  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.
