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Nonvarying sums of Lyapunov exponents of abelian differentials in low genus. (English) Zbl 1266.14018
Let $$\Omega M_g$$ denote the vector bundle of holomorphic one-forms over the moduli space $$M_g$$ of genus $$g$$ curves minus the zero section. The space $$\Omega M_g$$ is stratified according to the zeros of one-forms. Each stratum contains many Teichmüller curves, for which an algorithm given in [A. Eskin, M. Kontsevich and A. Zorich, “Sum of Lyapunov exponents of the Hodge bundle with respect to the Teichmüller geodesic flow”, arXiv:1112.5872] calculates the sum of Lyapunov exponents (one Teichmüller curve at a time). Based on a limited number of computer experiments about a decade ago, Kontsevich and Zorich observed that the sum of Lyapunov exponents is nonvarying among all the Teichmüller curves in a stratum roughly if the genus plus the number of zeros is less than seven, while the sum varies if this sum is greater than seven.
The authors of the paper under review find an explanation of this observation, and give a detailed description of what happens for all strata of the moduli space of genera less than or equal to five. More precisely, they give a list of strata with nonvarying sum of Lyapunov exponents, and for some other strata they give upper bounds for these sums. Closely related results are also obtained by the authors in [“Quadratic differentials in low genus: exceptional and non-varying”, arXiv:1204.1707] and by F. Yu and K. Zuo in [“Weierstrass filtration on Teichmüller curves and Lyapunov exponents”, arXiv:1203.6053].

MSC:
 14H10 Families, moduli of curves (algebraic) 37D40 Dynamical systems of geometric origin and hyperbolicity (geodesic and horocycle flows, etc.) 14H51 Special divisors on curves (gonality, Brill-Noether theory) 32G15 Moduli of Riemann surfaces, Teichmüller theory (complex-analytic aspects in several variables) 30F30 Differentials on Riemann surfaces
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