Typical properties of periodic Teichmüller geodesics: Lyapunov exponents

Consider a component ${\cal Q}$ of a stratum in the moduli space of area-one abelian differentials on a surface of genus g. Call a property ${\cal P}$ for periodic orbits of the Teichmüller flow on ${\cal Q}$ typical if the growth rate of orbits with property ${\cal P}$ is maximal. We show that the following property is typical. Given a continuous integrable cocycle over the Teichmüller flow with values in a vector bundle $V\to {\cal Q}$ , the logarithms of the eigenvalues of the matrix defined by the cocycle and the orbit are arbitrarily close to the Lyapunov exponents of the cocycle for the Masur–Veech measure.

Correction to: Masur–Veech volumes and intersection theory on moduli spaces of Abelian differentials

A Correction to this paper has been published: https://doi.org/10.1007/s00222-020-00969-4

Periods of Abelian Differentials With Prescribed Singularities

We give a necessary and sufficient condition for a representation of the fundamental group of a closed surface of genus at least $2$ to ${\mathbb{C}}$ to be the holonomy of a translation surface with a prescribed list of conical singularities. Equivalently, we determine the period maps of abelian differentials with prescribed list of multiplicities of zeros. Our main result was also obtained, independently, by Bainbridge, Johnson, Judge, and Park.

On Selberg’s eigenvalue conjecture for moduli spaces of abelian differentials

J.-C. Yoccoz proposed a natural extension of Selberg’s eigenvalue conjecture to moduli spaces of abelian differentials. We prove an approximation to this conjecture. This gives a qualitative generalization of Selberg’s $\frac{3}{16}$ theorem to moduli spaces of abelian differentials on surfaces of genus ${\geqslant}2$.