Representations of -manifolds groups in and their restriction to the boundary
Let be a cusped -manifold – e.g. a knot complement – and note the collection of its peripheral tori. Thurston [Thu79] gave a combinatorial way to produce hyperbolic structures via triangulation and the so-called gluing equations. This gives coordinates on the space of representations of to .
In their paper [NZ85], Neumann and Zagier showed how this coordinates are adapted to describe this space of representations as a lagragian subvariety lying inside a space equipped with a -form – now called Neumann-Zagier symplectic space. And they related this -form with a natural symplectic form on the space of representations of to : the Weil-Petersson form.
Subsequent works of Neumann [Neu92] and Kabaya [Kab07] extended the scope of the previous works. And, more recently, there has been generalizations of this strategy for representations to [BFG12] or [GTZ11, GGZ12, DGG13]. Unfortunately, in the -case, the program of Neumann-Zagier has not been fulfilled: indeed the second part (the link with the Weil-Petersson form) was not achieved. At the very end of the process of writing this paper, a paper of Garoufalidis and Zickert appeared on the ArXiv [GZ13]. Their result are very similar, though the point of view is slightly different.
Let be the -knot complement. Thurston [Thu79] explained the following program to construct its hyperbolic structure:
Triangulate , here thanks to the Riley’s triangulation.
Give a set of parameters to each tetrahedra, here cross-ratios, that describe their hyperbolic structure.
Glue back the tetrahedra, imposing the gluing equations. Those insure that the edge will not become singular.
Add a polynomial condition specifying that the structure is complete, by forcing the peripheral holonomy to be parabolic.
Hence the hyperbolic structure is described by the solution to a polynomial system. Moreover, relaxing the last condition, this parametrize a (Zariski-)open subset of a decorated version of the character variety:
This approach has proven very efficient and is followed in the computer program SnapPy to construct hyperbolic structures on ideally triangulated -manifolds.
This program was further developed by Neumann and Zagier in [NZ85]. By a careful analysis of items 2 and 3, they showed that there is a -vector space (denoted in [Neu92]) carrying an antisymmetric bilinear form such that
the character , through the parameters, is seen as a subvariety of tangent to the kernel of the -form 111More precisely, it is the decorated character variety..
the symplectic quotient of (the so-called Neumann-Zagier symplectic space) is isomorphic to the cohomology group with its Goldman-Weil- Peterson symplectic form ( denotes the peripheral torus).
This presentation uses the more precise version given by Neumann [Neu92]. This construction allows to understand the volume of the representations near the holonomy of the hyperbolic structure [NZ85]. It has been used to give a proof of the local rigidity of the holonomy of the hyperbolic structure [Cho04]. Kabaya [Kab07] investigated the case of being a compact hyperbolic manifold with higher genus boundary.
More recently, several new works revisited Neumann-Zagier strategy and generalized it to understand the character variety:
The reasons of this new interest seems to emanate from two very different fields. First, from a geometric point of view: the construction of representations , following the initial strategy of Thurston, has been undertaken by Falbel [Fal11] in order to investigate the possibility for to carry a CR-spherical structure. Using Neumann-Zagier approach, Bergeron, Falbel and the author [BFG12] gave a description of similar to the one of described above. This leads to a local rigidity result [BFGar] and actual computations (for ) [FKR13]. Those, in turn, leads to construction of geometric structures [DE13]. Another approach is via physical mathematics. I must confess my ignorance and refer to Dimofte and Garoufalidis [DG12] for a presentation. This motivated the works of Garoufalidis, Goerner, Thurston and Zickert [Zic, GTZ11, GGZ12]. They proposed a set of parameters for the case , and generalized partially Neumann-Zagier results for their setting. This also leads to actual computations (mainly when ) by the second named author. Dimofte, Gabella and Goncharov [DGG13] also analyzed the problem for from this point of view, giving a systematic account of a set of coordinates, together with the announcement that they are able to fulfill the Neumann-Zagier strategy. Unfortunately all the proofs are not given in their paper. As mentioned in the abstract, by the very end of the writing of this paper, Garoufalidis and Zickert [GZ13] published another version of this work. Their result and the one discussed in this paper are very similar. However, in my opinion, from a geometrical viewpoint, the approach here allows a better understanding222I think that the point raised in their remark 2.12 is answered here.. As an application of our approach, this gives a variational formula for the volume of a representation, as thoroughly discussed in [DGG13]. Here we present another, more geometric, application: we prove the local rigidity result generalizing [Cho04, BFGar].
This paper links the work of [DGG13] with [BFG12] to complete Neumann-Zagier program in the case of . My feeling is that the coordinates given in [DGG13] are very well adapted to understand of the "lagrangian part" of the strategy of Neumann-Zagier – i.e. describe the analog of the vector space with its form such that is tangent to its kernel in – and define the volume of those representations. But, in order to understand the "symplectic isomorphism part", a direct generalization of [BFG12] seems suitable.
After this rather long introduction, let me warn the reader that this paper heavily relies on three sources:
Fock and Goncharov combinatorics described in [FG06],
Dimofte, Gabella and Goncharov work in [DGG13],
Those works are not easily resumed. So I rather choosed to give precise references to them. This makes this paper absolutely not self-contained. I plan to write later on a more thorough presentation.
2. Triangulation, flags, affine flags and their configurations
2.1. Triangulated manifold
We will consider in this paper triangles and tetrahedra. Those will always be oriented: an orientation is an ordering of the vertices up to even permutations. Note that the faces of a tetrahedron inherits an orientation.
An abstract triangulation is defined as a pair where is a finite family of tetrahedra and is a matching of the faces of the ’s reversing the orientation. For any tetrahedron , we define as the tetrahedron truncated at each vertex. The space obtained from after matching the faces will be denoted by .
A triangulation of an oriented compact -manifold with boundary is an abstract triangulation together with an oriented homeomorphism .
Remark that a knot complement is homeomorphic to the interior of such a triangulated manifold [BFG12, Section 1.2]. And a theorem of Luo-Schleimer-Tillman [LST08] states that, up to passing to a finite cover, any complete cusped hyperbolic -manifold may be seen as the interior of a compact triangulated manifold.
From now on, we fix a triangulation of a compact manifold with boundary . We moreover add some combinatorial hypothesis on the triangulation: we assume that the link of any vertex is a disc, a torus or an annulus – [BFG12, Section 5.1] and [DGG13, Section 2.1]. Thus the boundary decomposes as a union of hexagons lying in the boundary of the complex and discs, tori and annuli lying in the links of the vertices. The latter are naturally triangulated by the traces of the tetrahedra.
2.2. Flags, Affine Flags
As in the work of Fock and Goncharov [FG06], the main technical tool will be the flags, affine flags, and their configuration.
Let , with its natural basis . All our flags will be complete: they are defined as "a line in a plane in a -dim plane… in a hyperplane".
More precisely, consider the exterior powers of and their projectivizations, for to :
Note that and , the dual of . We fix once for all the isomorphism by assigning to the element .
The space of flags in is a subset of . To describe it, recall that acts on each exterior power of , hence diagonally on the product. Moreover the standard flag is defined by:
Then the flag variety is the orbit of
As the stabilizer of is the Borel subgroup of the upper triangular matrices, we have .
The affine flag variety lies above . It is a subset of the product defined as the orbit under of the standard affine flag
As above, we get an isomorphism , where is the subgroup of unipotent upper triangular matrices.
We have a natural projection consisting in projectivizing each coordinates.
Let us introduce an additional notation: if is a flag (or affine flag) and , we denote by its -th coordinate in (or ).
2.3. Tetrahedra of affine flags
Coordinates for a triangle of affine flags may be defined following [FG06]. Consider the -triangulation (see [FG06, Section 1.16]) of a triangle : that is, suppose your triangle is define in the plane by
And consider the triangulation given by the lines or or , for to . Each of this line is oriented as the parallel edge of the triangle (see figure 1). The crossings of this line are the points with integer and non vanishing coordinates , , in the triangle. The oriented lines of the triangulation define a set of oriented edges between these crossings.
For a tetrahedron , we consider the -triangulation of its four faces. As in [BFG12, Section 4.1.1], let be the set of crossings of the lines. Once again, the oriented lines of the triangulation define a set of oriented edges between neighbor points in . We denote by the elements of .
Let be the free -module generated by and its natural basis. Define a -form by, for , :
where is the number of edges from to minus the number of edges from to .
Denote by its dual -module333See [BFG12, Section 4.1.2] for a presentation of -modules, duality and tensorization.. Then, a tetrahedron of affine flags in general position gives a point in by the following rule. Let be an element of . Let be an oriented face containing . Then can be written as the barycenter of , , and with nonnegative integer weights , , verifying . Then define:
The fact that the flags are in general position ensures that .
But there is a problem if lies on an edge and is even: whether we consider to belong to one or the other adjacent face, the relative coordinate may change sign. In order to fix it, we assign to the barycenter of and with weight the coordinate:
First, the less weighted coordinate.
Section 8 of [FG06] proves that a tetrahedra of affine flags is determined by the data:
Moreover, consider the new tetrahedron of affine flags given by multiplying the vector by some (for some and ). Then the vector is related to by:
where is the sum of the points of lying on the -th plane parallel to the face (counted from the face), see figure 2. One checks that the set of vectors generates the kernel .
2.4. Tetrahedra of flags
Consider the map:
given by . Let be its image, and be its dual -module. Then, one checks that this two spaces share the same dimension . Let be the dual map.
To a tetrahedron of flags , one associates a point in by the following way: let be a lift of as a tetrahedron of affine flags. And define
The considerations at the end of the previous section imply that is well-defined. This coincide (up to a sign) with the -coordinates of Fock and Goncharov defined using tri-ratios and cross-ratios, and with the coordinates defined in [BFG12] for .
Note that the space carries a natural -form defined by: if and belong to , then
This form is symplectic. Similarily, carries a symplectic form defined as the projection of to . The forms and match through duality.
Dimofte-Gabella-Goncharov [DGG13] gave coordinates (called octahedron coordinates) for a tetrahedron of flags and relate them to the -coordinates of Fock and Goncharov. They proved (see [DGG13, Section 4]) that the subset of consisting of vectors associated to an actual tetrahedron of flag form a lagrangian submanifold .
The space parametrize the space of framed flat -connections on the boundary of the tetrahdron (i.e. a sphere with four holes). Belonging to is a fillability condition: does the connection extend to the interior of the tetrahedra. In terms of representations of groups, describes the (decorated) representations of the fundamental group of the four-holed sphere that are unipotent (the loop around a hole is mapped to a unipotent element). The representations parametrized by equal the identity.
For now on, we will mostly forget about the lagrangian sub-manifold and work at the level of .
2.5. Holonomy in a tetrahedron
Consider a tetrahedron , and mark three points in each face, one near each vertex. Join the points in the same face and at the same vertex. The resulting graph may be realized as the 3d associahedron [DGG13, Section 4.3]. Then an element in defines a holonomy representation, that is a matrix of associated to each oriented edge of the graph. Indeed, from [FG06], such a parametrize a framed flat connection on the four-holed sphere and as such give an holonomy representation. More precisely, each of the above mentioned point defines a snake and thus a projective basis [FG06, Sections 9.7, 9.8] and [DGG13, Section 4]. The matrics are then base changes. The -case may be explained without the use of snakes, see [BFG12, Section 5.4].
3. Decorated complex and holonomy
We glue here tetrahedra together, in order to get information on the space of representation of . There are constraints, the analogous of the gluing equations.
3.1. Gluing equations
The gluing equations are the conditions we have to impose in order to glue the tetrahedra. So let ,, be the tetrahedra of the triangulation of , and be their coordinates as tetrahedra of flags. Denote by the vertices of the that remain after gluing in the interior of the complex . These vertices belong to the internal faces and edges of . Each element of may be seen as a subset of . This subset consists of two element if the vertex in is in a face of the complex and of several if it is on an edge.
The constraints have been described in [DGG13] and are natural generalization of those of [BFG12].
Indeed, when two faces of and are glued, one should ensures that the triangle of flags
decorating them match (up to the orientation). This translates into:
Faces equation: If the face point in the tetrahedron is glued to the point in , then
Another condition is that the holonomy of looping around an edge should be equal to the identity. This
translates (cf explanation for the holonomy below) into:
Edges equations: For an edge of the complex and , two integer with , let ,, be the tetrahedra abutting to the edge . Then, fix some integer and denote by the -th element of on the edge (counting from ) in any of the . Then we should impose:
3.2. Holonomy and the decorated variety of representations
Let be the orthogonal sum of the and still denote the symplectic form on it. Let us construct a graph by considering the associahedra associated to the tetrahedra and adding an edge between any pair of points lying on glued faces near the same vertex [DGG13, Section 4].
A point represents a set of framed flat -connection on each boundary of the tetrahedra. If it fulfills face and edge equations, this induce a holonomy representation for the graph constructed above. Here is how to compute this representation. First choose a loop in this graph and decompose it into three elementary steps [BFG12, Section 5.4]:
An edge between two vertices of the graph lying on the same face (say the vertices to in the face of a tetrahedron ).
Turning left around an edge in a tetrahedron and landing in the following tetrahedron. That is following the edge from the vertex near in the face of the tetrahedron to the vertex near in the face of the same tetrahedron, and then jump to the vertex near in the face of the glued tetrahedron.
Similarly, turning right around an edge in a tetrahedron and landing in the following tetrahedron.
Then, each of this step corresponds to a base change that can be computed. Indeed, let be the tetrahedron in which it takes place and let be its associated coordinates. Then there are three matrices , and corresponding to the three base changes.
We are not interested here in describing . In the case it is given in [BFG12, Section 5.4] and in the general case may be computed using either [FG06, Section 9] or [DGG13, Section 4]. From the same references, we compute the matrices and . Denote by the -th point in lying on the edge (counting from ). Then we get that is a diagonal matrices depending only on the edge coordinates (see [FG06, Lemma 9.3]):
The computation for is harder. But we are only interested here in its diagonal part. In order to describe it, define to be the product of all for lying at the level above the face and not in the face (see figure 3).
Remark that the fact that a point fulfills the edge and face conditions and the lagrangian constraint implies (and in fact is equivalent to) that if two loops in the graph based at the same vertex of the graph are homotopic in , then their holonomies are equal. This explain why such a parametrizes (decorated) representations of .
4. Coordinates for the boundary and the symplectic isomorphism
Fix once again an element , seen as a collection of framed flat -connections on the tetrahedra boundaries. If it fulfills the face an edge conditions, it should induce a framed flat -connection on . We explain here how to describe this connection using coordinates.
Recall that decomposes as the union of the boundary of the tetrahedra complex and discs, tori or annuli lying in the links of the vertices of . Discs will not need coordinates, as the associated moduli space is trivial. We describe first the coordinates for the boundary of and then for the tori/annuli part.
We define as the orthogonal sum of the , and we keep the notation for its -form.
4.1. Boundary of the complex
The boundary of the complex is homeomorphic to a punctured triangulated surface . We use the usual Fock and Goncharov coordinates for this surface [FG06, Section 9]. Namely, let be the vertex of the -triangulation of . Define . This -module carries a -form defined similarly to using the oriented edges of the -triangulation. Thus there is a map:
We denote by its image and the quotient of by the kernel . Restricting the framed flat -connection given by to the bounday of yields such connexion on . Its coordinates belong to . This operation defines a map that is so defined in coordinates: for , there is a subset of consisting of the that are identified to after gluing. For each of these , denote by the corresponding coordinate of . Then verifies:
4.2. Coordinates for tori and annuli
We choose, once for all, a symplectic basis of the homology for each tori and a generator of the homology together with a generator of the homology relative to the boundary for each annuli, with intersection number . Each of these tori and annuli is the link of a vertex of the tetrahedra complex. We choose for each of them a representative as a path as in section 3.2 which remains near the vertex. Denote by the number of torus links and the number of annulus links.
Using the rules of section 3.2, one may compute the holonomy of this paths. This is always a product of upper-triangular matrices. Denoting by the holonomy representation associated to , one may write:
and define accordingly the number , , for .
The coordinates associates to the boundary are these vectors:
We denote by this vector.
This spaces carry a natural symplectic form, the Goldman-Weil-Peterson form . It is formally defined as the coupling of the cup-product and the Killing form on . We will define it precisely later on.
The main result of our paper is stated as follows:
Restricted to the subvariety of defined by the face and edge conditions, the -form is the pull-back by the map
of a -form on .
Moreover coincide with the Weil-Petersson form in restriction on each torus or annulus and with in restriction to . For this form , the tori part is orthogonal to the annuli part and the boundary part. However there is a coupling between the annuli and boundary parts.
The form should be the Weil-Petersson form on the space of representation of . Unfortunaltely, this is not yet clear from the literature.
In order to prove this theorem, we remark that, let alone the lagrangian condition, every condition is expressed as "a product of -coordinates". So this is a good idea to linearize everything.
This section is a direct generalization of [BFG12, Section 7].
5.1. Face and edge conditions
We consider another -module444This corresponds to in [BFG12].: . Recall that is the set of vertices of the -triangulations of the tetrahedra that remain in the interior of the complex after gluing. Let be its natural basis. Any may be seen as a subset of . This yields a map:
By duality, one gets a line of applications:
From now on, we identify with its dual through the canonical basis.
The composition vanishes.
This is an inspection without difficulty. For example, if is inside a face of the complex , is of the form . Applying , for each neighbor of , we get a vector . Hence, the -component of vanishes, as well as every other component. ∎
Following closely [BFG12, Section 7.3], letting be the map induced by and be the map followed by the canonical projection from to , we get a complex:
Similarly, letting and be the restriction of to we get the dual complex:
We define the homology groups of these two complexes:
We note that:
The symplectic forms and thus induce skew-symmetric bilinear forms on and . These spaces are obviously dual spaces and the bilinear forms match through duality.
We claim that linearize the face and edge equations:
An element fulfills the face and edge equations if and only if:
Once again this is proved by inspection: the component of is the product of the component for belonging to . If sits on a face, this gives a face condition; if sits on an edge, this gives an edge condition. ∎
5.2. Coordinates for the links
The coordinates we have constructed for a torus may be seen as an element of . We construct now a map at the level of the chains. Once again, we are very close of [BFG12, Section 7.1].
5.2.1. Simplicial decompositions of the links
Each boundary surface in the link of a vertex is triangulated by the traces of the tetrahedra; from this we build the CW-complex whose edges consist of the inner edges of the first barycentric subdivision, see Figure 4. We denote by the dual cell division. Let and be the corresponding chain groups. Given two chains and we denote by the (integer) intersection number of and . This defines a bilinear form which induces the usual intersection form on (or between the homology and the homology relative to the boundary in the annulus case). In that way is canonically isomorphic to the dual of .
5.2.2. Goldman-Weil-Petersson form for tori
Here we equip
with the bilinear form defined by coupling the intersection form with the scalar product on seen as the space of roots of with its Killing form. We describe more precisely an integral version of this.
From now on we identify with the subspace via the map sending the -th vector of the canonical basis to , the entry being the -th.
We let be the standard lattice in where all the coordinates are in . We identify it with using the above basis of . The restriction of the usual euclidean product of gives a product, denoted , on (the “Killing form”)555In terms of roots of , the choosen basis is the usual basis of positive simple roots.. In other words, the matrix of the scalar product is the Cartan matrix: all entries are , except the diagonal which is filled with and the upper and lower-diagonals, filled with .
Identifying with using the scalar product , the dual lattice becomes a lattice in ; an element belongs to if and only if for every .
We consider and define by the formula
This induces a (symplectic) bilinear form on 666Or a coupling between the homology and the homology relative to the boundary in the annulus case. which we still denote by . Note that identifies with the dual of .
The canonical coupling identifies with . This last space is naturally equipped with the “Goldman-Weil-Petersson” form , dual to . Let be the natural scalar product on dual to : letting be the map defined by , we have . On the bilinear form wp induces a symplectic form — the usual Goldman-Weil-Petersson symplectic form — formally defined as the coupling of the cup-product and the scalar product .
5.3. Peripheral holonomy
To any decoration we now explain how to associate an element
We may represent any class in by an element in where is a closed path in (seen as the link of the corresponding vertex in the complex ). Using the decoration we may compute the holonomy of the loop , as explained in Section 3.2 (see also section 4.2): it is an upper triangular matrix. Let us write the diagonal part:
The application which maps to is the announced element of .
In the case of an annulus, we obtain in the same way a map, still denoted , from to .
The choice of a given longitude and meridian gives a basis of . It allows to identify with . This explains our definition of coordinates in section 4.2.
5.3.1. Linearization of the holonomy elements
We now linearize the map , i.e. we explain how the computations of the eigenvalues of the holonomy of the torus may be done in our framework of -modules.
We define the linear map on a basis. Let be the edge turning left around the edge in the tetrahedron , see Figure 5. For , let be the tensor of with the -th canonical basis vector of . Parametrize the points of in the two faces containing the edge : (for and is the point at the level from the face at the position (counted algebraically and rightward from the point on the edge ) – see figure 5. Then we define:
Remark that the second sum is empty for odd.
The claim that linearizes is given by the following:
Let . Seeing as an element of , we have:
The proof of this lemma goes along exactly the same lines as [BFG12, Lemma 7.2.1]. It is a lengthy inspection, whose major difficulty is to define reasonable notations. We postpone it until the last section.
Let be the map dual to . Note that for any and we have
Now composing with and identifying with using we get a map
and it follows from equation (5) that for any and we have
In the following we let and be the orthogonal sum of the ’s and ’s for each torus or annulus link . We abusively denote by and the product of the maps defined above on each .
6. Homologies and symplectic isomorphism in the closed case
Le us first assume that is a closed complex. In that case all links are tori. We will come back latter on the general case.
6.1. Homologies in the closed case
We defined the homology groups and and the chain groups of the simplicial decomposition. We claim here that induces well defined map in homology. This will allow to state our main technical theorem in the closed case.
Let and be the subspaces of cycles and boundaries in . The following lemma is easily checked by inspection.
In particular induces a map in homology. By duality, the map induces a map as follows from:
First of all, is the orthogonal of for the coupling . Moreover, by definition of , if , we have:
The last condition is given by the previous lemma. The second point is similar. ∎
Note that and are canonically isomorphic so that we identified them (to ) in the following.
The map is multiplication by .
Given and , we have
As a corollary, one understands the homology of the various complexes.
The map induces an isomorphism from to . Moreover we have .
The form on is the pullback of on by the map .
The proofs of the corollaries from the theorem is given in [BFG12, Section 7.4]. You just have to adapt the dimension of and : for , it is ( is the number of tori links).
6.2. Proof of theorem 2
We want to compute . Even if it seems simple, this is a point where a new approach was needed. In [BFG12], this was dealt with a direct computation, but did not show how to generalize it.
Let us first work in a single tetrahedron . We denote by the edge of corresponding to a (left) turn around the edge and we denote by its dual edge in , see Figure 4. We use the notation to denote the tensor product of with the -th canonical basis vector (still denoted ) of .
Then, for any vector , there exists a vector in such that the image decomposes as:
In view of equation 7, we need to compute on the image of .