Diagonal flow detects topology of strata

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.

Figures (22)

• Figure 4.1: A zippered rectangle construction belonging to the permutation $( 12312456354767 )$. The singularities are a regular point (blue dot) and a zero of cone angle $10\pi$ (yellow dot). Note that every zipper appears as a solid (green) arc exactly once. Also, all rectangles appear twice except$R_6$ which appears three times. This is because the zipper through the right endpoint of $I$ is taller than $R_7$.
• Figure 4.2: Three cases of singularity parameters.
• Figure 4.3: Illustration of the proof of \ref{['t:based-loops']}. Part of the loop $\gamma$ is depicted as a solid curve. The dotted lines represent the boundaries of the polytopes. Unlike the boxes $V_{k-1}$ and $V_{k+1}$, the box $V_k$ is not contained inside a polytope, so the diagonal flow must be applied to it. The resulting segment $\delta_k$ is shown as dashed curve.
• Figure 6.1: The domain and range of the piece $\mathop{\mathrm{RV}}\nolimits^\partial_\xi$.
• Figure 7.1: Example of the spanning set for the minus piece rendering the linear transformations coming from Rauzy moves equal to the Rauzy--Veech matrices. The original permutation is $(12123344)$ representing the stratum $\mathcal{Q}(2,-1,-1)$, which becomes $(12123344)$ after one bottom Rauzy move. In this case, the cycles in the spanning set can be tightened to saddle connections, so they are drawn in this manner. The general case is similar.

Theorems & Definitions (203)

• Definition 3.1
• Definition 3.2
• Definition 3.3
• Lemma 3.4
• Remark 3.5
• Definition 4.1
• Remark 4.2
• Definition 4.3
• Definition 4.4
• Lemma 4.5