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Expansion in perfect groups

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Abstract

Let Γ be a subgroup of \({{\rm GL}_d(\mathbb{Z}[1/q_0])}\) generated by a finite symmetric set S. For an integer q, denote by π q the projection map \({\mathbb{Z}[1/q_0] \to \mathbb{Z}[1/q_0]/q \mathbb{Z}[1/q_0]}\) . We prove that the Cayley graphs of π q (Γ) with respect to the generating sets π q (S) form a family of expanders when q ranges over square-free integers with large prime divisors if and only if the connected component of the Zariski-closure of Γ is perfect, i.e. it has no nontrivial Abelian quotients.

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References

  1. Abels H., Margulis G.A., Soifer G.A.: Semigroups containing proximal linear maps. Israel Journal of Mathematics 91, 1–30 (1995)

    Article  MathSciNet  MATH  Google Scholar 

  2. Alon N.: Eigenvalues and expanders. Combinatorica (2) 6, 83–96 (1986)

    Article  MATH  Google Scholar 

  3. N. Alon, A. Lubotzky, and A. Wigderson. Semidirect products in groups and zig-zag product in graphs: connections and applications. In: 42nd IEEE symposium on foundations of computer science (Las Vegas, NV, 2001) IEEE computer soc., Los Alamitos, CA (2001), pp. 630–637.

  4. Alon N., Milman V.D.: λ1, isoperimetric inequalities for graphs, and superconcentrators. Journal of Combinatorial Theory, Series B 38, 73–88 (1985)

    Article  MathSciNet  MATH  Google Scholar 

  5. Becker T., Weispfenning V.: Gröbner basis, a computational approach to commutative algebra. Springer, New York (1993)

    Google Scholar 

  6. Borel A.: Algebraic linear groups, second enlarged edition. Springer, New York (1991)

    Book  Google Scholar 

  7. Bourgain J., Gamburd A.: Uniform expansion bounds for Cayley graphs of \({SL_2(\mathbb{F}_p)}\) . Annals of Mathematics 167, 625–642 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  8. Bourgain J., Gamburd A.: Expansion and random walks in \({SL_d(\mathbb{Z}/p^n\mathbb{Z})}\) :I. Journal of the European Mathematical Society 10, 987–1011 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  9. Bourgain J., Gamburd A.: Expansion and random walks in \({SL_d(\mathbb Z/p^n\mathbb Z)}\) :II. With an appendix by J. Bourgain. Journal of the European Mathematical Society (5) 11, 1057–1103 (2009)

    MathSciNet  MATH  Google Scholar 

  10. Bourgain J., Gamburd A., Sarnak P.: Sieving and expanders. C. R. Math. Acad. Sci. Paris (3) 343, 155–159 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  11. Bourgain J., Gamburd A., Sarnak P.: Affine linear sieve, expanders, and sum-product. Inventiones Mathematicae (3) 179, 559–644 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  12. Bourgain J., Gamburd A., Sarnak P.: Generalization of Selberg’s 3/16 theorem and affine sieve. Acta Mathematica (2) 207, 255–290 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  13. Bourgain J., Varjú P.P.: Expansion in \({SL_d(\mathbb{Z}/q\mathbb{Z}), q}\) arbitrary. Inventiones Mathematicae. (2) 188, 151–173 (2012)

    Article  MATH  Google Scholar 

  14. Breuillard E., Gamburd A.: Strong uniform expansion in SL(2, p). Geometric and Functional Analysis 20, 1201–1209 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  15. Breuillard E., Green B.J., Tao T.C.: Approximate subgroups of linear groups. Geometric and Functional Analysis 21, 774–819 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  16. Breuillard E., Green B.J., Tao T.C.: Suzuki groups as expanders. Groups,Geometry,and Dynamics (2) 5, 281–299 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  17. E. Breuillard, B.J. Green, R.M. Guralnick, and T.C. Tao. Expansion in finite simple groups of Lie type (in preparation)

  18. Burger M., Sarnak P.: Ramanujan duals II. Inventiones Mathematicae (1) 106, 1–11 (1991)

    Article  MathSciNet  MATH  Google Scholar 

  19. Cano A., Seade J.: On the equicontinuity region of discrete subgroups of PU(1, n). Journal of Geometric Analysis (2) 20, 291–305 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  20. Clozel L.: Démonstration de la conjecture τ. Inventiones Mathematicae (2) 151, 133–150 (2003)

    MathSciNet  Google Scholar 

  21. Dinai O.: Poly-log diameter bounds for some families of finite groups. Proceedings of the American Mathematical Society (11) 134, 3137–3142 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  22. O. Dinai. Expansion properties of finite simple groups, PhD thesis, Hebrew University (2009). http://arxiv.org/abs/1001.5069

  23. Dodziuk J.: Difference equations, isoperimetric inequality and transience of certain random walks. Transactions of the American Mathematical Society (2) 284, 787–794 (1984)

    Article  MathSciNet  MATH  Google Scholar 

  24. Eskin A., Mozes S., Oh H.: On uniform exponential growth for linear groups. Inventiones Mathematicae 160, 1–30 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  25. Farah I.: Approximate homomorphisms II: group homomorphisms. Combinatorica 20, 47–60 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  26. H. Furstenberg. Boundary theory and stochastic processes on homogeneous spaces. In: Proceedings of Symposia in Pure Mathematics (Williamstown, MA, 1972) American Mathematical Society (1973), pp. 193–229.

  27. Gamburd A.: On the spectral gap for infinite index “congruence” subgroups of SL2(Z). Israel Journal of Mathematics 127, 157–200 (2002)

    Article  MathSciNet  Google Scholar 

  28. Gamburd A., Shahshahani M.: Uniform diameter bounds for some families of Cayley graphs. International Mathematics Research Notices 71, 3813–3824 (2004)

    Article  MathSciNet  Google Scholar 

  29. I.Ya. Goldsheid and G.A. Margulis. Lyapunov exponents of a product of random matrices (Russian). Uspekhi Matematicheskikh Nauk (5)44 (1989), 13–60 (translation in Russian Math. Surveys (5) 44 (1989), 11–71).

  30. Gowers W.T.: Quasirandom Groups. Combinatorics, Probability and Computing 17, 363–387 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  31. A. Grothendieck. EGA IV, triosiéme partie. Publications Mathematiques de IHES 28 (1996).

  32. Grunewald F., Segal D.: General algorithms, I: arithmetic groups. Annals of Mathematics, (2nd series) (3) 112, 531–583 (1980)

    MathSciNet  MATH  Google Scholar 

  33. Grunewald F., Segal D.: Decision problems concerning S-arithmetic groups. The Journal of Symbolic Logic (3) 50, 743–772 (1985)

    Article  MathSciNet  MATH  Google Scholar 

  34. Guralnick R.M.: Small representations are completely reducible. Jounal of Algebra 220, 531–541 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  35. Helfgott H.A.: Growth and generation in \({SL_2(\mathbb{Z}/p\mathbb Z)}\) . Annals of Mathematics 167, 601–623 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  36. Helfgott H.A.: Growth in \({SL_3(\mathbb{Z}/p\mathbb Z)}\) . Journal of the European Mathematical Society (3) 13, 761–851 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  37. Hoory S., Linial N., Widgerson A.: Expander graphs and their applications. Bulletin of the American Mathematical Society (4) 43, 439–561 (2006)

    Article  MATH  Google Scholar 

  38. Kazhdan D.A.: On the connection of the dual space of a group with the structure of its closed subgroups. (Russian) Funkcional. Anal. i Prilozen. 1, 71–74 (1967)

    Google Scholar 

  39. D. Kelmer and L. Silberman. A uniform spectral gap for congruence covers of a hyperbolic manifold, preprint. http://arxiv.org/abs/1010.1010

  40. Kim H.H., Sarnak P.: Refined estimates towards the Ramanujan and Selberg conjectures, Appendix to H. H. Kim. Journal of the American Mathematical Society. (1) 16, 139–183 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  41. Kesten H.: Symmetric random walks on groups. Transactions of the American Mathematical Society 92, 336–354 (1959)

    Article  MathSciNet  MATH  Google Scholar 

  42. E. Kowalski. Sieve in expansion preprint. http://arxiv.org/abs/1012.2793

  43. Lackenby M.: Heegaard splittings, the virtually Haken conjecture and property (τ). Inventiones Mathematicae (2) 164, 317–359 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  44. Landazuri V., Seitz G.M.: On the minimal degrees of projective representations of the finite Chevalley groups. Journal of Algebra 32, 418–443 (1974)

    Article  MathSciNet  MATH  Google Scholar 

  45. A. Logar. A computational proof of the Nöether normalization lemma. In: Proceedings of the 6th International Conference, on Applied Algebra, Algebraic Algorithms and Error-Correcting Codes, Lecture Notes in Computer Science. Springer, Berlin 357 (1989), 259–273.

  46. Long D.D., Lubotzky A., Reid A.W.: Heegaard genus and property τ for hyperbolic 3-manifolds. Journal of Topology 1, 152–158 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  47. A. Lubotzky. Cayley graphs: eigenvalues, expanders and random walks. In: Rowbinson, P. (ed.) Surveys in Combinatorics. London Mathematical Society Lecture Note Series 218, pp. 155–189. Cambridge University Press, Cambridge (1995).

  48. Lubotzky A.: Expander graphs in pure and applied mathematics. Bulletin of the American Mathematical Society 49, 113–162 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  49. Margulis G.A.: Explicit constructions of expanders. Problemy Peredaci Informacii (4) 9, 71–80 (1973)

    MathSciNet  MATH  Google Scholar 

  50. Nikolov N., Pyber L.: Product decompositions of quasirandom groups and a Jordan type theorem. Journal of the European Mathematical Society, (3) 13, 1063–1077 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  51. Nori M.V.: On subgroups of \({GL_n(\mathbb{F}_p)}\) . Inventiones Mathematicae 88, 257–275 (1987)

    Article  MathSciNet  MATH  Google Scholar 

  52. L. Pyber and E. Szabó. Growth in finite simple groups of Lie type of bounded rank, preprint. Available online: http://arxiv.org/abs/1005.1858

  53. Richardson D.: Multiplicative independence of algebraic numbers and expressions. Journal of Pure and Applied Algebra 164, 231–245 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  54. A. Salehi. Golsefidy and P. Sarnak. Affine Sieve, preprint. Available online: http://arxiv.org/abs/1109.6432

  55. Sarnak P., Xue X.X.: Bounds for multiplicities of automorphic representations. Duke Mathematical Journal (1) 64, 207–227 (1991)

    Article  MathSciNet  MATH  Google Scholar 

  56. A. Selberg. On the estimation of Fourier coefficients of modular forms. Proceedings of the Symposium on Pure Mathematics vol. VIII, American Mathematical Society, Providence, R.I. (1965), pp. 1–15.

  57. Shalom Y.: Expanding graphs and invariant means. Combinatorica (4) 17, 555–575 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  58. Y. Shalom. Expander graphs and amenable quotients. In: Emerging applications of number theory (Minneapolis, MN, 1996), IMA Vol. Mathematical Applications vol. 109, Springer, New York (1999), pp. 571–581.

  59. Springer T.: Linear algebraic groups. 2nd edn. Birkhäuser, Boston (1998)

    Book  MATH  Google Scholar 

  60. J. Tits. Reductive groups over local fields. Proceedings of symposia in pure mathematics 33 (1979), part 1, 29–69.

  61. Varjú P.P.: Expansion in SL d (O K /I), I square-free. Journal of the European Mathematical Society (1) 14, 273–305 (2012)

    Article  MATH  Google Scholar 

  62. W. Woess. Random walks on infinite graphs and groups. In: Cambridge Tracts in Mathematics vol. 138 Cambridge University Press, Cambridge (2000).

  63. R.B. Warfield. Nilpotent groups. In: Lecture Notes in Math. vol. 513, Springer, Berlin (1976).

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Authors and Affiliations

  1. Department of Mathematics, University of California, San Diego, CA, 92122, USA

    Alireza Salehi Golsefidy

  2. Department of Mathematics, Princeton University, Princeton, NJ, 08544, USA

    Péter P. Varjú

  3. Analysis and Stochastics Research Group of the Hungarian Academy of Sciences, University of Szeged, Szeged, Hungary

    Péter P. Varjú

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  1. Alireza Salehi Golsefidy
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  2. Péter P. Varjú
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Correspondence to Péter P. Varjú.

Additional information

A. Salehi Golsefidy was partially supported by the NSF grant DMS-1160472. P. P. Varjú was partially supported by the NSF grant DMS-0835373.

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Golsefidy, A.S., Varjú, P.P. Expansion in perfect groups. Geom. Funct. Anal. 22, 1832–1891 (2012). https://doi.org/10.1007/s00039-012-0190-7

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