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A coding-free simplicity criterion for the Lyapunov exponents of Teichmüller curves

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Abstract

In this note we show that the results of Furstenberg on the Poisson boundary of lattices of semisimple Lie groups allow to deduce simplicity properties of the Lyapunov spectrum of the Kontsevich–Zorich cocycle of Teichmüller curves in moduli spaces of Abelian differentials without the usage of codings of the Teichmüller flow. As an application, we show the simplicity of some Lyapunov exponents in the setting of (some) Prym Teichmüller curves of genus 4 where a coding-based approach seems hard to implement because of the poor knowledge of the Veech groups of these Teichmüller curves. Finally, we extend the discussion in this note to show the simplicity of Lyapunov exponents coming from (high weight) variations of Hodge structures associated to mirror quintic Calabi–Yau threefolds.

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Notes

  1. See, e.g., the survey [15] of Furman for a gentle introduction to the Poisson boundary.

  2. Recall that, by hypothesis, \((M,\omega )\) has no non-trivial automorphisms so that \(\text {SL}(M,\omega )\) injects into \(\text {Aff}(M,\omega )\).

  3. Recall that a Calabi–Yau \(n\)-fold is a compact Kähler manifold of complex dimension \(n\) with vanishing Ricci curvature.

  4. Actually, the same argument can be used to obtain an algebro-geometrical proof of Forni’s estimate.

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Acknowledgments

The authors are thankful to Alex Furman for suggesting the strategy of the proof of Theorem 1, to Pascal Hubert and Erwan Lanneau for sharing their insights on the geometry of Prym Teichmüller curves of genus 4, to Martin Möller and Jean-Christophe Yoccoz for useful exchanges around the Galois-theoretical simplicity criterion in the article [25], and to Pascal Hubert and Julien Grivaux for a careful reading of earlier versions of this work. Research of the first author is partially supported by NSF Grants DMS 0244542, DMS 0604251 and DMS 0905912. The second author was partially supported by the French ANR Grant “GeoDyM” (ANR-11-BS01-0004) and by the Balzan Research Project of J. Palis.

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  1. Department of Mathematics, University of Chicago, Chicago, IL, 60637, USA

    Alex Eskin

  2. LAGA, CNRS (UMR 7539), Université Paris 13, Sorbonne Paris Cité, 93430, Villetaneuse, France

    Carlos Matheus

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Correspondence to Carlos Matheus.

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Eskin, A., Matheus, C. A coding-free simplicity criterion for the Lyapunov exponents of Teichmüller curves. Geom Dedicata 179, 45–67 (2015). https://doi.org/10.1007/s10711-015-0067-7

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