Nonvarying sums of Lyapunov exponents of Abelian differentials in low genus

We prove that the moduli space of compact genus three Riemann surfaces contains only finitely many algebraically primitive Teichmüller curves. For the stratum M 3 (4), consisting of holomorphic one-forms with a single zero, our approach to finiteness uses the Harder-Narasimhan filtration of the Hodge bundle over a Teichmüller curve to obtain new information on the locations of the zeros of eigenforms. By passing to the boundary of moduli space, this gives explicit constraints on the cusps of Teichmüller curves in terms of cross-ratios of six points on P 1 .These constraints are akin to those that appear in Zilber and Pink's conjectures on unlikely intersections in diophantine geometry. However, in our case one is lead naturally to the intersection of a surface with a family of codimension two algebraic subgroups of G n m × G n a (rather than the more standard G n m ). The ambient algebraic group lies outside the scope of Zilber's Conjecture but we are nonetheless able to prove a sufficiently strong height bound.For the generic stratum M 3 (1, 1, 1, 1), we obtain global torsion order bounds through a computer search for subtori of a codimension-two subvariety of G 9 m . These torsion bounds together with new bounds for the moduli of horizontal cylinders in terms of torsion orders yields finiteness in this stratum. The intermediate strata are handled with a mix of these techniques.

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“…Using the hyperelliptic open-up of [CM12], these excluded forms only arise as cusps of Teichmüller curves in the hyperelliptic locus.…”

Section: Intermediate Strata: Using the Torsion Conditionmentioning

confidence: 99%

“…By the hyperelliptic open-up [CM12] then the whole Teichmüller curve parameterizes a family of hyperelliptic curves. This case has been dealt with in Section 6.3.…”

Section: Intermediate Strata: Using the Torsion Conditionmentioning

confidence: 99%

Teichmüller curves in genus three and just likely intersections in G m n × G a n $\mathbf{G}_{m}^{n}\times\mathbf{G}_{a}^{n}$

2016

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“…This has been fully proved by A. Avila and M. Viana [1] by using combinatorial properties of the so-called Rauzy-Veech induction, after an important partial answer by G. Forni [9] based on complex analytical and potential theoretical methods. Several authors have used algebro-geometric methods to compute individual values and/or sums of these exponents with respect to ergodic SL(2, R)-invariant probability measures: Bainbridge [2], Bouw and Möller [3], Chen and Möller [4], Yu and Zuo [23], and most importantly Eskin, Kontsevich and Zorich [8], [7].…”

Section: Annales De L'institut Fouriermentioning

confidence: 99%

Homology of origamis with symmetries

Matheus¹,

Yoccoz²,

Zmiaikou³

2014

Annales De l'Institut Fourier

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“…In the proof we see that if V = 0, then the divisor classes (m i + k)ψ i − η and k(κ+ψ)−(2g −2+n)η have degree zero on B. This is the case, for instance, when B corresponds to a Teichmüller curve in the strata of abelian or quadratic differentials, as degenerate differentials parameterized in the boundary of Teichmüller curves do not possess any identically zero component (see [CM1,CM2]).…”

Section: Special Families Of Differentialsmentioning

confidence: 97%