Diagonal flow detects topology of strata
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arXiv
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| Format: | Preprint |
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2021
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| author | Bell, Mark Delecroix, Vincent Gadre, Vaibhav Gutiérrez-Romo, Rodolfo Schleimer, Saul |
| author_facet | Bell, Mark Delecroix, Vincent Gadre, Vaibhav Gutiérrez-Romo, Rodolfo Schleimer, Saul |
| contents | 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2101_12197 |
| institution | arXiv |
| publishDate | 2021 |
| record_format | arxiv |
| spellingShingle | Diagonal flow detects topology of strata Bell, Mark Delecroix, Vincent Gadre, Vaibhav Gutiérrez-Romo, Rodolfo Schleimer, Saul Dynamical Systems Primary 30F30, Secondary 37D40, 32G15, 30F60, 37A20, 37F34, 57K20 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. |
| title | Diagonal flow detects topology of strata |
| topic | Dynamical Systems Primary 30F30, Secondary 37D40, 32G15, 30F60, 37A20, 37F34, 57K20 |
| url | https://arxiv.org/abs/2101.12197 |