Skip to main content
SpringerLink
Log in

Sector Estimates for Hyperbolic Isometries

  • Published:
Geometric and Functional Analysis Aims and scope Submit manuscript

Abstract

We prove various orbital counting statements for Fuchsian groups of the second kind. These are of independent interest, and also are used in the companion paper [BK] to produce primes in the Affine Linear Sieve.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Bourgain J., Gamburd A. (2008) Expansion and random walks in SL d (\({\mathbb{Z}/p^{n}\mathbb{Z}}\)), I. J. Eur. Math. Soc. (JEMS) 10(4): 987–1011

    Article  MATH  MathSciNet  Google Scholar 

  2. Bourgain J., Gamburd A. (2008) Uniform expansion bounds for Cayley graphs of SL2(F p ). Ann. of Math. (2) 167(2): 625–642

    Article  MATH  MathSciNet  Google Scholar 

  3. J. Bourgain, A. Gamburd, P. Sarnak, Generalization of Selberg’s theorem and Selberg’s sieve, preprint (2009).

  4. J. Bourgain, A. Kontorovich, On representations of integers in thin subgroups of SL(2, Z), GAFA, to appear in the same issue as the referencing paper; http://arxiv.org/abs/1001.4534

  5. Casselman W., Miličić D. (1982) Asymptotic behavior of matrix coefficients of admissible representations. Duke Math. J. 49(4): 869–930

    Article  MATH  MathSciNet  Google Scholar 

  6. Cowling M. (1978) The Kunze-Stein phenomenon. Ann. Math. (2) 107(2): 209–234

    Article  MathSciNet  Google Scholar 

  7. Cowling M., Haagerup U., Howe R. (1988) Almost L 2 matrix coefficients. J. Reine Angew. Math. 387: 97–110

    MATH  MathSciNet  Google Scholar 

  8. Duke W., Rudnick Z., Sarnak P. (1993) Density of integer points on affine homogeneous varieties. Duke Math. J. 71(1): 143–179

    Article  MATH  MathSciNet  Google Scholar 

  9. Eskin A., McMullen C. (1993) Mixing, counting and equidistribution in lie groups. Duke Math. J. 71: 143–180

    Article  MathSciNet  Google Scholar 

  10. Gamburd A. (2002) On the spectral gap for infinite index “congruence” subgroups of SL2(Z). Israel J. Math. 127: 157–2002

    Article  MathSciNet  Google Scholar 

  11. I.M. Gelfand, M.I. Graev, I.I. Pjateckii-Shapiro. Teoriya predstavlenii i avtomorfnye funktsii. Generalized Functions 6, Izdat. “Nauka”, Moscow, 1966.

  12. A. Good, Local Analysis of Selberg’s Trace Formula, Springer Lecture Notes in Mathematics 1040 (1983).

  13. Good A. (1983) On various means involving the Fourier coefficients of cusp forms. Math. Z. 183(1): 95–129

    Article  MATH  MathSciNet  Google Scholar 

  14. Howe R.E., Moore C.C. (1979) Asymptotic properties of unitary representations. J. Funct. Anal. 32(1): 72–96

    Article  MATH  MathSciNet  Google Scholar 

  15. A.W. Knapp, Representation Theory of Semisimple Groups, Princeton Mathematical Series 36 of Princeton University Press, Princeton, NJ (1986).

  16. A.W. Knapp, P.E. Trapa, Representations of semisimple Lie groups, in “Representation Theory of Lie Groups (Park City, UT, 1998)”, IAS/Park City Math. Ser. 8, Amer. Math. Soc., Providence, RI (2000), 7–87.

  17. Kontorovich A.V. (2009) The hyperbolic lattice point count in infinite volume with applications to sieves. Duke J. Math. 149(1): 1–36

    Article  MATH  MathSciNet  Google Scholar 

  18. A. Kontorovich, H. Oh, Almost prime Pythagorean triples in thin orbits, preprint (2009); http://arxiv.org/abs/1001.0370.

  19. Lax P.D., Phillips R.S. (1982) The asymptotic distribution of lattice points in Euclidean and non-Euclidean space. Journal of Functional Analysis, 46: 280–350

    Article  MATH  MathSciNet  Google Scholar 

  20. H. Oh, N. Shah, Equidistribution and counting for orbits of geometric finite hyperbolic groups, preprint (2010).

  21. Patterson S.J. (1975) The Laplacian operator on a Riemann surface. Compositio Math., 31(1): 83–107

    MATH  MathSciNet  Google Scholar 

  22. Patterson S.J. (1976) The limit set of a Fuchsian group. Acta Math. 136: 241–273

    Article  MATH  MathSciNet  Google Scholar 

  23. A. Selberg, On the estimation of Fourier coefficients of modular forms, Proc. of Symposia in Pure Math. VII (1965), 1–15.

  24. Sullivan D. (1984) Entropy, Hausdorff measures old and new, and limit sets of geometrically finite Kleinian groups. Acta Math. 153(3-4): 259–277

    Article  MATH  MathSciNet  Google Scholar 

  25. A. Venkatesh, Sparse equidistribution problems, period bounds and subconvexity, Annals Math. Online, to appear; http://arxiv.org/abs/math/0506224

Download references

Author information

Authors and Affiliations

  1. Institute of Advanced Study, School of Mathematics, Princeton, NJ, 08540, USA

    Jean Bourgain & Alex Kontorovich

  2. Department of Mathematics, Brown University, Providence, RI, 02912, USA

    Alex Kontorovich

  3. Department of Mathematics, Fine Hall, Washington Road, Princeton, NJ, 08544-1000, USA

    Peter Sarnak

  4. Institute of Advanced Study, School of Mathematics, Princeton, NJ, 08540, USA

    Peter Sarnak

Authors
  1. Jean Bourgain
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Alex Kontorovich
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Peter Sarnak
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Alex Kontorovich.

Additional information

Bourgain is partially supported by NSF grant DMS-0808042. Kontorovich is partially supported by NSF grants DMS-0802998 and DMS-0635607, and the Ellentuck Fund at IAS. Sarnak is partially supported by NSF grant DMS-0758299.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Bourgain, J., Kontorovich, A. & Sarnak, P. Sector Estimates for Hyperbolic Isometries. Geom. Funct. Anal. 20, 1175–1200 (2010). https://doi.org/10.1007/s00039-010-0092-5

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00039-010-0092-5

Keywords and phrases

2010 Mathematics Subject Classification

Search

Navigation