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Geometriae Dedicata

, Volume 159, Issue 1, pp 327–336 | Cite as

Sieve methods in group theory II: the mapping class group

  • Alexander Lubotzky
  • Chen MeiriEmail author
Original Paper

Abstract

We prove that the set of non-pseudo-Anosov elements in the Torelli group is exponentially small. This answers a question of Kowalski (2008).

Keywords

Sieve methods Mapping class group Random walks Pseudo-Anosov elements 

Mathematics Subject Classification (2000)

20F69 57M60 

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References

  1. 1.
    Bleiler, S.A., Casson, A.J.: Automorphisms of Surfaces After Nielsen and Thurston. London Mathematical Society Student Texts, vol. 9, iv+105 pp. Cambridge University Press, Cambridge (1988)Google Scholar
  2. 2.
    Bestvina M., Handel M.: Train tracks and automorphisms of free groups. Ann. Math. (2) 135(1), 1–51 (1992)MathSciNetzbMATHCrossRefGoogle Scholar
  3. 3.
    Chavdarov, N.I.: The generic irreducibility of the numerator of the zeta function in a family of curves with large monodromy. Thesis (Ph.D.) Princeton University. 56 pp, ProQuest LLC, Thesis More links (1995)Google Scholar
  4. 4.
    Farb, B., Margalit, D.: A primer on mapping class groups. (preprint). www.math.utah.edu/~margalit/primer/
  5. 5.
    Grunewald F., Lubotzky A.: Linear representations of the automorphism group of a free group. Geom. Funct. Anal. 18(5), 1564–1608 (2009)MathSciNetCrossRefGoogle Scholar
  6. 6.
    Grunewald, F., Lubotzky, A.: Arithmetic quotients of the mapping class group. (In preperation)Google Scholar
  7. 7.
    Hahn, A.J., O’Meara, O.T.: The Classical Groups and KK-Theory. Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 291, xvi+576 pp. Springer, Berlin (1989)Google Scholar
  8. 8.
    Ivanov N.V.: Subgroups of Teichmuller Modular Groups, Translations of Mathematical Monographs, vol. 115. American Mathematical Society, Providence, RI (1992)Google Scholar
  9. 9.
    Kowalski E.: The Large Sieve and Its Applications. Arithmetic Geometry, Random Walks and Discrete Groups. Cambridge Tracts in Mathematics vol. 175. Cambridge University Press, Cambridge (2008)CrossRefGoogle Scholar
  10. 10.
    Looijenga E.: Prym representations of mapping class groups. Geom. Dedicata 64(1), 69–83 (1997)MathSciNetzbMATHCrossRefGoogle Scholar
  11. 11.
    Lubotzky, A., Meiri, C.: Sieve methods in group theory II: Powers in linear groups. arXiv:1107.3666Google Scholar
  12. 12.
    Lubotzky, A.: Discrete Groups, Expanding Graphs and Invariant Measures, with an appendix by J. D. Rogawski. Reprint of the 1994 edition. Modern Birkhuser Classics. Birkhuser Verlag, Basel. (iii+192 pp) (2010)Google Scholar
  13. 13.
    Lubotzky, A.: Expanders in pure and applied mathematics, Bull. AMS (to appear). arXiv:1105.2389v1Google Scholar
  14. 14.
    Maher J.: Random walks on the mapping class group. Duke Math. J. 156(3), 429–468 (2011)MathSciNetzbMATHCrossRefGoogle Scholar
  15. 15.
    Malestein, J., Souto, J.: On genericity of pseudo-Anosovs in the Torelli group. arXiv:1102.0601Google Scholar
  16. 16.
    Rivin I.: Walks on groups, counting reducible matrices, polynomials, and surface and free group automorphisms. Duke Math. J. 142(2), 353–379 (2008)MathSciNetzbMATHCrossRefGoogle Scholar
  17. 17.
    Rivin I.: Zariski density and genericity. Int. Math. Res. Not. 19, 3649–3657 (2010)MathSciNetGoogle Scholar
  18. 18.
    Salehi-Golsefidy, A., Varju, P.: Expansion in perfect groups. arXiv:1108.4900Google Scholar
  19. 19.
    Tits, J.: Systèmes générateurs de groupes de congruence. (French) C. R. Acad. Sci. Paris Se’r. A-B 283(9), Ai, A693–A695 (1976)Google Scholar
  20. 20.
    Weisfeiler B.: Strong approximation for Zariski-dense subgroups of semisimple algebraic groups. Ann. Math. (2) 120(2), 271–315 (1984)MathSciNetzbMATHCrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media B.V. 2011

Authors and Affiliations

  1. 1.Einstein Institute of MathematicsHebrew UniversityJerusalemIsrael

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