Tetrahedra of flags, volume
and homology of
Abstract.
In the paper we define a “volume” for simplicial complexes of flag tetrahedra. This generalizes and unifies the classical volume of hyperbolic manifolds and the volume of CR tetrahedra complexes considered in [4, 6]. We describe when this volume belongs to the Bloch group and more generally describe a variation formula in terms of boundary data. In doing so, we recover and generalize results of NeumannZagier [13], Neumann [11], and Kabaya [10]. Our approach is very related to the work of Fock and Goncharov [7, 8].
Contents
 1 Introduction
 2 Configurations of flags and crossratios
 3 Tetrahedra of flags and volume
 4 Decoration of a tetrahedron and the preBloch group
 5 Decoration of a tetrahedra complex and its holonomy
 6 Some linear algebra and the unipotent case
 7 NeumannZagier bilinear relations for
 8 Homologies and symplectic forms
 9 Extension to the general case
 10 Examples
 11 Applications
1. Introduction
It follows from Mostow’s rigidity theorem that the volume of a complete hyperbolic manifold is a topological invariant. In fact, it coincides with Gromov’s purely topological definition of simplicial volume. If the complete hyperbolic manifold has cusps, Thurston showed that one could obtain complete hyperbolic structures on manifolds obtained from by Dehn surgery by gluing a solid torus with a sufficiently long geodesic. Thurston’s framed his results for more general deformations which are not complete hyperbolic manifolds, the volume of the deformation being the volume of its metric completion. Neumann and Zagier [13] and afterwards Neumann [11] provided a deeper analysis of these deformations and their volume. In particular, they showed that the variation of the volume depends only on the geometry of the boundary and they gave a precise formula for that variation in terms of the boundary holonomy.
It is natural to consider an invariant associated to a hyperbolic structure defined on the preBloch group which is defined as the abelian group generated by all the points in where is a field quotiented by the 5term relations (see section 3 for definitions and references). The volume function is well defined as a map using the dilogarithm. The Bloch group is a subgroup of the preBloch group . It is defined as the kernel of the map
given by . The volume and the ChernSimons invariant can then be seen through a function (the Bloch regulator)
The imaginary part being related to the volume and the real part related to ChernSimons mod invariant.
Several extensions of NeumannZagier results were obtained. Kabaya [10] defined an invariant in associated to a hyperbolic 3manifold with boundary and obtained a description of the variation of the volume function which depends only on the boundary data. Using different coordinates and methods Bonahon [3] showed a similar formula.
The volume function was extended in [4, 6] in order to deal with CauchyRiemann (CR) structures. More precisely, consider with the contact structure obtained as the intersection where is the multiplication by in . The operator restricted to defines the standard CR structure on . The group of CRautomorphisms of is and we say that a manifold has a spherical CR structure if it has a  geometric structure. Associated to a CR triangulation it was defined in [6] an invariant in which is in the Bloch group in case the structure has unipotent boundary holonomy. The definition of that invariant is valid for “crossratio structures” (which includes hyperbolic and CR structures) as defined in [4]. It turns out to be a coordinate description of the decorated triangulations described bellow and the invariant in coincides with the one defined before up to a multiple of four.
We consider in this paper a geometric framework which includes both hyperbolic structures and CR structures on manifolds. We are in fact dealing with representations of the fundamental group of the manifold in that are parabolic : the peripheral holonomy should preserve a flag in . Recall that a flag in is a line in a plane of . The consideration of these representations links us to the work of Fock and Goncharov [7, 8]. Indeed we make an intensive use of their combinatorics on the space of representations of surface groups in .
As in the original work of Thurston and NeumannZagier, we work with decorated triangulations. Namely, let be a triangulation of a 3manifold . To each tetrahedron we associate a quadruple of flags (corresponding to the four vertices) in . In the case of ideal triangulations, where the manifold is obtained from the triangulation by deleting the vertices, we impose that the holonomy around each vertex preserves the flag decorating this vertex. Such a decorated triangulation gives a set of flag coordinates, and more precisely two sets: affine flag coordinates and projective flag coordinates . Those are, in the Fock and Goncharov setting, the  and coordinates on the boundary of each tetrahedron, namely a fourholed sphere.
The main result in this paper is the construction of an element associated to a decorated triangulation and a description of a precise formula for in terms of boundary data. This formula is given in theorem 5.14.
The core of the proof of theorem 5.14 goes along the same lines of the homological proof of Neumann [11]. We nevertheless believe that the use of the combinatorics of Fock and Goncharov sheds some light on Neumann’s work. The two theories fit well together, allowing a new understanding, in particular, of the “NeumannZagier” symplectic form.
The organisation of the paper is as follows. In section 2 we describe flags and configurations of flags. Following [7], we define  and coordinates for configurations of flags. These data define a decorated tetrahedron. In section 3 we define an element in the preBloch group associated to a decorated tetrahedron (cf. also [6]). We then define the volume of a decorated tetrahedron and show how previous definitions in hyperbolic and CR geometry are included in this context. We moreover relate our work to Suslin’s work on , showing that our volume map is essentially Suslin map from to the Bloch group. This gives a geometric and intuitive construction of the latter. Here we are very close to the work of Zickert on the extended Bloch group [17]. In the next section 4 we associate to a decorated tetrahedron the element and compute it using both coodinates and coordinates.
This local work being done, we move on in section 5 to the framework of decorated simplicial complexes. The decoration consists of coordinates or coordinates associated to each tetrahedron and satisfying appropriate compatibility conditions along edges and faces. The main result is the computation of which turns out to depend only on boundary data (Theorem 5.14).
We first give a proof of Theorem 5.14 when the decoration is unipotent. We then deal with the proof of the general case. In doing so we have to develop a generalization of the NeumannZagier bilinear relations to the case. In doing so the GoldmanWeilPetersson form for tori naturally arises. We hope that our proof sheds some light on the classical case.
In section 10 we describe all unipotent decorations on the complement of the figure eight knot. It was proven by P.V. Koseleff that there are a finite number of unipotent structures and all of them are either hyperbolic or CR. The natural question of the rigidity of unitotent representation will be investigated in a forthcoming paper [1] (see also [9]).
Finally in section 11, we describe applications of theorem 5.14. First, we follow again NeumannZagier and obtain an explicit formula for the variation of the volume function which depends on boundary data. Then, relying on remarks of Fock and Goncharov, we describe a form on the space of representations of the boundary of our variety which coincides with WeilPetersson form in some cases (namely for hyperbolic structures and unipotent decorations).
We thank J. Genzmer, P.V. Koseleff and Q. Wang for fruitful discussions.
2. Configurations of flags and crossratios
We consider in this section the flag variety and the affine flag variety of over a field . We define coordinates on the configurations of flags (or affine flags), very similar to the coordinates used by Fock and Goncharov [7].
2.1. Flags, affine flags and their spaces of configuration
We set up here notations for our objects of interest. Let be a field and . A flag in is usually seen as a line and a plane, the line belonging to the plane. We give, for commodity reasons, the following alternative description using the dual vector space and the projective spaces and :
We define the spaces of affine flags and flags by the following:
The space of flags is identified with the homogeneous space , where is the Borel subgroup of uppertriangular matrices in . Similarly, the space of affine flags is identified with the homogeneous space , where is the subgroup of unipotent uppertriangular matrices in .
2.2.
Given a space , we classicaly define the configuration module of ordered points in as follows. For , let be the free abelian group generated by the set
of all ordered set of points in . The group acts on and therefore also acts diagonally on giving it a left module structure.
We define the differential by
then we can check that every is a module homomorphism and . Hence we have the complex
The augmentation map is defined on generators by for each . If is infinite, the augmentation complex is exact.
For a left module , we denote its group of coinvariants, that is,
Taking the coinvariants of the complex , we get the induced complex:
with differential induced by . We call the homology of this complex.
2.3.
We let now and . For every integer , the module of coinvariant configurations of ordered flags is defined by:
The natural projection gives a map
We will study in this paper the homology groups (which is the third group of discrete homology of ), and .
It is usefull to consider a subcomplex of of generic configurations which contains all the information about its homology. We leave to the reader the verification that indeed the definition below gives rise to subcomplexes of and
2.4.
A generic configuration of flags , is given by points in general position and lines in such that if . We will denote and the corresponding module of configurations and its coinvariant module by the diagonal action by .
A configuration of ordered points in is said to be in general position when they are all distinct and no three points are contained in the same line. Observe that the genericity condition of flags does not imply that the lines are in a general position.
2.5.
Since acts transitively on , we see that if , and the differential is zero.
In order to describe consider a configuration of generic flags . One can then define a projective coordinate system of : take the one where the point has coordinates , the point has coordinates , the point has coordinates and the intersection of and has coordinates . The line then has coordinates where
is the triple ratio. We have . Moreover the differential is given on generators by and therefore .
We denote by the triple ratio of a cyclically oriented triple of flags . Note that . Observe that when the three lines are not in general position.
2.6. Coordinates for a tetrahedron of flags
We call a generic configuration of flags a tetrahedron of flags. The coordinates we use for a tetrahedron of flags are the same as those used by Fock and Goncharov [7] to describe a flip in a triangulation. We may see it as a blowup of the flip into a tetrahedron. They also coincide with coordinates used in [4] to describe a crossratio structure on a tetrahedron (see also section 3.8).
Let be an element of . Let us dispose symbolically these flags on a tetrahedron (see figure 1). We define a set of 12 coordinates on the edges of the thetrahedron ( for each oriented edge) and four coordinates associated to the faces.
To define the coordinate associated to the edge , we first define and such that the permutation is even. The pencil of (projective) lines through the point is a projective line . We naturally have four points in this projective line: the line and the three lines through and one of the for . We define as the crossratio^{1}^{1}1Note that we follow the usual convention (different from the one used by Fock and Goncharov) that the crossratio of four points on a line is the value at of a projective coordinate taking value at , at , and at . So we employ the formula for the crossratio. of these four points:
We may rewrite this crossratio thanks to the following useful lemma.
2.7 Lemma.
We have . Here the determinant is w.r.t. the canonical basis on .
Proof.
Consider the following figure:
By duality, is the crossratio between the points and on the line . Now, is a linear form vanishing at and is a linear form vanishing at . Hence, on the line , the linear form is proportional to and is proportional to . This proves the formula. ∎
2.8.
Each face inherits a canonical orientation as the boundary of the tetrahedron . Hence to the face , we associate the ratio of the corresponding cyclically oriented triple of flags:
Observe that if the same face (with opposite orientation) is common to a second tetrahedron then
Figure 3 displays the coordinates.
2.9.
Of course there are relations between the whole set of coordinates. Fix an even permutation of . First, for each face , the ratio is the opposite of the product of all crossratios “leaving” this face:
(2.9.1) 
Second, the three crossratio leaving a vertex are algebraically related:
(2.9.2) 
Relations 2.9.2 are directly deduced from the definition of the coordinates , while relation 2.9.1 is a direct consequence of lemma 2.7.
At this point, we choose four coordinates, one for each vertex: , , , . The next proposition shows that a tetrahedron is uniquely determined by these four numbers, up to the action of . It also shows that the space of crossratio structures on a tetrahedron defined in [4] coincides with the space of generic tetrahedra as defined above.
2.10 Proposition.
A tetrahedron of flags is parametrized by the tuple of elements in .
Proof.
Let be the canonical basis of and its dual basis. Up to the action of , an element of is uniquely given, in these coordinates, by:

, ,

, ,

, and

, .
Observe that and by the genericity condition. Now we compute, using lemma 2.7 for instance, that , , , , completing the proof. ∎
We note that one can then compute on the generators of to be
2.11. Coordinates for affine flags
We will also need coordinates for a tetrahedron of affine flags (the coordinates in Fock and Goncharov [7]). Let be an element of . We also define a set of 12 coordinates on the edges of the thetrahedron (one for each oriented edge) and four coordinates associated to the faces:
We associate to the edge the number and to the face (oriented as the boundary of the tetrahedron) the number .
We remark that for a tetrahedron of affine flags, the coordinates are welldefined, and are ratios of the affine coordinates:
(2.11.1) 
3. Tetrahedra of flags and volume
In this section we define the volume of a tetrahedron of flags, generalizing and unifying the volume of hyperbolic tetrahedra (see section 3.7) and CR tetrahedra (see [4] and section 3.8). Via proposition 2.10, it coincides with the volume function on crossratio structures on a tetrahedron as defined in [4]. We then define the volume of a simplicial complex of flags tetrahedra. This volume is invariant under a change of triangulation of the simplicial complex (23 move) hence is naturally an element of the preBloch group and the volume is defined on the third homology group of flag configurations (see also [6]). Eventually we get a map, still called volume, from the third (discrete) homology group of to the Bloch group, through the natural projection from to . We conclude the section with the proof that this last map actually coincides with the Suslin map from to the Bloch group.
3.1. The preBloch and Bloch groups, the dilogarithm
We define a volume for a tetrahedron of flags by constructing an element of the preBloch group and then taking the dilogarithm map.
The preBloch group is the quotient of the free abelian group by the subgroup generated by the 5term relations
(3.1.1) 
For a tetrahedron of flags , let and be its coordinates.
3.2.
To each tetrahedron define the element
and extend it – by linearity – to a function
We emphasize here that depends on the ordering of the vertices of each tetrahedron^{2}^{2}2This assumption may be removed by averaging over all orderings of the vertices. In any case if is a chain in representing a cycle in we can represent by a closed cycle together with a numbering of the vertices of each tetrahedron of (see section 5.5). . The following proposition implies that is well defined on .
3.3 Proposition.
We have: .
Proof.
We have to show that is contained in the subgroup generated by the term relations. This is proven by computation and is exactly the content of [4, Theorem 5.2]. ∎
3.4.
We use wedge for skew symmetric product on Abelian groups. Consider , where is the multiplicative group of . It is the abelian group generated by the set factored by the relations
In particular, for any , and
3.5.
The Bloch group is the kernel of the homomorphism
which is defined on generators of by .
The BlochWigner dilogarithm function is
Here is the dilogarithm function. The function is welldefined and real analytic on and extends to a continuous function on by defining . It satisfies the 5term relation and therefore, for a subfield of , gives rise to a welldefined map:
given by linear extension as
3.6.
We finally define the volume map on via the dilogarithm (the constant will be explained in the next section):
From Proposition 3.3, Vol is well defined on .
3.7. The hyperbolic case
We briefly explain here how the hyperbolic volume for ideal tetrahedra in the hyperbolic space fits into the framework described above.
An ideal hyperbolic tetrahedron is given by points on the boundary of , i.e. . Up to the action of , these points are in homogeneous coordinates , , and – the complex number being the crossratio of these four points. So its volume is (see e.g. [16]).
Identifying with the Lie algebra , we have the adjoint action of on preserving the quatratic form given by the determinant, given in canonical coordinates by . The group preserves the isotropic cone of this form. The projectivization of this cone is identified to via the Veronese map (in canonical coordinates):
The first jet of that map gives a map from to the variety of flags . A convenient description of that map is obtained thanks to the identification between and its dual given by the quadratic form. Denote the bilinear form associated to the determinant. Then we have
Let be the tetrahedron , , and . An easy computation gives its coordinates:
It implies that and our function Vol coincide with the hyperbolic volume:
Remark.
Define an involution on the coordinates by:
on the faces and
on edges. The set of fixed points of correspond exactly with the hyperbolic tetrahedra.
3.8. The CR case
CR geometry is modeled on the sphere equipped with a natural action. More precisely, consider the group preserving the Hermitian form defined on by the matrix
and the following cones in ;
Let be the canonical projection. Then is the complex hyperbolic space and its boundary is
The group of biholomorphic transformations of is then , the projectivization of . It acts on by CR transformations.
An element gives rise to an element where corresponds to the unique complex line tangent to at . As in the hyperbolic case we may consider the inclusion map
and the first complex jet of that map gives a map
A generic configuration of four points in is given, up to , by the following four elements in homogeneous coordinates (we also give for each point , the corresponding dual to the complex line containing it and tangent to ):

, ,

, ,

, and

,
with and . Observe that acts doubly transitively on and for a generic triple of points the triple ratio of the corresponding flags is given by . One can easily compute the invariants of the tetrahedron:
The following proposition describes the space of generic configurations of four points in .
3.9 Proposition.
Configurations (up to translations by ) of four generic points in are parametrised by elements in with coordinates , satisfying the three complex equations
(3.9.1) 
with the exclusion of solutions such that and .
As in [4] (up to multiplication by 4) the volume of a CR tetrahedron is .
3.10. Relations with the work of Suslin
We show here how our map allows a new and more geometric way to interpret Suslin map (see [14]). First of all, recall that the natural projection gives a map .
3.11 Theorem.
We have .
Proof.
Let be the subgroup of diagonal matrices (in the canonical basis) of . Recall that is seen as a subgroup of via the adjoint representation (as in section 3.7). We find in the work of Suslin the three following results:
So we just have to understand the map on and . As is a subgroup of , the map vanishes on . And we have seen in the section 3.7 that, on a hyperbolic tetrahedron, coincide with times the crossratio.
This proves the theorem. ∎
Remark. After writing this section we became aware of Zickert’s paper [17]. In it (see §7.1) Zickert defines a generalization – denoted – of Suslin’s map. When specialized to our case his definition coincides with . We believe that the construction above sheds some light on the “naturality” of this map.
4. Decoration of a tetrahedron and the preBloch group
In this section we let be an ordered tetrahedron of flags and compute in two different ways . The first – and most natural – way uses coordinates associated to some lifting of as a tetrahedron of affine flags. In that respect we mainly follow Fock and Goncharov. The second way directly deals with coordinates and follows the approach of Neumann and Zagier. Finally we explain how the two ways are related; we will see in the remaining of the paper how fruitful it is to mix them.
4.1. Affine decorations and the preBloch group
We first let be an element of lifting . This allows us to associate coordinates to .
Let be the dimensional abstract free module where (see figure 4)
We denote the canonical basis of . It contains oriented edges (edges oriented from to ) and faces . Given and in we set:
4.2.
The coordinates of our tetrahedron of affine flags now define an element of where is any field which contains all the coordinates.
Let be a module equipped with a bilinear product
We consider on the module the bilinear product
defined on generators by
In particular letting be the bilinear skewsymmetric form on given by^{3}^{3}3Observe in particular that and so on, the logic being that the vector is the outgoing vector on the face and the vector (oriented from to ) turns around it in the positive sense.
we get:
4.3 Lemma.
We have:
(4.3.1) 
Proof.
To each ordered face of we associate the element
(4.3.2) 
The proof in the CR case of [6, Lemma 4.9] obviously leads to^{4}^{4}4Alternatively we may think of as a geometric realization of a mutation between two triangulations of the quadrilateral and apply [7, Corollary 6.15].:
Finally one easily sees that
∎
We let
Remark. 1. The element coincides with the invariant associated by Fock and Goncharov to the triangulation by a tetrahedron of a sphere with punctures. (The orientation of the faces being induced by the orientation of the sphere.)
2. Whereas – being a tetrahedron of flags – only depends on the flag coordinates, each associated to the faces depends on the affine flag coordinates.
In the next paragraph we make remark 2 more explicit by computing using the coordinates.
4.4. The NeumannZagier symplectic space
In this section we analyse an extension of NeumannZagier symplectic space introduced by J. Genzmer [9] in the space of coordinates associated to the edges of a tetrahedron. We reinterpret her definitions in our context of flag tetrahedra. Recall that we have associated coordinates to a tetrahedron of flags . These consists of edge coordinates and face coordinates subject to the relations (2.9.1) and (2.9.2). Recall that relation (2.9.1) is and note that (2.9.2) implies in particular that:
(4.4.1) 
We linearize (2.9.1) and (4.4.1) in the following way: We let be the module obtained as the quotient of by the kernel of . The latter is the subspace generated by elements of the form
for all such that for every . Equivalently it is the subspace generated by and . We will rather use as generators the elements
see Figure 5.
We let be the dual subspace which consists of the linear maps in which vanish on the kernel of . Note that (as well as ) is dimensional.
4.5.
The coordinates of our tetrahedron of flags now define an element