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Moduli Spaces of Surfaces: Spaces of translation surfaces

In the moduli space of Abelian differentials, big invariant subvarieties come from topology!- Paul Apisa (Yale University)

Abstract: It is a beautiful fact that any holomorphic one-form on a genus g Riemann surface can be presented as a collection of polygons in the plane with sides identified by translation. Since GL(2, R) acts on the plane (and polygons in it), it follows that there is an action of GL(2, R) on the collection of holomorphic one-forms on Riemann surfaces. This GL(2, R) action can also be described as the group action generated by scalar multiplication and Teichmuller geodesic flow. By work of McMullen in genus two, and Eskin, Mirzakhani, and Mohammadi in general, given any holomorphic one-form, the closure of its GL(2, R) orbit is an algebraic variety. While McMullen classified these orbit closures in genus two, little is known in higher genus. In the first part of the talk, I will describe the Mirzakhani-Wright boundary of an invariant subvariety (using mostly pictures) and a new result about reconstructing an orbit closure from its boundary. In the second part of the talk, I will define the rank of an invariant subvariety - a measure of size related to dimension - and explain why invariant subvarieties of rank greater than g/2 are loci of branched covers of lower genus Riemann surfaces. This will address a question of Mirzakhani. No background on Teichmuller theory or dynamics will be assumed. This material is work in progress with Alex Wright.

Video:

 

Translation surfaces with multiple short saddle connections- Ben Dozier (Stony Brook University)

Abstract: The SL_2(R) action on strata of translation surfaces allows us to answer many questions about the straight-line flow on individual translation surfaces (and this flow is in turn closely connected to billiards on rational polygons). By the pioneering work of Eskin-Mirzakhani, to understand dynamics on strata one is led to study "affine" measures. It is natural to ask about the interaction between volumes of certain subsets of surfaces and the geometric properties of the surfaces. I will discuss a proof of a bound on the volume, with respect to any affine measure, of the locus of surfaces that have multiple independent short saddle connections. This is a strengthening of the regularity result proved by Avila-Matheus-Yoccoz. A key tool is the new smooth compactification of strata due to Bainbridge-Chen-Gendron-Grushevsky-Moller, which gives a good picture of how a translation surface can degenerate.

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What does your average translation surface look like?- Anja Randecker (Heidelberg University)

Abstract: Almost every talk on translation surfaces starts with a double pentagon or with an octagon. But are these n-gons generic examples of translation surfaces? We will look at this question from the point of view of the geometry of the translation surfaces. Specifically, we prove an upper bound for the average diameter (resp. covering radius) in a stratum of translation surfaces of large genus. The first part of my talk assumes only basic knowledge about translation surfaces, the second part assumes more background to give a sketch of the proof. Both are based on joint work with Howard Masur and Kasra Rafi (available at https://arxiv.org/abs/1809.10769).

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20 replies on “Moduli Spaces of Surfaces: Spaces of translation surfaces”

Hi everyone,

I’m Anja, a researcher at Heidelberg University, and the speaker of the third talk above.
If you have questions related to my talk (of any level, really), come to my office hour on Monday, June 8, or simply ask them here in the comments.

Hope to connect with you at this conference,
Anja

Hey hey,

Thanks for watching! 🙂
In fact, sqrt(log(g))/g) is most likely not sharp at all. We have a lower bound which is of the order of sqrt(1/g). This is a direct calculation and holds for all translation surfaces.
So the truth might be lying anywhere between. We had an attempt to look at square-tiled surfaces to find the asymptotics but didn’t get new insides. If you have an idea in that direction, I’d be happy to hear it. 🙂

I suspected it might not be sharp.

I was also thinking about square-tiled surfaces and how analogous results could be obtained for squares. The Masur-Veech measure doesn’t see square-tiled surfaces, and so the model that I use is based on permutations. The model grossly overcounts surfaces, but it is easy to work with..

I wonder see if I can use the permutation model to calculate the covering radius. It could be interesting!

When I first saw Anja talk about this a while ago, I was also wondering whether you could at least heuristically derive the growth rate of covering radius/diameter from some fairly simple combinatorial model. So I’d also be interested to hear about your model.

Hi Anja,

Thanks for the talk, very clear!
And very nice result, too.

I had some questions:

1) Does the upper bound on the average covering radius give an upper bound on the average diameter?

2) Do your methods tell you anything about an upper bound for the average systolic ratio?

Hi Didac,

I’m glad you liked it!
Your questions are both very natural but unfortunately the short answer is “not yet” to both.
The upper bound on the average covering radius immediately gives you an upper bound on the average diameter but it goes to infinity in g… So that’s not what we expect.

In general, the methods should be useful to find bounds for averages of other geometric properties (such as the systolic ratio that you mention) but we still have to do that. 🙂

Hi Ben,

Loved the talk! Couple of questions on the first video:

1) Could you say a bit more about how the “non-parallel” condition is generalized to when you have more than two short saddle connections that you’re considering?

2) Is the measure of the collection of surfaces with n short saddle connections O(\prod_{i=1}^n \epsilon_i^2) in the more general version of your theorem?

Hi Sunrose,

Thanks for watching!
(1) To state the generalization, one needs to talk about the affine invariant manifold M that supports the measure. One defines a notion of a collection E of saddle connections on a surface in M being “M-independent”. The idea is that you don’t want any of the linear equations cutting out M to give a relation among the periods of the saddle connections in E. One way of saying this formally is that the saddle connections in E give linearly independent functionals on the tangent space of M.
(2) Yes, that is exactly the statement for any M-independent set of saddle connections.

Hi all!

I’m Paul Apisa, currently a postdoc at Yale on the verge of becoming a postdoc working at the University of Michigan. I tried to add a bit of drama to my talk by beginning with a problem whose stubborn openness was called “a bit of a scandal”. But don’t worry – the problem has since been resolved by Malestein and Putman – apologies for not putting this in the talk! Nevertheless, the problem provides an exciting springboard to the problems I end up discussing later. If you find yourself scandalized by the aforementioned problem or simply want to talk about anything in my talk, please drop by my office hours (https://yale.zoom.us/j/95201899376?pwd=MXJoK2VFVGNkMHZ4MDZtVGZia1o5dz09) on Wednesday June 10 from 4-5pm.

Hi Paul,

Great talk! Just so you know, your office hours are not listed in the “Live Events” calendar… I wasn’t sure if that was intentional.

I have a few questions:
1) Does geminal square-tiled surface (as you first defined with an asterisk) imply that the Veech group is SL_2(Z)?

2) In your geminal orbit closure example of the six squared surface, is it easy to realize the GL_2(R) matrices that perform the cylinder dilations that land you back in the orbit closure?

Hi Paul,

I think I just answered my own question in the negative.. I think I found a square-tiled surface with 4 squares in H(1,1) that is geminal and doesn’t have SL_2(Z) Veech group.

Hi Sunrose,

I’m so glad you enjoyed the talk! Thanks for letting me know that the office hours aren’t listed on the schedule; I’ll make inquiries about that. Regarding your questions:

(1) Every geminal square-tiled surface that I know has a Veech group that is a congruence subgroup. But there are examples of geminal surfaces whose Veech group has arbitrarily high index in SL(2,Z). I’m a bit perplexed by the claim that there is a geminal square-tiled surface in H(1,1); I seem to remember sketching a proof a while ago that no such surface could exist (although perhaps I’m misremembering or my sketch was mistaken). In any event, I’d love to talk more about the example you found if you’re interested.

(2) In the six-square surface example, since the orbit closure is not closed, the GL(2,R) orbit of the generic surface is only dense in the orbit closure. In particular, the orbit does not coincide with the orbit closure. So the best we could hope for is a sequence of matrices that takes our original surface closer and closer to the cylinder-dilated surface. The argument I know for why this is possible is due to Masur and follows by showing the ergodicity of the GL(2,R) action on the orbit closure M with respect to Lebesgue measure on M. That proof in turn relies on the Hopf argument (a flexible technique for establishing the ergodicity of a flow when there is an expanding and contracting foliation of the manifold being studied). But I don’t think that this argument can be used to produce the desired explicit sequence of matrices.

I hope that’s useful – thanks again for the comments!

Dear Paul,

1) I apologize. I got the definition of geminal wrong. I just found a surface that has “a” pair of cylinders that are isometric and cover the surface, but I missed that it should be the case that every cylinder should have such a twin. My bad, sorry.

2) I see. It would certainly be super nice to be able to get such a sequence of matrices, but maybe that is hoping too much.

Thank you!

Hi Anja,

Thanks! The result I mentioned is here (https://arxiv.org/abs/1708.06486); Putman and Malestein show that for any genus g there is a branched cover X of a genus g surface so that simple-curve-homology, i.e. the subspace of first homology of X generated by components of preimages of simple closed curves, is strictly smaller than the first homology of X. When g = 1, the Eierlegende-Wollmilchsau, Ornithorynque, and Matheus-Yoccoz examples are examples of this behavior. The translation surfaces that Alex and I built are examples where “cylinder homology”, which is a subset of simple-curve-homology, is as small as can be. Thanks again!

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