The flow group of rooted abelian or quadratic differentials

Mark Bell, Vincent Delecroix, Vaibhav Gadre, Rodolfo Gutiérrez-Romo, Saul Schleimer

Published 2021-01-28Version 1

We define the flow group of any component of any stratum of rooted abelian or quadratic differentials (those marked with a horizontal separatrix) to be the group generated by almost-flow loops. We prove that the flow group is equal to the fundamental group of the component. As a corollary, we show that the plus and minus modular Rauzy--Veech groups are finite-index subgroups of their ambient modular monodromy groups. This partially answers a question of Yoccoz. Using this, and recent advances on algebraic hulls and Zariski closures of monodromy groups, we prove that the Rauzy--Veech groups are Zariski dense in their ambient symplectic groups. Density, in turn, implies the simplicity of the plus and minus Lyapunov spectra of any component of any stratum of quadratic differentials. Thus, we establish the Kontsevich--Zorich conjecture.