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Large SL(2,R)–Orbit Closures in the Moduli of Translation Surfaces

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Large SL(2,R)–Orbit Closures in the Moduli of Translation Surfaces Large SL(2; R){orbit closures in the moduli of translation surfaces Alex Wright University of Chicago Alex Wright (University of Chicago) Large SL(2; R){orbit closures 1 / 27 The moduli of translation surfaces 2 A translation surface can be thought of as a collection of polygons in R , with parallel edge identifications. SL(2; R) acts linearly on such collections of polygons, inducing an action on the moduli of translation surfaces. A translation surface can also be defined as a pair (X ;!), where X is a closed Riemann surface and ! is a holomorphic one-form on X . 2 The simplest example of a translation surface is the flat torus (C=Z ; dz). Alex Wright (University of Chicago) Large SL(2; R){orbit closures 2 / 27 Strata In genus two, a translation surface can have either a single singularity with cone angle 6π, or two singularities each with cone angle 4π. In every genus there is a similar stratification of the moduli space of translation surfaces, according to the number and type of singularities. The SL(2; R){action preserves this decomposition. Alex Wright (University of Chicago) Large SL(2; R){orbit closures 3 / 27 Period coordinates Let Σ be the set of singularities of (X ;!), and let γ1; : : : ; γn be a basis of H1(X ; Σ; Z). The map Φ defined by Z Z Φ(X ;!) = !; : : : ; ! γ1 γn defines local period coordinates on a neighborhood (X ;!) inside the stratum. Alex Wright (University of Chicago) Large SL(2; R){orbit closures 4 / 27 Period coordinates continued Period coordinates provide each stratum with a system of coordinate n charts with values in C and transition maps in GL(n; Z). The set of unit area translation surfaces in each stratum has a natural probability measure (Masur, Veech). Alex Wright (University of Chicago) Large SL(2; R){orbit closures 5 / 27 Motivation Many properties of a translation surfaces are determined to a certain extent by its SL(2; R){orbit closure. The dynamics of straight line flow, for example the Veech dichotomy and deviations of ergodic averages (Veech, Masur, Forni, Eskin-Kontsevich-Zorich) Counting problems (Veech, Eskin-Masur) Flat geometry, for example which translation surfaces can be written as a convex polygon with edge identifications (Veech) Alex Wright (University of Chicago) Large SL(2; R){orbit closures 6 / 27 Water water everywhere, but not a drop to drink Call a translation surface generic if its orbit closure is as big as possible, that is equal to (the set of unit area translation surfaces in) a connected component of a stratum. Masur and Veech showed that the SL(2; R){action is ergodic on each stratum. So almost every translation surface is generic. However, we know relatively few explicit examples. Alex Wright (University of Chicago) Large SL(2; R){orbit closures 7 / 27 Known examples McMullen has classified orbit closures in genus two! Hubert-Lanneau-M¨ollerand Nguyen have provided some extremely interesting examples in genus three. Their methods probably also give some examples in some other strata, but even in genus three produces only a measure zero set of examples. Alex Wright (University of Chicago) Large SL(2; R){orbit closures 8 / 27 Explicit generic translation surfaces Let QR ⊂ R be the algebraic closure of Q inside of R. Theorem (W) Inside of any stratum, let G denote the set of translation surfaces whose period coordinates are linearly independent over QR. Then G is a full measure set of generic points. 2 3 An example of a set linearly independent over QR is f1; π; π ; π ;:::g. Alex Wright (University of Chicago) Large SL(2; R){orbit closures 9 / 27 Strategy Rule out smaller orbit closures. To do this, we have to prove new theorems about orbit closures. What sort of object is an orbit closure? Alex Wright (University of Chicago) Large SL(2; R){orbit closures 10 / 27 Affine invariant submanifolds An affine invariant submanifold is a closed subset of a stratum defined in local period coordinates by real linear equations. Theorem (Eskin-Mirzakhani-Mohammadi) Every SL(2; R){orbit closure is equal to the set of unit area translation surfaces in an affine invariant submanifold. There is an analogy to ergodic theory on homogeneous spaces, and this result is a long sought after partial analogue of Ratner Theory. Alex Wright (University of Chicago) Large SL(2; R){orbit closures 11 / 27 Field of definition Definition We say that an affine invariant submanifold is defined over a number field if, in local period coordinates, it can be defined by real linear equations with coefficients in a number field. Theorem (W) Every affine invariant submanifold is defined over a number field. Alex Wright (University of Chicago) Large SL(2; R){orbit closures 12 / 27 Back to generic surfaces Theorem (W) Inside of any stratum, let G denote the set of translation surfaces whose period coordinates are linearly independent over QR. Then G is a full measure set of generic points. Proof. The orbit closure is defined in period coordinates by linear equations with coefficients in QR. But the period coordinates of (X ;!) 2 G don't satisfy any such equations. So there aren't any equations, and the orbit closure is as big as possible. Alex Wright (University of Chicago) Large SL(2; R){orbit closures 13 / 27 Warm up on field of definition n If V ⊂pC is a linear subspace which is spanned by points with coordinates in pQ[ 2], then V may be defined by linear equations with coefficients in Q[ 2]. We will show that affine invariant submanifolds are defined over a number field by producing lots of points whose period coordinates lie in a number field. Alex Wright (University of Chicago) Large SL(2; R){orbit closures 14 / 27 From closed orbits to number fields The Teichm¨ullergeodesic flow is given by et 0 g = ⊂ SL(2; ): t 0 e−t R Theorem (Kenyon-Smillie) If (X ;!) lies on a closed gt {orbit, then (up to rescaling Re(!); Im(!)), the period coordinates lie in a number field. We will discuss a proof which highlights ideas which recur. In the end, Re(!); Im(!) will be eigenvectors of simple eigenvalues of an integer matrix, which gives the result. Alex Wright (University of Chicago) Large SL(2; R){orbit closures 15 / 27 The action of a pseudo-Anosov Proof of Kenyon-Smillie result. If gT (X ;!) = (X ;!), then the action of gT on (X ;!) is via a pseudo-Anosov. For example, 2 1 ( = 2; dz) = ( = 2; dz): 1 1 C Z C Z The induced action of gT on relative cohomology has a simple largest and smallest eigenvalue, with eigenvectors Re(!) and Im(!). The action on relative cohomology is via an integer matrix (the action on integral relative cohomology), so up to scaling Re(!) and Im(!) have period coordinates in a number field. Alex Wright (University of Chicago) Large SL(2; R){orbit closures 16 / 27 Finding closed orbits A Closing Lemma of Eskin-Mirzakhani-Rafi provides lots of closed gt {orbits in every affine invariant submanifold. (See also Hamenstadt. Uses seminal work of Forni, Veech.) By ergodicity, there are lots of gt {trajectories which almost close up, and the Closing Lemma allows us to find nearby closed orbits if we make some extra assumptions. Alex Wright (University of Chicago) Large SL(2; R){orbit closures 17 / 27 Putting it all together Theorem (W) Every affine invariant submanifold is defined over a number field. Proof. n In local period coordinates, the submanifold is a linear subspace V ⊂ C . The Closing Lemma provides lots of points in V which generate a closed orbit. So there are lots of points in V whose coordinates lie in a number field (Kenyon-Smillie). Alex Wright (University of Chicago) Large SL(2; R){orbit closures 18 / 27 From dynamics to orbit closures We used dynamics (the Closing Lemma) to prove something about orbit closures. This is a major theme, and goes back at least to Smillie's proof that closed SL(2; R) orbits have finite volume. Alex Wright (University of Chicago) Large SL(2; R){orbit closures 19 / 27 The field of definition Definition The field of definition of an affine invariant submanifold is the smallest subfield of R such that the submanifold can be defined in local period coordinates by linear equations with coefficients in this field. Theorem (W) The field of definition is always a real number field of degree at most the genus, and can be calculated as the intersection of all holonomy fields of translation surfaces in the affine invariant submanifold. Alex Wright (University of Chicago) Large SL(2; R){orbit closures 20 / 27 Degree bound Getting the bound on degree is not hard once the computation of the field of definition in terms of holonomy field is established. Corollary (W) In every stratum, there is a dense set of generic translation surfaces whose period coordinates lie in a number field. Alex Wright (University of Chicago) Large SL(2; R){orbit closures 21 / 27 Holonomy field The holonomy field of a translation surface is the smallest subfield k ⊂ R so that something in SL(2; R) orbit has period coordinates in k[i], up to scaling. The hard part is to show that if an affine invariant submanifold contains a translation surface with period coordinates in k[i], then the submanifold is defined over k. Alex Wright (University of Chicago) Large SL(2; R){orbit closures 22 / 27 Tangent space If M is an affine invariant submanifold, its tangent bundle T (M) is a flat 1 subbundle of Hrel .
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