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

Abstract

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(ρ2) 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.