Corpus ID: 235593346

Effective mapping class group dynamics III: Counting filling closed curves on surfaces

@inproceedings{AranaHerrera2021EffectiveMC,
  title={Effective mapping class group dynamics III: Counting filling closed curves on surfaces},
  author={Francisco Arana-Herrera},
  year={2021}
}
We prove a quantitative estimate with a power saving error term for the number of filling closed geodesics of a given topological type and length ≤ L on an arbitrary closed, orientable, negatively curved surface. More generally, we prove estimates of the same kind for the number of free homotopy classes of filling closed curves of a given topological type on a closed, orientable surface whose geometric intersection number with respect to a given filling geodesic current is ≤ L. The proofs rely… Expand
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References

SHOWING 1-10 OF 46 REFERENCES
On the rigidity of discrete isometry groups of negatively curved spaces
Abstract. We prove an ergodic rigidity theorem for discrete isometry groups of CAT(-1) spaces. We give explicit examples of divergence isometry groups with infinite covolume in the case of trees,Expand
A primer on mapping class groups
Given a compact connected orientable surface S there are two fundamental objects attached: a group and a space. The group is the mapping class group of S, denoted by Mod(S). This group is defined byExpand
Thurston's Work on Surfaces
This book is an exposition of Thurston’s theory of surfaces: measured foliations, the compactification of Teichmuller space and the classification of diffeomorphisms. The mathematical content isExpand
Deviation of ergodic averages for area-preserving flows on surfaces of higher genus
We prove a substantial part of a conjecture of Kontsevich and Zorich on the Lyapunov exponents of the Teichmuller geodesic flow on the deviation of ergodic averages for generic conservative flows onExpand
Foliations and laminations on hyperbolic surfaces
GEODESIC LAMINATIONS on surfaces have been introduced by Thurston in his study[l7] of 3-manifolds with a hyperbolic structure (i.e. a Riemannian metric of constant curvature -1). He has also noticedExpand
Effective mapping class group dynamics I: Counting lattice points in Teichm\"uller space
We prove a quantitative estimate with a power saving error term for the number of points in a mapping class group orbit of Teichm\"uller space that lie within a Teichm\"uller metric ball of givenExpand
The asymptotic geometry of teichmuller space
INTRODUCTION TEICHMULLER SPACE is the space of conformal structures on a topological surface MR of genus g where two are equivalent if there is a conformal map between them which is homotopic to theExpand
Skinning maps are finite-to-one
We show that Thurston’s skinning maps of Teichmüller space have finite fibers. The proof centers around a study of two subvarieties of the $${{\rm SL}_2(\mathbb{C})}$$SL2(C) character variety of aExpand
SHORTENING CURVES ON SURFACES
METHODS of shortening a curve in a manifold have been used to establish the existence of closed geodesics, and in particular of simple closed geodesics on 2-spheres. For this purpose, a curveExpand
Quadratic differentials and foliations
This paper concerns the interplay between the complex structure of a Riemann surface and the essentially Euclidean geometry induced by a quadratic differential. One aspect of this geometry is the "Expand
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