In this post (with title inspired by the “What is …” column in Notices of the AMS), I would like to record some conversations I had with Jean-Christophe Yoccoz (mostly by the time we wrote our joint paper with David Zmiaikou) about a little technical issue arising when one tries to see the so-called Kontsevich-Zorich cocycle as a linear cocycle (in the usual sense of Dynamical Systems) over the Teichmüller flow (and/or -action) on moduli spaces of translation surfaces.
Of course, there are several ways to come around this little technical subtlety (from the dynamical point of view) in the definition of Kontsevich-Zorich cocycle and this is the main purpose of this post. Evidently, the content of this post is well-known (especially among experts), but I hope that this post will benefit the reader with some background in Dynamical Systems wishing to know the answer to the following question:
Does the Kontsevich-Zorich cocycle (as it is classically defined) qualifies as a genuine example of linear cocycle in the usual sense in Dynamical Systems?
Disclaimer. Even though this post benefited from my conversations with Jean-Christophe Yoccoz, all errors and mistakes below are my sole responsibility.
1. The Kontsevich-Zorich cocycle
Let be a fixed compact orientable topological surface of genus , let be a non-empty finite set of points of cardinality and let be a list of “ramification indices” such that .
Recall that a translation surface structure on is a maximal atlas of charts on such that all changes of coordinates are translation in and, for each , there are neighborhoods , and a ramified cover of degree such that every injective restriction of is a chart of the maximal atlas .
Remark 1 Equivalently, we can think of translation structures as the data of a Riemann surface structure on together with an Abelian differential (holomorphic one-form) possessing zeroes of orders at for . However, for the purposes of this post, we will not need this alternative point of view.
Remark 2 Since the usual Euclidean area form on is translation invariant, it makes sense to talk about the total area of a translation structure. From now on, we will always implicitly assume that our translation structures are normalized, i.e., they have unit total area. Here, it is worth to point out that this normalization is not important for the definition of Teichmüller and moduli spaces, but it is important for the discussion of the dynamics of the Teichmüller flow on moduli spaces.
We denote by the group of orientation-preserving homeomorphisms of fixing (pointwise), by the connected component in of the identity element, and by the mapping class group (sometimes also called modular group).
Note that the group acts (by pre-composition) on the set of translation surfaces: given and a translation surface structure on , we get a translation structure by defining .
In this setting, the Teichmüller space is the quotient of the set of translation structures on by the action of and the moduli space is the quotient of the set of translation structures on by the action of . By definition, the moduli space is the quotient of Teichmüller space by the action of the mapping class group .
Remark 3 The Teichmüller space is a manifold, but the moduli space is an orbifold (not a manifold) in general. We will come back to this point later in this post.
The group acts (by post-composition) on the Teichmüller space : given and a translation structure , we define (note that this action is well-defined because the conjugation of a translation in by the linear action of the matrix is still a translation). Furthermore, since acts by pre-composition and acts by post-composition, these actions commute and, hence, the action of descends to the moduli space .
The actions of the diagonal subgroup of on Teichmüller and moduli spaces are called Teichmüller flow.
The dynamics of the Teichmüller flow and/or -action on moduli spaces of (normalized) translation surfaces is a rich subject with interesting applications to the Ergodic Theory of some parabolic systems (such as interval exchange transformations and billiards in rational tables): see, for example, these posts here and here for more details.
A main character in the investigation of the -action on moduli spaces of (normalized) translation surfaces is the so-called Kontsevich-Zorich cocycle. Very roughly speaking, this cocycle was introduced by Kontsevich and Zorich as a practical way to extract the “interesting part” of the derivative cocycle of the Teichmüller flow.
Formally, the Kontsevich-Zorich (KZ) cocycle is usually defined as follows (compare with Forni’s paper). Let be the vector bundle over Teichmüller space whose fibers are the absolute homology group with real coefficients. One usually refers to as the Hodge bundle over .
Remark 4 The reader with some background in Complex Geometry might have thought that this notion is very similar to the Hodge bundle over Teichmüller and moduli spaces of algebraic curves (Riemann surfaces) obtained by attaching the space of holomorphic -forms to a Riemann surface . In fact, this is no coincidence and the nomenclature “Hodge bundle” for is a “popular” abuse of notation in the literature about the Teichmüller flow. In fact, this abuse of notation goes beyond this: one could also construct (trivial) bundles over Teichmüller spaces by taking the fibers to be the absolute homology group with complex coefficients or the absolute cohomology group or with real or complex coefficients. These variants are closely related to each other (because and the first absolute homology and cohomology groups of a surface are dual [by Poincaré duality]) and they are also called Hodge bundle in the literature (depending on the author’s taste).
The vector bundle is well-defined and trivial, i.e., : in a nutshell, this is a consequence of the fact that a homeomorphism that is isotopic to the identity (such as the elements of ) act trivially on homology.
By taking the quotient of by the natural action of the mapping class group on both factors, we get the so-called Hodge bundle
over the moduli space .
In this context, the (trivial) cocycle
over the Teichmüller flow on Teichmüller space given by
for and descends to the so-called Kontsevich-Zorich cocycle on the Hodge bundle over moduli space (by taking the quotient by the action of ). Here, it is worth to observe that the Kontsevich-Zorich cocycle is well-defined (i.e., we can take this quotient) because of the fact that acts by pre-composition and acts by post-composition on Teichmüller spaces (so that these actions commute).
2. Is the KZ cocycle a linear cocycle?
The reader with some familiarity with Dynamical Systems might have noticed some similarities between the notions of Kontsevich-Zorich cocycle and a linear cocycle over (discrete or continuous time) dynamical system.
In fact, let us recall that a linear cocycle , , over a flow , , is a flow on a vector bundle such that (i.e., projects onto ) and is a vector bundle automorphism, i.e., for all , the restriction of to is a linear map from the fiber on the fiber .
Example 1 The trivial cocycle on the trivial bundle over a flow is . In particular, the cocycle on defined above is an example of trivial cocycle.
Example 2 The derivative map of a smooth flow on a smooth manifold is an important class of examples of linear cocycles.
Given that the Kontsevich-Zorich cocycle on moduli spaces projects to the Teichmüller flow on moduli spaces and it acts on the fibers of via the (symplectic) action on homology of the elements of the mapping class group , one might be tempted to qualify the Kontsevich-Zorich cocycle as a linear cocycle.
However, a closer inspection of the definitions reveals that:
The Kontsevich-Zorich cocycle is not always a linear cocycle!
Actually, the fact that KZ cocycle is not a linear cocycle in general is not its fault: in order to talk about linear cocycles one needs vector bundles, and, as it turns out, the fibers of the Hodge bundle over moduli space are not vector spaces over the orbifold points of moduli spaces.
More precisely, we see from the definition that the fiber of at a translation surface is the quotient where is the group of automorphisms of , that is, the group of homeomorphisms of fixing pointwise whose local expressions in the charts are translations of .
Note that is a finite group: for instance, any element of is holomorphic (with respect to the Riemann surface structure underlying ) and, hence, by Hurwitz’s automorphisms theorem, we have that has cardinality . (Actually, even though Hurwitz’s theorem is sharp, this estimate of is not optimal: see, e.g., this paper of Schlage-Puchta and Weitze-Schmithuesen)
Therefore, the fiber of at is not very far from a vector space: it differs from by the quotient by (the action on homology of) the finite group (of “symplectic matrices”).
Nevertheless, when the translation surface is an orbifold point of moduli space (i.e., when is non-trivial), the fiber is not necessarily a vector space. (A simple concrete example of such a situation is the cone obtained from the quotient of by the finite group generated by the rotation of angle )
In summary, KZ cocycle is not always a linear cocycle because the Hodge bundle over moduli space is not always a vector bundle.
In other terms, the moduli space is an orbifold (but not a manifold in general), the Hodge bundle is an orbifold vector bundle (in general) and, thus, KZ cocycle is an orbifold linear cocycle (in general).
Example 3 One of my favorite examples of translation surface with a non-trivial group of automorphisms is the so-called Eierlegende Wollmilchsau.A concrete description of the Eierlegende Wollmilchsau is the following. We consider the quaternion group , we take an unit square in for each , and we glue (by translation) the vertical rightmost side of to the vertical leftmost side of and we glue (by translation) the horizontal top side of to the horizontal bottom side of . In this way, one obtais a translation surface where has genus , consists of four points and .
A simple argument (see, e.g., this paper here) shows that the group of automorphisms of the Eierlegende Wollmilchsau is isomorphic to the quaternion group .
Example 4 Some moduli spaces are manifolds and the corresponding Hodge bundles are vector bundles. For instance, the so-called minimal stratum of translation surfaces on a genus surface with a single marked point is a manifold because it can be shown (see, e.g., Proposition 2.4 in this paper here) that the automorphism group of any translation surface is trivial.
3. Dynamics of the KZ cocycle?
From the point of view of Topology and Algebraic Geometry, the “orbifoldic” nature of KZ cocycle is not surprising. Indeed, this kind of object is very common when studying monodromy representations and, also, one can overcome the “orbifoldic” nature of KZ cocycle by taking covers of moduli spaces in order to “kill” orbifold points. In particular, an orbifold linear cocycle is as good as a linear cocycle for topological considerations.
On the other hand, for dynamical considerations, the classical definition of KZ cocycle as an orbifold linear cocycle deserves further discussion.
For example, given an ergodic Teichmüller flow invariant probability measure on , it is desirable to apply Oseledets theorem to KZ cocycle in order to talk about its Lyapunov exponents and/or Oseledets subspaces. However, the Oseledets theorem deals only with linear cocycles and, thus, the fact that the KZ cocycle is merely an orbifold linear cocycle, or, more precisely, the fibers of Hodge bundle are not vector spaces, imposes some technical difficulties.
Remark 6 For ergodic-theoretical purposes, the technical point pointed out above only shows up when -almost every translation surface is an orbifold point. In particular, the discussion in this section does not concern the so-called Masur-Veech probability measures on moduli spaces (because its generic points have trivial group of automorphisms). This explains why the orbifoldic nature of KZ cocycle is never discussed in earlier papers in the literature (such as Forni’s paper): in those paper, the authors were concerned exclusively with the behavior of almost every trajectory with respect to Masur-Veech measures.
Remark 7 The orbifoldic nature is discussed (in an implicit way at least) in the literature on Veech surfaces. Recall that a Veech surface is a translation surface whose -orbit in moduli space is closed. As it turns out, the stabilizer in of a Veech surface is a lattice, so that its -orbit is naturally isomorphic to the unit cotangent bundle of the finite area hyperbolic surface . If the Veech surface has a non-trivial group of automorphisms, the Hodge bundle over its -orbit (and hence the corresponding KZ cocycle) is orbifoldic, but one can get around this by studying the group of so-called affine diffeomorphisms , a sort of “finite cover” of in view of a natural exact sequence . See, e.g., this survey paper of P. Hubert and T. Schmidt (or our joint paper with J.-C.Yoccoz).
Fortunately, for the sake of the definition of Lyapunov exponents of KZ cocycle with respect to an ergodic Teichmüller flow invariant probability measure , the possible ambiguity coming from the fact that the KZ cocycle is a “linear cocycle up to ” is irrelevant (cf. Section 4.3 of our paper with J.-C. Yoccoz and D. Zmiaikou). In fact, the Lyapunov exponents are defined by measuring the exponential rate of growth
of vectors (along typical trajectories), and the ambiguity caused by the fact that is a well-defined linear operator only up to the matrices in (action on homology of) does not change these rates because is a finite group and, hence, the possible values of (after composing with the elements of ) are uniformly related to each other by universal multiplicative constants (whose effects disappear when considering the expression ). In other terms, the Lyapunov exponents of orbifold linear cocycles are well-defined!
Unfortunately, there is no “cheap” solution (similar to the previous paragraph) for the definition of Oseledets subspaces of KZ cocycle: one needs linear structures on the fibers of the Hodge bundle to talk about them. Logically, this is an annoying situation because Oseledets subspaces are useful: for example, the analysis of these subspaces plays a major role in the recent breakthrough paper of A. Eskin and M. Mirzakhani about Ratner-like theorems for the -action on moduli spaces.
As it was already mentioned in the beginning of this section, the way out of this dilemma is to pick a cover of moduli space where all orbifold points disappear and to lift the Hodge bundle to this cover.
Of course, there are several ways of picking such a cover, but the whole point of this post is that certain covers are better than others depending on our purposes.
For example, the Teichmüller space is a cover of having no orbifold points (because an automorphism of a translation surface of genus that is isotopic to the identity is the identity), and the Hodge bundle is a vector bundle. Nevertheless, the Teichmüller space of (normalized) translation structures is not dynamically interesting: for example, there are no finite Teichmüller invariant measure (and, thus, we can not use the standard tools from Ergodic Theory). This situation is very similar to the case of geodesic flows on the unit cotangent bundle of a finite-area hyperbolic surfaces (if we think of as moduli space, as Teichmüller space and the geodesic flow as Teichmüller flow): even though there are plenty of finite geodesic flow invariant measures on , there are no finite geodesic flow invariant measures on the cover (and, in fact, the dynamics of the geodesic flow on unit cotangent of the hyperbolic plane is rather boring). In summary, despite the fact that the Hodge bundle is an honest vector bundle over Teichmüller space , we can not use it to define Oseledets subspaces (or, in general, do non-trivial dynamics) because the Teichmüller flow on is not dynamically rich.
The “failure” (from the dynamical point of view) in the previous paragraph suggests that we should try picking finite covers of moduli spaces (having no orbifold points). Indeed, the lift of Teichmüller flow invariant probability measures leads to a finite measures in that case and we are in good position to discuss Ergodic Theory.
In this direction, J.-C. Yoccoz, D. Zmiaikou and myself considered the cover of obtained by marking an horizontal separatrix issued of a conical singularity of the translation surface. This is a finite cover because there are exactly outgoing horizontal separatrices at a conical singularity with total angle . Furthermore, has no orbifold points: indeed, any automorphism of a translation surface that fixes an horizontal separatrix issued of a conical singularity is the identity. Moreover, the diagonal subgroup (Teichmüller flow) still acts on because the matrices respect the horizontal direction (and, thus, horizontal separatrices).
In particular, we can talk about the Oseledets subspaces of the KZ cocycle over Teichmüller flow at the level of the lift of the Hodge bundle to (because this “lifted KZ cocycle” over Teichmüller flow is a genuine linear cocycle over a probability measure preserving flow and hence we can apply Oseledets theorem).
For the purposes of our joint paper with J.-C. Yoccoz and D. Zmiaikou, the lift of the KZ cocycle to the Hodge bundle over the finite cover of moduli space was adequate (and this is why we decided to stick to it in our paper).
However, we should confess that we were not completely happy with because does not act on (even though its diagonal subgroup do act!). Therefore, the more general version of the KZ cocycle over the -action on moduli spaces in Remark 5 above (also very important for the applications of the Teichmüller flow) can not be lifted to the Hodge bundle over .
Actually, the reason why does not act on is very simple: the action of the rotation subgroup where is ill-defined. In order to see this, let us consider the following translation surface of genus with a marked (in blue) horizontal separatrix issued of its unique conical singularity :
Let us try to define the action of on by letting vary from to . Starting from , let us slowly apply the rotation matrices to the translation surface for small positive values of :
In this way, we get a new translation surface and the horizontal separatrix is sent into a non-horizontal separatrix . Thus, and, a fortiori, the natural “reflex” of posing fails.
Evidently, for any small positive angle , the non-horizontal separatrix is very close to the horizontal separatrix (obtained by “rotating” by an angle inside ) indicated below
If we do this, then, by letting vary from to , we would be force to impose where is the “previous” horizontal separatrix issued from the singularity in the “natural cyclic order” (obtained by rotating around the singularity) indicated (in red) below However, since the horizontal separatrices and are distinct, we have that and are distinct points of , a contradiction with the fact that the rotation matrix must act by the identity map on .
A simple inspection of the argument above shows that the finite cover of (obtained by “replacing” the rotation (circle) group by its -fold cover , but keeping the “same” diagonal subgroup ) does act on ! In other words, “almost acts” on . Nevertheless, since the (non-trivial) finite cover is not an algebraic group (unlike itself), the natural action on the Hodge bundle over is not so useful from the point of view of Dynamical Systems.
In summary, despite the usefulness of for the study of the KZ cocycle over the Teichmüller flow on moduli spaces, it is desirable to find an alternative finite cover of moduli spaces having no orbifold points where still acts.
One solution to this problem is to take the quotient of Teichmüller space by a torsion-free finite index subgroup of : indeed, is a finite cover of moduli space because has finite index, has no orbifoldic points because is torsion-free and acts on because acts by post-composition and acts by pre-composition on .
Here, a result of J.-P. Serre (see also the book of B. Farb and D. Margalit for a “modern” exposition of Serre’s result [in the context of moduli spaces of Riemann surfaces]) produces many of those subgroups with the desired properties: in fact, given any integer , Serre showed that the subgroup
consisting of elements of the mapping class group acting trivially on the homology of with coefficients in is torsion-free. (This finite cover of moduli space was already mentionned in this blog: see this post here)
The lift of the KZ cocycle to the Hodge bundle over defines a linear cocycle over the -action on moduli spaces that we could (should?) call KZ cocycle in dynamical discussions (where certain specific notions such as Oseledets subspaces are needed).
Remark 8 For the sake of comparison, the finite cover might be somewhat big relative to . In fact, is a cover of degree in general, while, from the fact that the action on homology of the mapping class group surjects into , we have that in general is a cover of degree (a quantity of the form for prime).