text: We introduce a space of stable meromorphic differentials with poles of prescribed orders and define its tautological cohomology ring. This space, just as the space of holomorphic differentials, is stratified according to the set of multiplicities of zeros of the differential. The main goal of this paper is to compute the Poincar 'e-dual cohomology classes of all strata. We prove that all these classes are tautological and give an algorithm to compute them. In a second part of the paper we study the Picard group of the strata. We use the tools introduced in the first part to deduce several relations in these Picard groups.
  @cite_24: Abstract We study the algebro-geometric aspects of Teichmuller curves parameterizing square-tiled surfaces with two applications. (a) There exist infinitely many rigid curves on the moduli space of hyperelliptic curves. They span the same extremal ray of the cone of moving curves. Their union is a Zariski dense subset. Hence they yield infinitely many rigid curves with the same properties on the moduli space of stable n -pointed rational curves for even n . (b) The limit of slopes of Teichmuller curves and the sum of Lyapunov exponents for the Teichmuller geodesic flow determine each other, which yields information about the cone of effective divisors on the moduli space of curves.
  @cite_16: We compute the sum of the positive Lyapunov exponents of the Hodge bundle with respect to the Teichmuller geodesic flow. The computation is based on the analytic Riemann-Roch Theorem and uses a comparison of determinants of flat and hyperbolic Laplacians when the underlying Riemann surface degenerates.
  @cite_7: In the 80's H. Masur and W. Veech defined two numerical invariants of strata of abelian differentials: the volume and the Siegel-Veech constant. Based on numerical experiments, A. Eskin and A. Zorich proposed a series of conjectures for the large genus asymptotics of these invariants. By a careful analysis of the asymptotic behavior of quasi-modular forms, D. Chen, M. Moeller, and D. Zagier proved this conjecture for strata of differentials with simple zeros. Here, we prove that the conjecture holds for the other extreme case, i.e. for strata of differentials with a unique zero. Our main ingredient is the expression of the numerical invariants of these strata in terms of Hodge integrals on moduli spaces of curves.
  @cite_2: 
summary: Our computation of cohomology classes of strata of differentials could be useful to compute these numerical invariants. This idea is developed for example in @cite_2 and @cite_24 based on the work of Eskin, Kontsevich, and Zorich @cite_16 (see ). This has been explored in the subsequent paper (see @cite_7 )