We prove some ergodic-theoretic rigidity properties of the action of SL(2,R) on moduli space. In particular, we show that any ergodic measure invariant under the action of the upper triangular subgroup of SL(2,R) is supported on an invariant affine submanifold. 
The main theorems are inspired by the results of several authors on unipotent flows on homogeneous spaces, and in particular by Ratner's seminal work.

Invariant and stationary measures for the SL(2,R) action on Moduli space

Bounded cohomology of subgroups of mapping class groups (双曲空間及び離散群の研究 研究集会報告集)

We prove some ergodic-theoretic rigidity properties of the action of on moduli space. In particular, we show that any ergodic measure invariant under the action of the upper triangular subgroup of is supported on an invariant affine submanifold. The main theorems are inspired by the results of several authors on unipotent flows on homogeneous spaces, and in particular by Ratner’s seminal work.

Invariant and stationary measures for the action on Moduli space

We prove that affine invariant manifolds in strata of flat surfaces are algebraic varieties. The result is deduced from a generalization of a theorem of Moller. Namely, we prove that the image of a certain twisted Abel-Jacobi map lands in the torsion of a factor of the Jacobians. This statement can be viewed as a splitting of certain mixed Hodge structures.

Splitting mixed Hodge structures over affine invariant manifolds

The Siegel–Veech formula establishes such a connection asymptotically in counting problems; see Veech [26] and Eskin and Masur [5]. The Veech dichotomy shows in particular that for translation surfaces in closed orbits, a very strong form of complete periodicity holds [25]. Work of Calta [4] and also very recent work of Lanneau and Nguyen [13] show that in certain special orbit closures in low genus, every translation surface is completely periodic; see also McMullen [17].

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Cylinder deformations in orbit closures of translation surfaces

This paper is devoted to the classification of the infinite families of Teichmuller curves generated by Prym eigenforms of genus 3 having a single zero. These curves were discovered by McMullen. The main invariants of our classification is the discriminant D of the corresponding quadratic order, and the generators of this order. It turns out that for D sufficiently large, there are two Teichmueller curves when D=1 modulo 8, only one Teichmueller curve when D=0,4 modulo 8, and no Teichmueller curves when D=5 modulo 8. For small values of D, where this classification is not necessarily true, the number of Teichmueller curves can be determined directly. The ingredients of our proof are first, a description of these curves in terms of prototypes and models, and then a careful analysis of the combinatorial connectedness in the spirit of McMullen. As a consequence, we obtain a description of cusps of Teichmueller curves given by Prym eigenforms. We would like also to emphasis that even though we have the same statement compared to, when D=1 modulo 8, the reason for this disconnectedness is different. The classification of these Teichmueller curves plays a key role in our investigation of the dynamics of SL(2,R) on the intersection of the Prym eigenform locus with the stratum H(2,2), which is the object of a forthcoming paper.

Teichmüller curves generated by Weierstrass Prym eigenforms in genus 3 and genus 4

We show that in any non-arithmetic rank 1 orbit closure of translation surfaces, there are only finitely many
Teichm\"uller curves. We also show that in any non-arithmetic rank 1 orbit closure, any completely parabolic
surface is Veech.

Finiteness of Teichmüller Curves in Non-Arithmetic Rank 1 Orbit Closures

We show that every surface in the component \({\mathcal{H}^{\rm hyp}(4)}\), that is the moduli space of pairs \({(M,\omega)}\) where M is a genus three hyperelliptic Riemann surface and \({\omega}\) is an Abelian differential having a single zero on M, is either a Veech surface or a generic surface, i.e. its \({{\rm GL}^{+}(2,\mathbb{R})}\)-orbit is either a closed or a dense subset of \({\mathcal{H}^{\rm hyp}(4)}\). The proof develops new techniques applicable in general to the problem of classifying orbit closures, especially in low genus. Combined with work of Matheus and the second author, a corollary is that there are at most finitely many non-arithmetic Teichmuller curves (closed orbits of surfaces not covering the torus) in \({\mathcal{H}^{\rm hyp}(4)}\).

Non-Veech surfaces in {\mathcal{H}^{\rm hyp}(4)} are generic

This paper is devoted to the classification of the infinite families of Teichmuller curves generated by Prym eigenforms of genus 3 having a single zero. These curves were discovered by McMullen. The main invariants of our classification is the discriminant D of the corresponding quadratic order, and the generators of this order. It turns out that for D sufficiently large, there are two Teichmueller curves when D=1 modulo 8, only one Teichmueller curve when D=0,4 modulo 8, and no Teichmueller curves when D=5 modulo 8. For small values of D, where this classification is not necessarily true, the number of Teichmueller curves can be determined directly. The ingredients of our proof are first, a description of these curves in terms of prototypes and models, and then a careful analysis of the combinatorial connectedness in the spirit of McMullen. As a consequence, we obtain a description of cusps of Teichmueller curves given by Prym eigenforms. 
We would like also to emphasis that even though we have the same statement compared to, when D=1 modulo 8, the reason for this disconnectedness is different. 
The classification of these Teichmueller curves plays a key role in our investigation of the dynamics of SL(2,R) on the intersection of the Prym eigenform locus with the stratum H(2,2), which is the object of a forthcoming paper.

Teichmueller curves generated by Weierstrass Prym eigenforms in genus three and genus four

The moduli space of genus 3 translation surfaces with a single zero has two connected components. We show that in the odd connected component H^{odd}(4) the only GL^+(2,R) orbit closures are closed orbits, the Prym locus Q(3,-1^3), and H^{odd}(4). 
Together with work of Matheus-Wright, this implies that there are only finitely many non-arithmetic closed orbits (Teichmuller curves) in H^{odd}(4) outside of the Prym locus.

Duc-Manh Nguyen

Papers

Teichmüller curves generated by Weierstrass Prym eigenforms in genus 3 and genus 4

Finiteness of Teichmüller Curves in Non-Arithmetic Rank 1 Orbit Closures

Non-Veech surfaces in {\mathcal{H}^{\rm hyp}(4)} are generic

Teichmueller curves generated by Weierstrass Prym eigenforms in genus three and genus four

Classification of higher rank orbit closures in H^{odd}(4)