SL(2,R)-invariant probability measures on the moduli spaces of translation surfaces are regular

Abstract

In the moduli space $H_g$ of normalized translation surfaces of genus $g$, consider, for a small parameter $\rho >0$, those translation surfaces which have two non-parallel saddle-connections of length $\leq \rho$. We prove that this subset of $H_g$ has measure $o(\rho^2)$ w.r.t. any probability measure on $H_g$ which is invariant under the natural action of $SL(2,R)$. This implies that any such probability measure is regular, a property which is important in relation with the recent fundamental work of Eskin-Kontsevich-Zorich on the Lyapunov exponents of the KZ-cocycle.