Growth of the number of simple closed geodesics on hyperbolic surfaces

Mirzakhani, Maryam

doi:10.4007/annals.2008.168.97

Cited by 157 publications

(201 citation statements)

References 20 publications

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“…where c(S) is a constant depending only on the geometry of S [Mir08]. Rivin extended this result to geodesics with at most one self-intersection, to get that…”

Section: Question 2 If K = K(l) Is a Function Of L What Is The Asymmentioning

confidence: 94%

“…Thus, l max (P) ≤ l(P) + s X for all pairs of pants embedded in S. By [Mir08], the number of pairs of pants P with length at most L grows asymptotically like c(X)L 6g−6+2n . So there is some constant c ′ (X) so that this number is bounded below by c ′ (X)L 6g−6+2n for all L > l sys , where l sys is the length of the shortest closed geodesic on S. If l(P) < l 0 − s X then l max (P) < l 0 .…”

Section: Area(x) Lsysmentioning

confidence: 98%

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Lower bound on the number of non-simple closed geodesics on surfaces

Sapir

2016

Geom Dedicata

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Abstract. We give a lower bound on the number of non-simple closed curves on a hyperbolic surface, given upper bounds on both length and self-intersection number. In particular, we carefully show how to construct closed geodesics on pairs of pants, and give a lower bound on the number of curves in this case. The lower bound for arbitrary surfaces follows from the lower bound on pairs of pants. This lower bound demonstrates that as the self-intersection number K = K(L) goes from a constant to a quadratic function of L, the number of closed geodesics transitions from polynomial to exponential in L. We show upper bounds on the number of such geodesics in a subsequent paper.

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“…where c(S) is a constant depending only on the geometry of S [Mir08]. Rivin extended this result to geodesics with at most one self-intersection, to get that…”

Section: Question 2 If K = K(l) Is a Function Of L What Is The Asymmentioning

confidence: 94%

Section: Area(x) Lsysmentioning

confidence: 98%

Lower bound on the number of non-simple closed geodesics on surfaces

Sapir

2016

Geom Dedicata

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“…The definition, and in particular the introduction of the factor 1 2 , is designed so that when n=2 the function 1 coincides with Thurston's length function : ML(S)!R for the hyperbolic metric on S associated with ∈Hit 2 (S), which plays a fundamental role in hyperbolic geometry; see for instance [40], [17], [39], [3], [33] for a few applications of this length function .…”

Section: Smentioning

confidence: 99%

Hitchin characters and geodesic laminations

Bonahon

Dreyer

2017

Acta Mathematica

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For a closed surface S, the Hitchin component Hit n (S) is a preferred component of the character variety consisting of group homomorphisms from the fundamental group π 1 (S) to the Lie group PSL n (R). We construct a parametrization of the Hitchin component that is well-adapted to a geodesic lamination λ on the surface. This is a natural extension of Thurston's parametrization of the Teichmüller space T(S) by shear coordinates associated to λ, corresponding to the case n = 2. However, significantly new ideas are needed in this higher dimensional case. The article concludes with a few applications. ContentsIntroduction 0.1. Background and motivation 0.2. Main results 1. Generic configurations of flags 1.1. Flags 1.2. Wedge-product invariants of generic flag triples 1.3. Quadruple ratios 1.4. Double ratios 1.5. Positivity 2. Geodesic laminations 3. Triangle invariants 3.1. The flag curve 3.2. Triangle invariants of Hitchin characters 4. Tangent cycles for a geodesic lamination 4.1. Tangent cycles 4.2. Train track neighborhoods 4.3. Homological interpretation of tangent cycles 4.4. Tangent cycles relative to the slits 4.5. Homological interpretation of tangent cycles relative to the slits 4.6. Twisted relative tangent cycles 4.7. Relative tangent cycles from another viewpoint 5. The shearing tangent cycle of a Hitchin character 5.1. Slithering 5.2. The shearing cycle 6. Hitchin characters are determined by their invariants 1 2 FRANCIS BONAHON AND GUILLAUME DREYER 6.1. Revisiting the slithering map 37 6.2. Reconstructing a Hitchin homomorphism from its invariants 38 7. Length functions 41 7.1. Length functions associated to Hitchin characters 42 7.2. Shearing cycles and length functions 44 8. Parametrizing Hitchin components 50 8.1. Constraints between invariants 50 8.2. An estimate from the Positive Intersection Condition 55 8.3. Realization of invariants, and parametrization of Hit n (S) 58 8.4. Constraints among triangle invariants, and on shearing cycles 60 9. The action of pseudo-Anosov homeomorphisms on the Hitchin component 62 10. Length functions of measured laminations 64 References 66

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“…In this respect, one should mention the name of Maryam Mirzakhani, who dedicated a large part of her work to this subject, and more generally to the dynamics and the ergodic theory of the Teichmüller flows. See in particular [23], [24], [25], and [1]. Connections with number theory were also discovered through the number fields of dilatations of pseudo-Anosov homeomorphisms, which are related to the length spectrum of the Teichmüller geodesic flow.…”

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confidence: 99%

Book Review: Teichmüller theory and applications to geometry, topology, and dynamics. Volume 1: Teichmüller theory; Teichmüller theory and applications to geometry, topology, and dynamics. Volume 2: Surface homeomorphisms and rational functions

Papadopoulos¹

2018

Bull. Amer. Math. Soc.

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Growth of the number of simple closed geodesics on hyperbolic surfaces

Cited by 157 publications

References 20 publications

Lower bound on the number of non-simple closed geodesics on surfaces

Lower bound on the number of non-simple closed geodesics on surfaces

Hitchin characters and geodesic laminations

Book Review: Teichmüller theory and applications to geometry, topology, and dynamics. Volume 1: Teichmüller theory; Teichmüller theory and applications to geometry, topology, and dynamics. Volume 2: Surface homeomorphisms and rational functions

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