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

In the moduli space ${{\mathcal {H}}_g}$ of normalized translation surfaces of genus g, consider, for a small parameter ρ > 0, those translation surfaces which have two non-parallel saddle-connections of length ? ρ. We prove that this subset of ${{\mathcal {H}}_g}$ has measure o(ρ ²) w.r.t. any probability measure on ${{\mathcal {H}}_g}$ which is invariant under the natural action of ${SL(2,\mathbb{R})}$ . 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.