Lattice Point Asymptotics and Volume Growth on Teichmuller space.

@article{Athreya2006LatticePA,
  title={Lattice Point Asymptotics and Volume Growth on Teichmuller space.},
  author={Jayadev S. Athreya and Alexander I. Bufetov and A. V. Eskin and Maryam Mirzakhani},
  journal={Duke Mathematical Journal},
  year={2006},
  volume={161},
  pages={1055-1111}
}
We apply some of the ideas of the Ph.D. Thesis of G. A. Margulis (Mar70) to Teichmuller space. Let X be a point in Teichmuller space, and let BR(X) be the ball of radius R centered at X (with distances measured in the Teichmuller metric). We obtain asymptotic formulas as R tends to infinity for the volume of BR(X), and also for the cardinality of the intersection of BR(X) with an orbit of the mapping class group. 

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References

SHOWING 1-10 OF 39 REFERENCES
Extremal length estimates and product regions in Teichm
We study the Teichm\"uller metric on the Teichm\"uller space of a surface of finite type, in regions where the injectivity radius of the surface is small. The main result is that in such regions the
The asymptotic geometry of teichmuller space
A Combinatorial Model for the Teichmüller Metric
Abstract.We study how the length and the twisting parameter of a curve change along a Teichmüller geodesic. We then use our results to provide a formula for the Teichmüller distance between two
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 on
Connected components of the moduli spaces of Abelian differentials with prescribed singularities
Consider the moduli space of pairs (C,ω) where C is a smooth compact complex curve of a given genus and ω is a holomorphic 1-form on C with a given list of multiplicities of zeroes. We describe
THICK-THIN DECOMPOSITION FOR QUADRATIC DIFFERENTIALS
Quadratic differentials arise naturally in the study of Teichmuller space and Teichmuller geodesics, in particular. A quadratic differential q on a surface S defines a singular Euclidean metric on S.
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 "
Quantitative recurrence and large deviations for Teichmüller geodesic flow
We prove quantitative recurrence and large deviations results for the Teichmuller geodesic flow on connected components of strata of the moduli space Qg of holomorphic unit-area quadratic
Combinatorics of Train Tracks.
Measured geodesic laminations are a natural generalization of simple closed curves in surfaces, and they play a decisive role in various developments in two-and three-dimensional topology, geometry
...
1
2
3
4
...