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.