Equidistribution of families of expanding horospheres on moduli spaces of hyperbolic surfaces

Arana-Herrera, Francisco

doi:10.1007/s10711-020-00534-6

Cited by 10 publications

(6 citation statements)

References 15 publications

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“…The integration of B(X) defines the constant b g,n . The above result is extended to arbitrary closed curves or multi-curves by Mirzakhani [Mir16], see also [Ara21].…”

Section: Introductionmentioning

confidence: 89%

Counting mapping class group orbits under shearing coordinates

Sicheng¹,

Su²

2021

Preprint

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Let S g,n be an oriented surface of genus g with n punctures, where 2g − 2 + n > 0 and n > 0. Any ideal triangulation of S g,n induces a global parametrization of the Teichmüller space T g,n called the shearing coordinates. We study the asymptotics of the number of the mapping class group orbits with respect to the standard Euclidean norm of the shearing coordinates. The result is based on the works of Mirzakhani.

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“…The integration of B(X) defines the constant b g,n . The above result is extended to arbitrary closed curves or multi-curves by Mirzakhani [Mir16], see also [Ara21].…”

Section: Introductionmentioning

confidence: 89%

Counting mapping class group orbits under shearing coordinates

Sicheng¹,

Su²

2021

Preprint

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“…Mirzakhani proved in [Mi5, Theorem 1.2] that the same asymptotic length statistics is valid for any individual hyperbolic surface in M g (and not only in average, as we do). F. Arana Herrera and M. Liu independently proved in [AH2], [AH3] and in [Liu] a generalization of this result to arbitrary multicurves. Namely, for any stable graph Γ ∈ G g,n , any associated collection of positive integer weights H and any hyperbolic surface X ∈ M g,n , the asymptotic statistics of normalized lengths of components of hyperbolic geodesic multicurves in Mod g,n •γ(Γ, H) coincides (up to a global normalization constant depending only on g and n) with X (x, H)(P Γ ) dx.…”

Section: Introduction and Statements Of Main Theoremsmentioning

confidence: 95%

Masur-Veech volumes, frequencies of simple closed geodesics and intersection numbers of moduli spaces of curves

Delecroix,

Goujard,

Zograf

et al. 2020

Preprint

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We express the Masur-Veech volume and the area Siegel-Veech constant of the moduli space Qg,n of genus g meromorphic quadratic differentials with at most n simple poles and no other poles as polynomials in the intersection numberswith explicit rational coefficients, where g ′ < g and n ′ < 2g + n. The formulae obtained in this article are derived from lattice point counts involving the Kontsevich volume polynomialsalso appear in Mirzakhani's recursion for the Weil-Petersson volumes of the moduli spaces M g ′ ,n ′ (b 1 , . . . , b n ′ ) of bordered hyperbolic surfaces with geodesic boundaries of lengths b 1 , . . . , b n ′ .A similar formula for the Masur-Veech volume (though without explicit evaluation) was obtained earlier by M. Mirzakhani through a completely different approach. We prove a further result: the density of the mapping class group orbit Modg,n •γ of any simple closed multicurve γ inside the ambient set MLg,n(Z) of integral measured laminations computed by Mirzakhani, coincides with the density of square-tiled surfaces having horizontal cylinder decomposition associated to γ among all square-tiled surfaces in Qg,n.We study the resulting densities (or, equivalently, volume contributions) in more detail in the special case when n = 0. In particular, we compute explicitly the asymptotic frequencies of separating and non-separating simple closed geodesics on a closed hyperbolic surface of genus g for all small genera g and we show that in large genera the separating closed geodesics are 2 3πg • 1 4 g times less frequent.

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“…In particular, Arana-Herrera proves a much more general version of our Theorem 5.1 ([1, Theorem 1.3]), which is one of the key ingredients allowing to attack the counting problem and the length statistics. We learned from [1] that this kind of statistics was initially conjectured by S. Wolpert. Papers [1] and [2] established results closely related to Theorem 1.2.…”

Section: Introductionmentioning

confidence: 99%

“…We learned from [1] that this kind of statistics was initially conjectured by S. Wolpert. Papers [1] and [2] established results closely related to Theorem 1.2.…”

Section: Introductionmentioning

confidence: 99%

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Length statistics of random multicurves on closed hyperbolic surfaces

Liu

2022

Groups Geom. Dyn.

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A multicurve can be decomposed into components. The ratios of the length of each component to the total length give a hint of the shape of the multicurve. In this paper, we determine the distribution of these ratios of a random multicurve with a given topological type on a closed hyperbolic surface, using the methods of Margulis' thesis and Mirzakhani's equidistribution theorem for horospheres. This distribution admits a polynomial density whose coefficients can be expressed explicitly in terms of intersection numbers of psi-classes on the Deligne-Mumford compactification of the moduli space of complex curves, and in particular it does not depend on the hyperbolic metric. This result generalizes prior work of M. Mirzakhani in the case of random pants decompositions. Results very close to ours were obtained independently and simultaneously by F. Arana-Herrera.

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Equidistribution of families of expanding horospheres on moduli spaces of hyperbolic surfaces

Cited by 10 publications

References 15 publications

Counting mapping class group orbits under shearing coordinates

Counting mapping class group orbits under shearing coordinates

Masur-Veech volumes, frequencies of simple closed geodesics and intersection numbers of moduli spaces of curves

Length statistics of random multicurves on closed hyperbolic surfaces

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