. Then we have
Now, the first term in the summation is , while the second term equals
where we have applied the equivariance of and of dev for the holonomy representation , and the property that the infinitesimal hyperbolic translation along the geodesic coincides with the infinitesimal translation along composed with . This shows that the second term in (16) coincides with and thus concludes the claim.
It now only remains to show that the definition of does not depend on the choice of the basepoint . In fact, given another basepoint , let be a path connecting and and let be obtained by applying the definition (13) to . Then one has
where is the quantity obtained by a summation, exactly as in (13), along a transverse path which connects and . This shows that is well-defined in .
5. Geometric description
In this section we will give two types of interpretations of the infinitesimal deformation 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 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 on . As in Example 4.2, one can turn into a balanced geodesic graph , with constant weight . If we put , then is the infinitesimal left twisting, or infinitesimal left earthquake, along the simple closed geodesic , 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 is a geodesic graph, which disconnects in convex geodesic faces. Then there are (differentiable) deformations of the hyperbolic metric so that contains a geodesic graph 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 can be divided into black faces and white faces, and the black faces can be flipped to obtain a new hyperbolic metric . The metric is also endowed with a geodesic graph, which is a right flippable tiling. As , the metrics and converge to the original metric . The black faces of and collapse continuously to the vertices of the original graph , while the white faces converge to the connected components of . Moreover, the derivatives of the lengths of the edges of the black faces satisfy the balance condition, and thus define (positive) weights such that is a balanced geodesic graph. See Figure 7.
A deformation of 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 , and coincides with the quantity we defined. Namely,
where is the holonomy of and is the holonomy of .
Polyhedral surfaces in Minkowski space
Let us now move to the second interpretation of the map . Let be the holonomy representation of a hyperbolic surface and let . Using the isomorphism introduced at the end of Section 3, and the equivariance of for the natural -actions, we have an isomorphism, which we still denote by , between the spaces of cocycles:
Analogously, it induces an isomorphism between the spaces of coboundaries, and thus we have an isomorphism
Recall that the identity component of the isometry group of is isomorphic to
and that, given a representation with linear part , the translation part of such representation is a cocycle in . Now, let and suppose is a convex polyhedral surface in with spacelike faces, invariant by the action of , where the linear part is and the translation part is . (This corresponds to the lift to 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 is a set-valued map equivariant with respect to the action of on , and of on . The image of the faces of are points in , the image of edges of are geodesic edges. Therefore defines a geodesic graph on .
Moreover, we can define a set of weights on . Given an edge