During the graduate workshop on moduli of curves (organized by Samuel Grushevsky, Robert Lazarsfeld, and Eduard Looijenga last July 2014), Alex Wright gave a minicourse on the -orbits on moduli spaces of translation surfaces (the videos of the lectures and the corresponding lecture notes are available here and here).
These lectures by Alex Wright made Eduard Looijenga ask if some “remarkable” translation surfaces could help in solving the following question.
Let be a ramified finite cover of the two-torus (say branched at only one point ). Denote by the subspace of generated by the homology classes of all simple closed loops on covering such a curve on .
Question 1. Is it true that one always has in this setting?
By following Alex Wright’s advice, Eduard Looijenga wrote me asking if I knew the answer to this question. I replied to him that my old friend Eierlegende Wollmilchsau provided a negative answer to his question, and I directed him to the papers of Forni (from 2006), Herrlich-Schmithüsen (from 2008) and our joint paper with Yoccoz (from 2010) for detailed explanations.
In a subsequent email, Eduard told me that my answer was a good indication that notable translation surfaces could be interesting for his purposes: indeed, the Eierlegende Wollmilchsau is precisely the example described in the appendix of a paper by Andrew Putman and Ben Wieland from 2013 where Question 1 was originally solved.
After more exchanges of emails, I learned from Eduard that his question was motivated by the attempts of Putman-Wieland (in the paper quoted above) to attack the following conjecture of Nikolai Ivanov (circa 1991):
Conjecture (Ivanov). Let and . Consider a finite-index subgroup of the mapping-class group of isotopy classes of homeomorphisms of a genus surface fixing pointwise a set of marked points. Then, there is no surjective homomorphism from to .
Remark 1 This conjecture came from the belief that mapping-class groups should behave in many aspects like lattices in higher-rank Lie groups and it is known that such lattices do not surject on because they satisfy Kazhdan property (T). Nevertheless, Jorgen Andersen recently proved that the mapping-class groups do not have Kazhdan property (T) when .
In fact, Putman-Wieland proposed the following strategy to study Ivanov’s conjecture. First, they introduced the following conjecture:
Conjecture (Putman-Wieland). Fix and . Given a finite-index characteristic subgroup of the fundamental group of a surface of genus with punctures , denote by the associated finite cover, and let be the compact surface obtained from by filling its punctures.
Then, the natural action on of the group of lifts to of isotopy classes of diffeomorphisms of fixing pointwise has no finite orbits.
Remark 3 This conjecture is closely related (for reasons that we will not explain in this post) to a natural generalization of Question 1 to general ramified finite covers .
Remark 4 The analog of Putman-Wieland conjecture in genus is false: the same counterexample to Question 1 (namely, the Eierlegende Wollmilchsau) serves to answer negatively this genus 1 version of Putman-Wieland conjecture.
Remark 5 In the context of Putman-Wieland conjecture, one has a representation (induced by the lifts of elements of to ). This representation is called a higher Prym representation by Putman-Wieland. In this language, Putman-Wieland conjecture asserts that higher Prym representations have no non-trivial finite orbits when and .
Secondly, they proved that:
Theorem 1 (Putman-Wieland) Fix and .
- (a) If Putman-Wieland conjecture holds for every finite-index characteristic subgroup of , then Ivanov conjecture is true for any finite-index sugroup of .
- (b) If Ivanov conjecture holds for every finite-index subgroup of , then Putman-Wieland conjecture is true for any finite-index characteristic subgroup of .
Moreover, if Ivanov conjecture is true for all finite-index subgroups of for all , then it is also true for all finite-index subgroups of with , .
In other words, Putman-Wieland proposed to approach an algebraic problem (Ivanov conjecture) via the study of a geometric problem (Putman-Wieland conjecture) because these two problems are “essentially” equivalent.
In particular, this gives the following concrete route to establish Ivanov conjecture:
- (I) if we want to show that Ivanov conjecture is true for all and , then it suffices to prove Putman-Wieland conjecture for (and all ); indeed, this is so because item (a) of Putman-Wieland theorem would imply that Ivanov conjecture is true for (and all ) in this setting, and, hence, the last paragraph of Putman-Wieland theorem would allow to conclude the validity of Ivanov conjecture in general.
- (II) if we want to show that Ivanov conjecture is false for some and , then it suffices to construct a counterexample to Putman-Wieland conjecture for and .
Once we got at this point in our email conversations, Eduard told me that Question 1 was just a warmup towards his main question:
Question 2. Are there remarkable translation surfaces giving counterexamples to Putman-Wieland conjecture?
By inspecting my list of “preferred” translation surfaces, I noticed that I knew such an example: in fact, there is exactly one member in a family of translation surfaces that I’m studying with Artur Avila and Jean-Christophe Yoccoz (for other purposes) which is a counterexample to Putman-Wieland conjecture in genus (and ).
In other words, one of the translation surfaces in a forthcoming paper joint with Artur and Jean-Christophe answers Question 2.
Remark 6 This shows that Putman-Wieland’s strategy (I) above does not work (because their conjecture is false in genus ). Of course, this does not mean that Ivanov conjecture is false: in fact, by Putman-Wieland strategy (II), one needs a counterexample to Putman-Wieland conjecture in genus (rather than in genus ). Here, it is worth to point out that Artur, Jean-Christophe and I have no good candidates of counterexamples to Putman-Wieland conjecture in genus and/or Ivanov conjecture.
Below the fold, we focus on the case and of Putman-Wieland conjecture.
Update (September 7, 2015): Last June 2015, Eduard gave a talk on algebro-geometrical aspects of mapping-class groups, and he wrote the following summary here where the connections between (a natural generalization of) Question 1 and the conjectures of Ivanov and Putman-Wieland are discussed.
1. A genus cover of a genus surface
Let be the genus surface associated to the Riemann surface . The genus surface corresponding to the Riemann surface has the structure of a triple cover given by . Observe that is unramified off the six (Weierstrass) points of located at the five roots of unit , , and the point at infinity .
Recall that the ramified finite cover corresponds to a finite-index subgroup of where is a genus surface with punctures and is a point of located at .
It is possible to check that is not a characteristic subgroup of . Nevertheless, we can easily construct a subgroup of such that is a finite-index characteristic subgroup of . Indeed,
is a subgroup of which is characteristic in . Furthermore, has finite-index in because has the same (finite) index of for all and has only finitely many subgroups of a given index (since is finitely generated).
Denote by the compact surface associated to the finite ramified cover of induced by , and let be the mapping-class group of . Since is characteristic, we can lift any element of to a mapping-class of , so that we have a higher Prym representation .
We will deduce this theorem as a consequence of the following result:
Theorem 3 There exist a eight-dimensional subspace and a finite-index subgroup of with the following properties. Any element of lifts to a mapping-class of and the orbit of any under the corresponding representation is finite.
2. Proof of Theorem 2 assuming Theorem 3
Since is a subgroup of , we have that the cover associated to factors through the cover (associated to ), that is, we have a cover such that the composition is the cover corresponding to .
Given a eight-dimensional subspace and a finite-index subgroup of as in the statement of Theorem 3, let be the subspace of cohomology cycles projecting to which are also invariant under the whole group of deck transformations of .
By Theorem 3, the natural action of on factors a finite group of matrices. By construction, all orbits of the action of on are finite. Since is a finite-index subgroup of , it follows that all orbits of the action of on are also finite.
3. Proof of Theorem 3
Let be the automorphism where generating the group of deck transformations of the cover where is the Riemann sphere and .
Note that factors : indeed, where is the natural projection from to the quotient of by its hyperelliptic involution , .
Since has genus , the elements of commute with the hyperelliptic involution : this is a very special property of the genus setting whose proof follows from the results in this paper here (see also page 77 of Farb-Margalit book [while paying attention that our convention differs from them because our mapping-class groups are required to fix pointwise each puncture]).
It follows that the elements with form a finite-index subgroup of such that the lift to of any such commutes with the automorphism (in fact, this is so because projects under to the hyperelliptic involution of ). [Update (August 24, 2015): See also the exchange between Ben Wieland and myself in the comments below.]
By construction, acts on and our task is to show the existence of a eight-dimensional subspace of such that the -orbit of any is finite.
For this sake, we start by analyzing the action of on . Here, the crucial point is that was built in such a way that all of its elements commute with . In particular, the action of preserves each summand of the decomposition
into the eigenspaces associated to the eigenvalues of . (Note that the eigenspace is trivial because ).
Recall that the action of on preserves the intersection form . Since each eigenspace has a Hodge decomposition
and the intersection form is positive definite on the space of holomorphic -forms and negative definite on the space of anti-holomorphic -forms, we have that acts on via a indefinite unitary group of a pseudo-Hermitian form of signature where
In our context, is associated to the curve , so that
is an explicit basis of the space of holomorphic -forms on . From this, we infer that and (and, in general, for each ).
In other words, , , and acts on the eight-dimensional complex subspace
via (a subgroup of) the compact group .
Next, we study the action of on . We begin by noticing that the eight-dimensional complex subspace is defined over . In fact, this is a consequence of the following elementary observation (from Galois theory): is the sum of all eigenspaces associated to all primitive th roots of unity .
Since is defined over , it intersects into a lattice of rank . In particular, acts on via (a subgroup of) the symplectic group because respects the symplectic intersection form on .
In summary, we proved that:
- on one hand, acts on via the compact group ;
- on the other hand, acts on via .
In other terms, acts on through a compact subgroup of the discrete group , i.e., acts on the eight-dimensional subspace through a finite subgroup of symplectic matrices.
It follows that the -orbit of any is finite, so that the proof of Theorem 3 is complete.