Duke Mathematical Journal

Lattice point asymptotics and volume growth on Teichmüller space

Jayadev Athreya, Alexander Bufetov, Alex Eskin, and Maryam Mirzakhani

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Abstract

We apply some of the ideas of Margulis’s Ph.D. dissertation to Teichmüller space. Let $X$ be a point in Teichmüller space, and let $B_{R}(X)$ be the ball of radius $R$ centered at $X$ (with distances measured in the Teichmüller metric). We obtain asymptotic formulas as $R$ tends to infinity for the volume of $B_{R}(X)$, and also for the cardinality of the intersection of $B_{R}(X)$ with an orbit of the mapping class group.

Article information

Source
Duke Math. J. Volume 161, Number 6 (2012), 1055-1111.

Dates
First available in Project Euclid: 5 April 2012

Permanent link to this document
https://projecteuclid.org/euclid.dmj/1333633316

Digital Object Identifier
doi:10.1215/00127094-1548443

Mathematical Reviews number (MathSciNet)
MR2913101

Zentralblatt MATH identifier
1246.37009

Subjects
Primary: 37A25: Ergodicity, mixing, rates of mixing
Secondary: 30F60: Teichmüller theory [See also 32G15]

Citation

Athreya, Jayadev; Bufetov, Alexander; Eskin, Alex; Mirzakhani, Maryam. Lattice point asymptotics and volume growth on Teichmüller space. Duke Math. J. 161 (2012), no. 6, 1055--1111. doi:10.1215/00127094-1548443. https://projecteuclid.org/euclid.dmj/1333633316.


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References

  • [A] R. D. M. Accola, Differentials and extremal length on Riemann surfaces, Proc. Nat. Acad. Sci. USA 46 (1960), 540–543.
  • [B] C. Blatter, Über Extremallängen auf geschlossenen Flächen, Comment. Math. Helv. 35 (1961), 153–168.
  • [Du] D. Dumas, Skinning maps are finite-to-one, preprint, arXiv:1203.0273v1 [math.GT]
  • [EMa] A. Eskin and H. Masur, Asymptotic formulas on flat surfaces, Ergodic Theory Dynam. Systems 21 (2001), 443–478.
  • [EMc] A. Eskin and C. McMullen, Mixing, counting, and equidistribution in Lie groups, Duke Math. J. 71 (1993), 181–209.
  • [FM] B. Farb and D. Margalit, A Primer on Mapping Class Groups, Princeton Math. Ser. 49, Princeton Univ. Press, Princeton, 2012.
  • [FK] H. Farkas and I. Kra, Riemann Surfaces, Grad. Texts in Math. 71, Springer, New York, 1980.
  • [FLP] A. Fathi, F. Laudenbach, and V. Poenaru, Travaux de Thurston sur les surfaces, Astérisque 66, 67, Soc. Math. France, Paris, 1979.
  • [Fa] J. D. Fay, Theta functions on Riemann surfaces, Lecture Notes in Math. 352, Springer, Berlin, 1973.
  • [Fo] G. Forni, Deviation of ergodic averages for area-preserving flows on surfaces of higher genus, Ann. of Math. (2) 155 (2002), 1–103.
  • [HMa] J. Hubbard and H. Masur, Quadratic differentials and foliations, Acta Math. 142 (1979), 221–274.
  • [Ka1] V. A. Kaimanovich, “Invariant measures of the geodesic flow and measures at infinity on negatively curved manifolds” in Hyperbolic Behaviour of Dynamical Systems (Paris, 1990), Ann. Inst. H. Poincaré Phys. Théor. 53 (1990), 361–393.
  • [Ka2] V. A. Kaimanovich, “Bowen–Margulis and Patterson measures on negatively curved compact manifolds” in Dynamical Systems and Related Topics (Nagoya, 1990), Adv. Ser. Dynam. Systems 9, World Sci. Publ., River Edge, NJ, 1991, 223–232.
  • [Ke] S. P. Kerckhoff, The asymptotic geometry of Teichmüller space, Topology 19 (1980), 23–41.
  • [LM] E. Lindenstrauss and M. Mirzakhani, Ergodic theory of the space of measured laminations, Int. Math. Res. Not. IMRN 2008, no. 4, art. ID rnm126.
  • [Mar] G. A. Margulis, On Some Aspects of the Theory of Anosov Systems, Springer, Berlin, 2004.
  • [M] B. Maskit, Comparison of hyperbolic and extremal lengths, Ann. Acad. Sci. Fenn. Math. 10 (1985), 381–386.
  • [Ma1] H. Masur, Extension of the Weil–Peterson metric to the boundary of Teichmüller space, Duke Math. J. 43 (1976), 623–635.
  • [Ma2] H. Masur, Interval exchange transformations and measured foliations, Ann. of Math. (2) 115 (1982), 169–200.
  • [MaS] H. Masur and J. Smillie, Hausdorff dimension of sets of nonergodic measured foliations, Ann. of Math. (2) 134 (1991), 455–543.
  • [Mi1] Y. N. Minsky, Teichmüller geodesics and ends of hyperbolic 3-manifolds, Topology 32 (1993), 625–647.
  • [Mi2] Y. N. Minsky, Extremal length estimates and product regions in Teichmüller space, Duke Math. J. 83 (1996), 249–286.
  • [PH] R. C. Penner and J. L. Harer, Combinatorics of Train Tracks, Ann. of Math. Stud. 125, Princeton Univ. Press, Princeton, 1992.
  • [R1] K. Rafi, A combinatorial model for the Teichmüller metric, Geom. Funct. Anal. 17 (2007), 936–959.
  • [R2] K. Rafi, Thick-thin decomposition for quadratic differentials, Math. Res. Lett. 14 (2007), 333–341.
  • [V1] W. A. Veech, Gauss measures for transformations on the space of interval exchange maps, Ann. of Math. (2) 115 (1982), 201–242.
  • [V2] W. A. Veech, The Teichmüller geodesic flow, Ann. of Math. (2) 124 (1986), 441–530.
  • [W] S. A. Wolpert, “Geometry of the Weil–Petersson completion of Teichmüller space” in Surveys in Differential Geometry, VIII (Boston, 2002), Surv. Differ. Geom. 8, Int. Press, Somerville, Mass., 2003, 357–393.