Lattice point asymptotics and volume growth on Teichmüller space

Athreya, Jayadev S.; Bufetov, Alexander I.; Eskin, Alex; Mirzakhani, Maryam

doi:10.1215/00127094-1548443

Cited by 65 publications

(89 citation statements)

References 24 publications

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“…This result implies Theorem 1.2. Then we use the basic properties of the Hodge norm [5] to prove a closing lemma for the Teichmüller geodesic flow in §6. But the Hodge norm behaves badly near smaller strata, i.e., near points with degenerating zeros of the quadratic differential.…”

Section: Remarksmentioning

confidence: 99%

Counting closed geodesics in moduli space

Eskin¹,

Mirzakhani²

2011

Journal of Modern Dynamics

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Section: Remarksmentioning

confidence: 99%

Counting closed geodesics in moduli space

Eskin¹,

Mirzakhani²

2011

Journal of Modern Dynamics

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“…Though enormous hyperbolic characteristics have been observed in Teichmüller space, we would like to notice a remarkable fact proven by Athreya, Bufetov, Eskin and Mirzakhani in [1]. Indeed, they observed that the volume of the metric balls in Teichmüller space has exponential growth.…”

Section: Corollary 12 (Criterion For Parallelism)mentioning

confidence: 77%

“…The author would like to express his hearty gratitude to Professor Ken'ichi Ohshika for fruitful discussions and his constant encouragements. He also thanks Professor Yair Minsky for informing him about a work by Athreya, Bufetov, Eskin and Mirzakhani in [1]. The author also thanks Professor Athanase Papadopoulos for his kindness and useful comments.…”

Section: Corollary 12 (Criterion For Parallelism)mentioning

confidence: 94%

Geometry of the Gromov product: Geometry at infinity of Teichmüller space

Miyachi¹

2017

J. Math. Soc. Japan

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Abstract. This paper is devoted to study of transformations on metric spaces. It is done in an effort to produce qualitative version of quasi-isometries which takes into account the asymptotic behavior of the Gromov product in hyperbolic spaces. We characterize a quotient semigroup of such transformations on Teichmüller space by use of simplicial automorphisms of the complex of curves, and we will see that such transformation is recognized as a "coarsification" of isometries on Teichmüller space which is rigid at infinity. We also show a hyperbolic characteristic that any finite dimensional Teichmüller space does not admit (quasi)-invertible rough-homothety.

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“…By either [Fay73,Yam80] or [ABEM12,§3], there exists N sufficiently large such 14 D. Aulicino that the derivative of the period matrix (X n , ω n ) ∈ M has a 2 × 2 minor of full rank as well. This contradicts the fact that M has maximal Forni subspace and shows that every boundary translation surface is connected.…”

Section: Definition a Multicomponent Translation Surface Is A Collecmentioning

confidence: 99%

Affine invariant submanifolds with completely degenerate Kontsevich–Zorich spectrum

Aulicino¹

2016

Ergod. Th. Dynam. Sys.

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Link to this article: http://journals.cambridge.org/abstract_S0143385716000262How to cite this article: DAVID AULICINO Afne invariant submanifolds with completely degenerate Kontsevich-Zorich spectrum.Abstract. We prove that if the Lyapunov spectrum of the Kontsevich-Zorich cocycle over an affine SL 2 (R)-invariant submanifold is completely degenerate, i.e. if λ 2 = · · · = λ g = 0, then the submanifold must be an arithmetic Teichmüller curve in the moduli space of Abelian differentials over surfaces of genus three, four, or five. As a corollary, we prove that there is at most a finite number of such Teichmüller curves. IntroductionThe Lyapunov exponents of the Kontsevich-Zorich (KZ) cocycle give information about the dynamics of numerous systems such as billiards in polygons with rational angles, interval exchange transformations and the Teichmüller geodesic flow on the moduli space of Abelian differentials on Riemann surfaces. More precisely, we consider the Lyapunov exponents of the KZ-cocycle on the absolute cohomology bundle over affine SL 2 (R)-invariant submanifolds of the moduli space of Abelian differentials on genus g surfaces. These exponents were studied extensively in [For02], [AV07] and [EKZ14]. In [For02], it was proved that the smallest non-negative exponent is always positive with respect to the canonical measures on strata of the moduli space of Abelian differentials. In [AV07], the KZ conjecture was verified using techniques independent of [For02], i.e. when the spectrum of 2g exponents of the cocycle is simple with respect to the canonical measures on strata. Explicit formulae for sums of the positive Lyapunov exponents of the KZ-cocycle were given in [EKZ14]. On the contrary, there are examples of orbit closures where the Lyapunov exponents are zero. An affine invariant submanifold with λ 2 = · · · = λ g = 0 is said to have completely degenerate KZ-spectrum. In this paper, we prove the following theorem.

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Lattice point asymptotics and volume growth on Teichmüller space

Cited by 65 publications

References 24 publications

Counting closed geodesics in moduli space

Counting closed geodesics in moduli space

Geometry of the Gromov product: Geometry at infinity of Teichmüller space

Affine invariant submanifolds with completely degenerate Kontsevich–Zorich spectrum

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