Diagonal flow detects topology of strata

TL;DR

This paper investigates the topology of strata of quadratic differentials using diagonal flow, proving the density of Rauzy-Veech groups and confirming the Kontsevich-Zorich conjecture on Lyapunov spectra.

Contribution

It establishes the equality of the flow group and the fundamental group, and proves the Zariski density of Rauzy-Veech groups, advancing understanding of monodromy and Lyapunov spectra.

Findings

Flow group equals the fundamental group.

Rauzy-Veech groups are Zariski dense in symplectic groups.

Confirmed the simplicity of Lyapunov spectra.

Abstract

We study the interplay between the diagonal flow on, and the topology of, a stratum component of a space of rooted quadratic differentials. We prove that the flow group -- the subgroup of the fundamental group generated by almost-flow loops -- equals the fundamental group. 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 symplectic 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. We thus establish the Kontsevich -- Zorich conjecture.