Dynamics and topology of absolute period foliations of strata of holomorphic 1-forms

Abstract

Let $S_g$ be a closed oriented surface of genus $g$, and let $\Omega\mathcal{M}_g(\kappa)$ be a stratum of the moduli space of holomorphic $1$-forms of genus $g$. In this thesis, we study dynamical and topological properties of the absolute period foliation of $\Omega\mathcal{M}_g(\kappa)$. We show that in most cases, the absolute period foliation is ergodic on the area-$1$ locus, and we give an explicit full measure set of dense leaves. These dynamical results are obtained as an application of a topological result on the connectedness of the space of holomorphic $1$-forms in $\Omega\mathcal{M}_g(\kappa)$ representing a given cohomology class in $H^1(S_g;\mathbb{C})$. As another application, we give a new proof that in most cases, the monodromy representation of $\pi_1(\Omega\mathcal{M}_g(\kappa))$ on absolute homology surjects onto ${\rm Sp}(2g,\mathbb{Z})$.