Tight upper bounds on the number of invariant components on translation surfaces

We present several results pertaining to Veech surfaces and completely periodic translation surfaces in genus two. A translation surface is a pair ( M , ω ) (M, \omega ) where M M is a Riemann surface and ω \omega is an Abelian differential on M M . Equivalently, a translation surface is a two-manifold which has transition functions which are translations and a finite number of conical singularities arising from the zeros of ω \omega . A direction v v on a translation surface is completely periodic if any trajectory in the direction v v is either closed or ends in a singularity, i.e., if the surface decomposes as a union of cylinders in the direction v v . Then, we say that a translation surface is completely periodic if any direction in which there is at least one cylinder of closed trajectories is completely periodic. There is an action of the group S L ( 2 , R ) SL(2, \mathbb {R}) on the space of translation surfaces. A surface which has a lattice stabilizer under this action is said to be Veech. Veech proved that any Veech surface is completely periodic, but the converse is false. In this paper, we use the J J -invariant of Kenyon and Smillie to obtain a classification of all Veech surfaces in the space H ( 2 ) {\mathcal H}(2) of genus two translation surfaces with corresponding Abelian differentials which have a single double zero. Furthermore, we obtain a classification of all completely periodic surfaces in genus two.

The flow in a fixed direction on a translation surface S determines a decomposition of S into closed invariant sets, each of which is either periodic or minimal. We study this decomposition for translation surfaces in the hyperelliptic connected components Hhyp(2g − 2) and Hhyp(g − 1, g − 1) of the corresponding strata of the moduli space of translation surfaces. Specifically, we characterize the pairs of nonnegative integers (p,m) for which there exists a translation surface in Hhyp(2g−2) or Hhyp(g−1, g−1) with precisely p periodic components and m minimal components. This extends results by Naveh ([Nav08]), who obtained tight upper bounds on numbers of invariant components for each stratum.

We compute many new classes of effective divisors in M¯g,n coming from the strata of Abelian differentials. Our method utilizes maps between moduli spaces and the degeneration of Abelian differentials.

We study the transcendence of periods of abelian differentials, both at the arithmetic and functional level, from the point of view of the natural bi-algebraic structure on strata of abelian differentials. We characterize geometrically the arithmetic points, study their distribution, and prove that in many cases the bi-algebraic curves are the linear ones.

This paper focuses on the interplay between the intersection theoryand the Teichmüller dynamics on the moduli space of curves. Asapplications, we study the cycle class of strata of the Hodge bundle,present an algebraic method to calculate the class of the divisorparameterizing abelian differentials with a nonsimple zero, andverify a number of extremal effective divisors on the moduli space ofpointed curves in low genus.

The dynamics of transcendental functions in the complex plane has received a significant amount of attention. In particular much is known about the description of Fatou components. Besides the types of periodic Fatou components that can occur for polynomials, there also exist so-called Baker domains, periodic components where all orbits converge to infinity, as well as wandering domains. In trying to find analogues of these one dimensional results, it is not clear which higher dimensional transcendental maps to consider. In this paper we find inspiration from the extensive work on the dynamics of complex Hénon maps. We introduce the family of transcendental Hénon maps, and study their dynamics, emphasizing the description of Fatou components. We prove that the classification of the recurrent invariant Fatou components is similar to that of polynomial Hénon maps, and we give examples of Baker domains and wandering domains.

Abelian differentials on Riemann surfaces can be seen as translation surfaces, which are flat surfaces with cone-type singularities. Closed geodesics for the associated flat metrics form cylinders whose number under a given maximal length generically has quadratic asymptotics in this length, with a common coefficient constant for the quadratic asymptotics called a Siegel--Veech constant which is shared by almost all surfaces in each moduli space of translation surfaces. Square-tiled surfaces are specific translation surfaces which have their own quadratic asymptotics for the number of cylinders of closed geodesics. It is an interesting question whether, as n tends to infinity, the Siegel--Veech constants of square-tiled surfaces with n tiles tend to the generic constants of the ambient moduli space. We prove that this is the case in the moduli space H(2) of translation surfaces of genus two with one singularity.

The moduli space of curves endowed with a nonzero abelian differential admits a natural stratification according to the configuration of its zeroes. We give a description of these strata for genus 3 in terms of root system data. For each non-open stratum we obtain a presentation of its orbifold fundamental group.

We compute a closed formula for the class of the closure of the locus of curves in $\overline{\mathcal{M}}_g$ that admit an abelian differential of signature $\kappa=(k_1,...,k_{g-2})$.

Let $w$ be an Abelian differential on compact Riemann surface of genus $g\geq 1$. We obtain an explicit holomorphic factorization formula for $\zeta$-regularized determinant of the Laplacian in flat conical metrics with trivial holonomy $|w|^2$, generalizing the classical Ray-Singer result in $g=1$.

We introduce a twisted cohomology cocycle over the Teichmüller flow and prove a “spectral gap” for its Lyapunov spectrum with respect to the Masur–Veech measures. We then derive Hölder estimates on spectral measures and bounds on the speed of weak mixing for almost all translation flows in every stratum of Abelian differentials on Riemann surfaces, as well as bounds on the deviation of ergodic averages for product translation flows on the product of a translation surface with a circle.

Cette note donne une preuve élémentaire que les strates des différentiels abéliens ne contiennent pas de variétés algébriques complètes.

On the space of ergodic measures for the horocycle flow on strata of Abelian differentials

We show that an algebraic subvariety of the moduli space of genus g Riemann surfaces is coarsely dense with respect to the Teichmüller metric (or Thurston metric) if and only if it has full dimension. We apply this to determine which strata of abelian differentials have coarsely dense projection to moduli space. Furthermore, we prove a result on coarse density of projections of GL 2 (ℝ)-orbit closures in the space of abelian differentials.

A cyclic cover of the complex projective line branched at four appropriate points has a natural structure of a square-tiled surface. We describe the combinatorics of such a square-tiled surface, the geometry of the corresponding Teichmüller curve, and compute the Lyapunov exponents of the determinant bundle over the Teichmüller curve with respect to the geodesic flow. This paper includes a new example (announced by G. Forni and C. Matheus in [17] of a Teichmüller curve of a square-tiled cyclic cover in a stratum of Abelian differentials in genus four with a maximally degenerate Kontsevich--Zorich spectrum (the only known example in genus three found previously by Forni also corresponds to a square-tiled cyclic cover [15]. We present several new examples of Teichmüller curves in strata of holomorphic and meromorphic quadratic differentials with a maximally degenerate Kontsevich--Zorich spectrum. Presumably, these examples cover all possible Teichmüller curves with maximally degenerate spectra. We prove that this is indeed the case within the class of square-tiled cyclic covers.