Volumes of strata of Abelian differentials and Siegel-Veech constants in large genera

We state conjectures on the asymptotic behavior of the volumes of moduli spaces of Abelian differentials and their Siegel-Veech constants as genus tends to infinity. We provide certain numerical evidence, describe recent advances and the state of the art towards proving these conjectures.Comment: Some background material is added on request of the referee. To appear in Arnold Math. Journa

Semisimplicity of the Lyapunov spectrum for irreducible cocycles

Let $G$ be a semisimple Lie group acting on a space $X$, let $\mu$ be a compactly supported measure on $G$, and let $A$ be a strongly irreducible linear cocycle over the action of $G$. We then have a random walk on $X$, and let $T$ be the associated shift map. We show that the cocycle $A$ over the action of $T$ is conjugate to a block conformal cocycle. This statement is used in the recent paper by Eskin-Mirzakhani on the classifications of invariant measures for the SL(2,R) action on moduli space. The ingredients of the proof are essentially contained in the papers of Guivarch and Raugi and also Goldsheid and Margulis.Comment: 27 pages. Final version based on the referee's report. To appear in Israel J. Mat