Hitchin characters and geodesic laminations
For a closed surface , the Hitchin component is a preferred component of the character variety consisting of group homomorphisms from the fundamental group to the Lie group . We construct a parametrization of the Hitchin component that is well-adapted to a geodesic lamination on the surface. This is a natural extension of Thurston’s parametrization of the Teichmüller space by shear coordinates associated to , corresponding to the case . However, significantly new ideas are needed in this higher dimensional case. The article concludes with a few applications.
- 0.1 Background and motivation
- 0.2 Main results
- 1 Generic configurations of flags
- 2 Geodesic laminations
- 3 Triangle invariants
- 4 Tangent cycles for a geodesic lamination
- 5 The shearing tangent cycle of a Hitchin character
- 6 Hitchin characters are determined by their invariants
- 7 Length functions
- 8 Parametrizing Hitchin components
- 9 The action of pseudo-Anosov homeomorphisms on the Hitchin component
- 10 Length functions of measured laminations
0.1. Background and motivation
For a closed, connected, oriented surface of genus , the Hitchin component is a preferred component of the character variety
consisting of group homomorphisms from the fundamental group to the Lie group (equal to the special linear group if is odd, and to if is even), where acts on these homomorphisms by conjugation. The quotient should normally be taken in the sense of geometric invariant theory [MFK94], but this subtlety is irrelevant here as this quotient construction coincides with the usual topological quotient on the Hitchin component.
When , the Lie group is also the orientation-preserving isometry group of the hyperbolic plane , and the Hitchin component of consists of all characters represented by injective homomorphisms whose image is discrete in and for which the natural homotopy equivalence has degree . The Hitchin component is in this case called the Teichmüller component, and can also be described as the space of isotopy classes of hyperbolic metrics on .
When , there is a preferred homomorphism coming from the unique –dimensional representation of (or, equivalently, from the natural action of on the vector space of homogeneous polynomials of degree in two variables). This provides a natural map , and the Hitchin component is the component of that contains the image of . The terminology is motivated by the following fundamental result of Hitchin [Hit92], who was the first to single out this component.
Theorem 0.1 (Hitchin).
The Hitchin component is diffeomorphic to .
A Hitchin character is an element of the Hitchin component , and a Hitchin homomorphism is a homomorphism representing a Hitchin character. We will use the same letter to represent the Hitchin homomorphism and the corresponding Hitchin character .
About 15 years after [Hit92], Labourie [Lab06] showed that Hitchin homomorphisms satisfy many important geometric and dynamical properties, and in particular are injective with discrete image; see also [FG06].
Hitchin’s construction of the parametrization of given by Theorem 0.1 is based on geometric analysis techniques that provide little information on the geometry of the Hitchin homomorphisms themselves; see [Lof01, Lab07, Lab14] for different geometric analytic parametrizations when . The current article is devoted to developing another parametrization of the Hitchin component which is much more geometric, and has the additional advantage of being well-behaved with respect to a geodesic lamination. Geodesic laminations were introduced by Thurston to develop a continuous calculus for simple closed curves on the surface , and provide very powerful tools for many topological and geometric problems in dimensions 2 and 3. See §§9 and 10 for two simple applications of our parametrization, one to the dynamics of the action of a pseudo-Anosov homomorphism of on the Hitchin component, and another one to the length functions defined by a Hitchin character on Thurston’s space of measured laminations on .
Our construction is a natural extension of Thurston’s parametrization of the Teichmüller component by shear coordinates [Thu86, Bon96]. It draws its inspiration from this classical case where , but also from work of Fock-Goncharov [FG06] on a variant of the Hitchin component where the surface has punctures, and where these punctures are endowed with additional information. As in the classical case when , the situation is conceptually and analytically much more complicated for a closed surface than in the case considered in [FG06]. Many arguments, such as those of §§5.1, 6.2 and 8.2, are new even for the case .
The companion article [BD14] is devoted to a special case of our parametrization, when the geodesic lamination has only finitely many leaves. The situation is much simpler in that case, and in particular the arguments of [BD14] tend to be very combinatorial in nature. The current article has a much more analytic flavor. It is also more conceptual, and provides a homological interpretation of some of the invariants and phenomena that were developed in a purely computational way in [BD14]. And of course the framework of general geodesic laminations, possibly with uncountably many leaves, considered in this article is better suited for applications.
The article [Dre13b] was developed, to a large extent, as a first step towards the more general results of the current paper. It investigates all deformations of a Hitchin character that respect its triangle invariants, as discussed in the next section.
0.2. Main results
We can now be more specific. Let be a maximal geodesic lamination in . See §2 for precise definitions. What we need to know here is just that, for an arbitrary auxiliary metric of negative curvature on the surface, is decomposed as a union of disjoint geodesic leaves, and that its complement consists of infinite triangles with geodesic boundary. Some maximal geodesic laminations, such as the ones considered in [BD14], have only a finite number of leaves, but generic examples have uncountably many leaves.
Given a Hitchin character , the rich dynamical structure for discovered by Labourie [Lab06] associates a triple of three flags of to each triangle component of . In addition, Fock and Goncharov [FG06] prove that this flag triple is positive, in a sense discussed in §1.5, and is determined by invariants . Since has components, these flag triple invariants can be collected into a single triangle invariant .
The really new feature introduced in this article describes how to glue these flag triples across the (possibly uncountably many) leaves of the lamination, and simultaneously involves analytic and combinatorial arguments. The analytic part of this analysis is based on the slithering map constructed in §5.1, which is a higher dimensional analogue of the horocyclic foliation that is at the basis of the case [Thu86, Bon96]. This slithering map enables us to control the gluing by elements of the homology of a train track neighborhood for , which we now briefly describe. The precise definition of train track neighborhoods can be found in §4.2 (and is familiar to experts); at this point, it suffices to say that is obtained from by removing disjoint disks, one in each component of ; in addition, the boundary is decomposed into a horizontal boundary and a vertical boundary , in such a way that each component of is a hexagon made up of three arc components of and three arc components of .
The geodesic lamination has a well-defined 2–fold orientation cover , whose leaves are continuously oriented, and the covering map uniquely extends to a 2–fold cover . In particular, is a geodesic lamination in the surface .
Our new invariant for a Hitchin character is a certain shearing class . This shearing class has the property that , for the covering involution of the cover and for the involution of that associates to . In particular, can also be interpreted as a twisted homology class valued in a suitable coefficient bundle over with fiber .
The triangle invariant and shearing class satisfy two types of constraints. The first constraint is a homological equality.
Proposition 0.2 (Shearing Cycle Boundary Condition).
The boundary of the shearing class of a Hitchin character is completely determined by the triangle invariant , by an explicit linear formula given in §5.2.
The second constraint is a positivity property, proved as Corollary 7.10 in §7.2. Because the leaves of the orientation cover are oriented, a famous construction of Ruelle and Sullivan [RS75] interprets every transverse measure for the orientation cover as a 1–dimensional de Rham current in . In particular, such a transverse measure determines a homology class .
Proposition 0.3 (Positive Intersection Condition).
For every transverse measure for the orientation cover , the algebraic intersection vector of the shearing class with is positive, in the sense that all its coordinates are positive.
The Shearing Cycle Boundary and Positive Intersection Conditions restrict the pair to a convex polyhedral cone in . The main result of the article, proved as Theorem 8.13 in §8.3, shows that these are the only restrictions on the triangle and shearing invariants, and that these provide a parametrization of the Hitchin component .
Theorem 0.4 (Parametrization of the Hitchin component).
The map , which to a Hitchin character associates the pair formed by its triangle invariant and its shearing class , is a homeomorphism.
The Shearing Cycle Boundary Condition provides some unexpected constraints on the triangle invariants of Hitchin characters, as well as on their shearing classes. The following two statements are abbreviated expressions of more specific computations given in §8.4. These restrictions are somewhat surprising when one considers the relatively large dimension of .
An element is the triangle invariant of a Hitchin character if and only if it belongs to a certain explicit subspace of codimension of .
A relative homology class is the shearing class of a Hitchin character if and only if it belongs to a certain open convex polyhedral cone in an explicit linear subspace of dimension if , of dimension if , and of dimension if .
The dimensions in Proposition 0.6 should be compared to the dimension of the twisted homology space , consisting of those such that .
At first, the relative homology group of a train track neighborhood may not appear very natural. In fact, although we decided to privilege this more familiar point of view in this introduction, it occurs as a space of tangent cycles for the orientation cover relative to its slits, where the slits of are lifts of the spikes of the complement ; Proposition 4.5 then provides an isomorphism . A relative tangent cycle assigns a vector to each arc transverse to , in a quasi-additive way: If is split into two subarcs and , then is equal to the sum of , and of a correction factor depending on the slit of facing the point along which was split. In particular, depends only on the maximal geodesic lamination , and not on the train track neighborhood .
The lack of additivity of a relative tangent cycle has a nice expression in terms of the boundary map , and is at the basis of the Shearing Cycle Boundary Condition of Proposition 0.2. In the classical case where , the Shearing Cycle Boundary Condition says that the shearing class has boundary 0, and in particular that the corresponding tangent cycle is additive with no correction factors; such objects were called “transverse cocycles” in [Bon97b, Bon96].
This point of view enables us to shed some light on the Positive Intersection Condition of Proposition 0.3. Given a Hitchin character , Labourie [Lab06] shows that for every nontrivial the matrix is diagonalizable, and that its eigenvalues can be ordered in such a way that . If we define by the property that its –th coordinate is , the second author showed in [Dre13a] that this formula admits a continuous linear extension to the space of Hölder geodesic currents of , a topological vector space that contains all conjugacy classes of in a natural way; this continuous extension is unique on the subspaces of that are of interest to us in this paper (see Remark 7.3).
Theorem 0.7 (Length and Intersection Formula).
If is the shearing cycle of a Hitchin character , and if is a tangent cycle for the orientation cover , then
is the algebraic intersection vector of the homology classes and in the train track neighborhood of .
In the special case where is a transverse measure for , the Positive Intersection Condition of Proposition 0.3 is then equivalent to the property that all coordinates of the vector are positive. In this version, this statement is an immediate consequence of the Anosov Property that is central to [Lab06] (see Proposition 7.4).
The article concludes, in §§9 and 10, with two brief applications of Theorems 0.4 and 0.7. The first one is concerned with the dynamics of the action of a pseudo-Anosov diffeomorphism on the Hitchin component ; applying the parametrization of Theorem 0.4 to the case of a maximal geodesic lamination containing the stable lamination of shows that the dynamics of the action of on are concentrated on submanifolds of of relatively large codimension. The second application considers the restriction of the length function to Thurston’s space of measured laminations on ; a consequence of Theorem 0.7 is that, at each , the tangent map is linear on each face of the piecewise linear structure of .
These results can be put in a broader perspective. Indeed, the properties of the Hitchin component remain valid when the Lie group is replaced by any split real algebraic group [Hit92, Lab06, FG06]. In this more general framework, our triangle invariant associates to each component of a positive triple in the flag space , where is a Borel subgroup. The shearing class is now a relative homology class valued in the Cartan algebra of , and equivariant with respect to the covering involution and to minus the opposition involution of . The Shearing Cycle Boundary Condition then states that the boundary is completely determined by the triangle invariant , while the Positive Intersection Condition requires that the algebraic intersection vector belong to the principal Weyl chamber of . The output of these constructions is perhaps not as explicit as in the case of , but extending the proofs to this more general context is only a matter of using the right vocabulary.
Acknowledgement: The authors are very pleased to acknowledge very helpful conversations with Antonin Guilloux and Anne Parreau, at a time when they (the authors) were very confused. They are also grateful to Giuseppe Martone for many useful comments on the manuscript.
1. Generic configurations of flags
Flags in play a fundamental rôle in our construction of invariants of Hitchin characters. This section is devoted to certain invariants of finite families of flags, borrowed from [FG06].
A flag in is a family of nested linear subspaces of where each has dimension .
A pair of flags is generic if every subspace of is transverse to every subspace of . This is equivalent to the property that for every .
Similarly, a triple of flags is generic if each triple of subspaces , , , respectively in , , , meets transversely. Again, this is equivalent to the property that for every , , with .
1.2. Wedge-product invariants of generic flag triples
Elementary linear algebra shows that, for any two generic flag pairs and , there is a linear isomorphism sending to and to . However, the same is not true for generic flag triples. Indeed, there is a whole moduli space of generic flag triples modulo the action of , and this moduli space can be parametrized by invariants that we now describe. These invariants are expressed in terms of the exterior algebra of .
Consider the discrete triangle
represented in Figure 1.
A function is balanced if, for every , , ,
namely if the sum of the over each line parallel to one side of the triangle is equal to 0.
Such a balanced function defines an invariant of a generic flag triple as follows. For each , , between and , the spaces , and are each isomorphic to . Choose non-zero elements , and . We will use the same letters to denote their images , and . We then define
where we choose an isomorphism to interpret each term in the product as a real number. The fact that the flag triple is generic guarantees that these numbers are non-zero, while the property that is balanced is exactly what is needed to make sure that this product is independent of the choices of the elements , and and of the isomorphism . We say that is the wedge-product invariant of generic flag triples associated to the balanced function .
We now consider a fundamental special case. For , , with , namely for a point in the interior of the triangle , the –hexagon cycle is the balanced function defined by
where denotes the Kronecker function such that if and otherwise. The terminology is explained by the fact that the support of is a small hexagon in the discrete triangle , centered at the point ; see Figure 1 for the case where and . The wedge-product invariant associated to the hexagon cycle is the –triple ratio
Note the elementary property of triple ratios under permutation of the flags.
The natural action of the linear group on the flag variety descends to an action of the projective linear group , quotient of by its center consisting of all non-zero scalar multiples of the identity. Note that the projective special linear group is equal to if is odd, and is an index 2 subgroup of otherwise.
Two generic flag triples and are equivalent under the action of if and only if for every , , with .
In addition, for any set of non-zero numbers , there exists a generic flag triple such that for every , , with .
See [FG06, §9]. ∎
In particular, the moduli space of generic flag triples under the action of is homeomorphic to .
Corollary 1.4 below partially accounts for the important rôle played by the triple ratios in Proposition 1.2. We will not really need this property, but it explains why we will always be able to express in terms of triple ratios the various wedge-product invariants that we will encounter in the paper.
The hexagon cycles form a basis for the free abelian group consisting of all balanced function .
The proof is elementary, by induction on . ∎
Lemma 1.3 immediately implies:
Every wedge-product invariant can be uniquely expressed as a product of integer powers of triple ratios. ∎
1.3. Quadruple ratios
In addition to triple ratios, the following wedge-product invariants of generic flag triples will play a very important rôle in this article.
For , 2, …, , the –th quadruple ratio of the generic flag triple is the wedge-product invariant
where, as usual, we consider arbitrary non-zero elements , and , and where the ratios are computed in .
Note that , but that this quadruple ratio usually does not behave well under the other permutations of the flags , and , as plays a special rôle in .
For this wedge-product invariant, we can explicitly determine the formula predicted by Corollary 1.4.
For , , …, ,
where the product is over all integers , with . In particular, and .
When computing the right hand side of the equation, most terms cancel out and we are left with the eight terms of . ∎
1.4. Double ratios
We now consider quadruples of flags , , , . Such a flag quadruple is generic if each quadruple of subspaces , , , meets transversely. As usual, we can restrict attention to the cases where .
For , the –th double ratio of the generic flag quadruple is
where we choose arbitrary non-zero elements , , and . As usual, is independent of these choices.
The following computation gives a better feeling of what is actually measured by this double ratio.
For a generic flag quadruple , consider the decomposition where . For arbitrary non-zero vectors and , let , be the respective projections of and to the line parallel to the other lines with . Then
where the ratios are measured in the lines .
Note that does not really depend on the whole flags and , but only on the lines and . The following elementary properties indicate how it behaves under transposition of and , or of and .
The minus sign in the definition of is justified by the positivity property of the next section, and in particular by Proposition 1.8.
An ordered family of flags is positive if:
for every distinct , , and for every , , with , the triple ratio is positive.
for every distinct , , , with or , and for every , the double ratio is positive.
Proposition 1.8 ([Fg06]).
If the flag –tuple is positive, any flag –tuple obtained by dihedral permutation of the is also positive. ∎
Recall that a dihedral permutation is, either a cyclic permutation, or the composition of the order reversal with a cyclic permutation.
2. Geodesic laminations
Geodesic laminations are a now very classical tool in 2–dimensional topology and geometry. They occur in many different contexts, for instance when one takes limits of sequences of simple closed curves. We state here a few basic definitions and facts, and refer to [Thu81, CB88, PH92, Bon01] for proofs and background.
To define geodesic laminations, one first chooses a metric of negative curvature on the surface .
An –geodesic lamination is a closed subset that can be decomposed as a disjoint union of simple complete –geodesics, called its leaves. Recall that a geodesic is complete if it cannot be extended to a longer geodesic, and it is simple if it has no transverse self-intersection point. The leaves of a geodesic laminations can be closed or bi-infinite. A geodesic lamination can have finitely many leaves (as in the case considered in [BD14]), or uncountably many leaves.
An –geodesic lamination has measure 0, and in fact Hausdorff dimension 1 [BS85], and its decomposition as a union of leaves is unique. The complement of an –geodesic lamination is a surface of finite topological type, bounded by finitely many leaves of . The completion of for the path metric induced by is a finite area surface with geodesic boundary; it is the union of a compact part and of finitely many spikes homeomorphic to , where is contained in two leaves of . The width of these spikes decreases exponentially in the sense that the parametrization by can be chosen so that its restriction to each has speed 1 and so that the length of each arc decreases exponentially with .
Because the leaves of are disjoint, every point of has a neighborhood homeomorphic to for which the intersection corresponds to for some totally disconnected compact subset ; beware that, in general, the homeomorphism cannot be made differentiable, only Hölder bicontinuous.
We will make heavy use of transverse arcs for . These are arcs differentiably immersed in that are transverse to the leaves of . In addition, we require that the endpoints of such a transverse arc be disjoint from .
The notion of geodesic lamination is independent of the choice of the negatively curved metric in the sense that, if is another negatively curved metric on , there is a natural one-to-one correspondence between –geodesic laminations and –geodesic laminations.
A geodesic lamination is maximal if it is contained in no other geodesic lamination. This is equivalent to the property that each component of its complement is a triangle, bounded by three infinite leaves of and containing three spikes of . If the surface has genus , an Euler characteristic argument shows that the number of triangle components of the complement of a maximal geodesic lamination is equal to .
Every geodesic lamination is contained in a maximal geodesic lamination.
We can think of maximal geodesic laminations as some kind of triangulations of the surface , where the edges are geodesic and where the vertices have been pushed to infinity. This point of view explains why maximal geodesic laminations are powerful tools for many problems, such as the ones considered in the current article.
3. Triangle invariants
Let be a Hitchin homomorphism. We will use a maximal geodesic lamination to construct invariants of the corresponding character .
3.1. The flag curve
The key to the definition of these invariants is the following construction of Labourie [Lab06].
Let and be the unit tangent bundles of the surface and of its universal cover , respectively. For convenience, lift the homomorphism to a homomorphism . The fact that such a lift exists is classical when , and therefore when comes from a discrete representation ; the existence of the lift in the general case follows by connectedness of the Hitchin component , and by homotopy invariance of the obstruction to lift. We can then consider the twisted product
where the fundamental group acts on by its usual action on the universal cover , and acts on by . The natural projection presents as a vector bundle over with fiber .
Endow the surface with an arbitrary metric of negative curvature. This defines a circle at infinity for the universal cover , and a geodesic flow on the unit tangent bundle . It is well known (see for instance [Gro87, BH99, GdlH90]) that these objects are actually independent of the choice of the negatively curved metric, at least if we do not care about the actual parametrization of the geodesic flow (which is the case here).
The geodesic flow of has a natural flat lift to a flow