Diagonal flow, topology and applications to Lyapunov exponents.

Given a flow on a manifold, we may define the flow group to be the subgroup of the fundamental group generated by the almost flow loops, namely, by based loops that are obtained from flow segments that start and end in a fixed contractible open set. For the diagonal (Teichmüller) flow on a linear invariant submanifold in a stratum of abelian differentials, we prove that the flow group equals the fundamental group. For components of strata of abelian/ quadratic differentials, we use the flow group result to derive simplicity of Lyapunov exponents for the plus and minus Kontsevich—Zorich cocycles, a generalisation in statement and in approach of simplicity for abelian strata by Avila—Viana. In the process, we answer in the affirmative several questions by Yoccoz regarding Rauzy diagrams for interval exchange maps. This is (variously) joint work with Arana-Herrera, Bell, Delecroix, Gutierrez-Romo and Schleimer.