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Uniform Uniform Exponential Growth of Subgroups of the Mapping Class Group

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Abstract

Let Mod(S) denote the mapping class group of a compact, orientable surface S. We prove that finitely generated subgroups of Mod(S) which are not virtually abelian have uniform exponential growth with minimal growth rate bounded below by a constant depending only, and necessarily, on S. For the proof, we find in any such subgroup explicit free group generators which are “short” in any word metric. Besides bounding growth, this allows a bound on the return probability of simple random walks.

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References

  1. J. Anderson, J. Aramayona, K. Shackleton, Uniformly exponential growth and mapping class groups of surfaces, In the Tradition of Ahlfors–Bers. IV, 1–6, Contemp. Math. 432, Amer. Math. Soc., Providence, RI (2007).

  2. J. Behrstock, Asymptotic geometry of the mapping class group and Teichmüller space, PhD Thesis, SUNY at Stony Brook (2004).

  3. S. Bigelow, Braid groups are linear, J. Amer. Math. Soc. 14:2 (2001), 471–486 (electronic).

  4. Birman J., Lubotzky A., McCarthy J.: Abelian and solvable subgroups of the mapping class groups. Duke Math. J. 50(4), 1107–1120 (1983)

    Article  MATH  MathSciNet  Google Scholar 

  5. Bowditch B.: Tight geodesics in the curve complex. Invent. Math. 171(2), 281–300 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  6. E. Breuillard, A strong Tits alternative, arXiv preprint (2008).

  7. Breuillard E., Gelander T.: Cheeger constant and algebraic entropy of linear groups. Int. Math. Res. Not. 56, 3511–3523 (2005)

    Article  MathSciNet  Google Scholar 

  8. Eskin A., Mozes S., Oh H.: Uniform exponenial growth for linear groups. Int. Math. Research Notices 31, 1675–1683 (2002)

    Article  MathSciNet  Google Scholar 

  9. A. Fathi, F. Laudenbach, V. Poenaru, et al., Travaux de Thurston sur les surfaces, Astérisque 66-67 (1979).

  10. K. Fujiwara, Subgroups generated by two pseudo-Anosov elements in a mapping class group. I. Uniform exponential growth, Groups of Diffeomorphisms, to appear.

  11. R. Grigorchuk, P. de la Harpe, Limit behaviour of exponential growth rates for finitely generated groups, in “Essays on Geometry and Related Topics”, Vol. 1,2, Monogr. Enseign. Math. 38, Enseignement Math., Geneva (2001), 351–370.

  12. Hamidi-Tehrani H.: Groups generated by positive multi-twists and the fake lantern problem. Algebr. Geom. Topol. 2, 1155–1178 (2002)

    Article  MATH  MathSciNet  Google Scholar 

  13. Hempel J.: 3-manifolds as viewed from the curve complex. Topology 40(3), 631–657 (2001)

    Article  MATH  MathSciNet  Google Scholar 

  14. N.V. Ivanov, Subgroups of Teichmüller modular groups, Transl. Math. Monogr. 115, Amer. Math. Soc., Providence, RI (1992).

  15. Kaimanovich V., Vershik A.: Random walks on discrete groups: boundary and entropy. Ann. Probab. 11(3), 457–490 (1983)

    Article  MATH  MathSciNet  Google Scholar 

  16. Kesten H.: Symmetric random walks on groups. Trans. Amer. Math. Soc. 92, 336–354 (1959)

    Article  MATH  MathSciNet  Google Scholar 

  17. Krammer D.: Braid groups are linear. Ann. of Math. (2) 155(1), 131–156 (2002)

    Article  MATH  MathSciNet  Google Scholar 

  18. Masur H.A., Minsky Y.N.: Geometry of the complex of curves. I. Hyperbolicity. Invent. Math. 138(1), 103–149 (1999)

    Article  MATH  MathSciNet  Google Scholar 

  19. Masur H.A., Minsky Y.N.: Geometry of the complex of curves. II. Hierarchical structure. Geom. Funct. Anal. 10(4), 902–974 (2000)

    Article  MATH  MathSciNet  Google Scholar 

  20. McCarthy J.: A “Tits-alternative” for subgroups of surfacemapping class groups. Trans. Amer. Math. Soc. 291(2), 583–612 (1985)

    Article  MATH  MathSciNet  Google Scholar 

  21. Paris L., Rolfsen D.: Geometric subgroups of mapping class groups. J. Reine Angew. Math. 521, 47–83 (2000)

    MATH  MathSciNet  Google Scholar 

  22. Shalen P.B., Wagreich P.: Growth rates, Z p -homology, and volumes of hyperbolic 3-manifolds. Trans. Amer. Math. Soc. 331(2), 895–917 (1992)

    Article  MATH  MathSciNet  Google Scholar 

  23. Wilson J.: On exponential growth and uniformly exponential growth for groups. Invent. Math. 155(2), 287–303 (2004)

    Article  MATH  MathSciNet  Google Scholar 

  24. W. Woess, Random Walks on Infinite Graphs and Groups, Cambridge Tracts in Mathematics, 138. Cambridge University Press, Cambridge (2000).

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Authors and Affiliations

  1. Department of Mathematics, University of Michigan, 2074 East Hall, 530 Church Street, Ann Arbor, MI, 48109-1043, USA

    Johanna Mangahas

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  1. Johanna Mangahas
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Correspondence to Johanna Mangahas.

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The author is partially supported by NSF RTG grant #0602191.

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Mangahas, J. Uniform Uniform Exponential Growth of Subgroups of the Mapping Class Group. Geom. Funct. Anal. 19, 1468–1480 (2010). https://doi.org/10.1007/s00039-009-0038-y

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  • DOI: https://doi.org/10.1007/s00039-009-0038-y

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