Sieve methods in group theory I: Powers in linear groups

Lubotzky, Alexander; Meiri, Chen

doi:10.1090/S0894-0347-2012-00736-X

Sieve methods in group theory I: Powers in linear groups
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by Alexander Lubotzky and Chen Meiri

J. Amer. Math. Soc. 25 (2012), 1119-1148

DOI: https://doi.org/10.1090/S0894-0347-2012-00736-X

Published electronically: April 11, 2012

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Abstract:

A general sieve method for groups is formulated. It enables one to “measure” subsets of a finitely generated group. As an application we show that if $\Gamma$ is a finitely generated non-virtually solvable linear group of characteristic zero, then the set of proper powers in $\Gamma$ is exponentially small. This is a far-reaching generalization of a result of Hrushovski, Kropholler, Lubotzky, and Shalev.

References

Jean Bourgain and Alex Gamburd, Uniform expansion bounds for Cayley graphs of $\textrm {SL}_2(\Bbb F_p)$, Ann. of Math. (2) 167 (2008), no. 2, 625–642. MR 2415383, DOI 10.4007/annals.2008.167.625
Jean Bourgain and Alex Gamburd, Expansion and random walks in $\textrm {SL}_d(\Bbb Z/p^n\Bbb Z)$. I, J. Eur. Math. Soc. (JEMS) 10 (2008), no. 4, 987–1011. MR 2443926, DOI 10.4171/JEMS/137
Jean Bourgain and Alex Gamburd, Expansion and random walks in $\textrm {SL}_d(\Bbb Z/p^n\Bbb Z)$. II, J. Eur. Math. Soc. (JEMS) 11 (2009), no. 5, 1057–1103. With an appendix by Bourgain. MR 2538500, DOI 10.4171/JEMS/175
Jean Bourgain, Alex Gamburd, and Peter Sarnak, Affine linear sieve, expanders, and sum-product, Invent. Math. 179 (2010), no. 3, 559–644. MR 2587341, DOI 10.1007/s00222-009-0225-3
J. Bourgain, A. Gamburd and P. Sarnak, Generalization of Selberg’s $3/16$ Theorem and Affine Sieve, arXiv:0912.5021
Emmanuel Breuillard, Ben Green, and Terence Tao, Approximate subgroups of linear groups, Geom. Funct. Anal. 21 (2011), no. 4, 774–819. MR 2827010, DOI 10.1007/s00039-011-0122-y
Roger W. Carter, Simple groups of Lie type, Pure and Applied Mathematics, Vol. 28, John Wiley & Sons, London-New York-Sydney, 1972. MR 0407163
Zoé Chatzidakis, Lou van den Dries, and Angus Macintyre, Definable sets over finite fields, J. Reine Angew. Math. 427 (1992), 107–135. MR 1162433
Harold Davenport, Multiplicative number theory, 2nd ed., Graduate Texts in Mathematics, vol. 74, Springer-Verlag, New York-Berlin, 1980. Revised by Hugh L. Montgomery. MR 606931, DOI 10.1007/978-1-4757-5927-3
Michael D. Fried, Dan Haran, and Moshe Jarden, Effective counting of the points of definable sets over finite fields, Israel J. Math. 85 (1994), no. 1-3, 103–133. MR 1264342, DOI 10.1007/BF02758639
John Friedlander and Henryk Iwaniec, Opera de cribro, American Mathematical Society Colloquium Publications, vol. 57, American Mathematical Society, Providence, RI, 2010. MR 2647984, DOI 10.1090/coll/057
E. S. Golod, On nil-algebras and finitely approximable $p$-groups, Izv. Akad. Nauk SSSR Ser. Mat. 28 (1964), 273–276 (Russian). MR 0161878
E. S. Golod and I. R. Šafarevič, On the class field tower, Izv. Akad. Nauk SSSR Ser. Mat. 28 (1964), 261–272 (Russian). MR 0161852
H. A. Helfgott, Growth and generation in $\textrm {SL}_2(\Bbb Z/p\Bbb Z)$, Ann. of Math. (2) 167 (2008), no. 2, 601–623. MR 2415382, DOI 10.4007/annals.2008.167.601
Shlomo Hoory, Nathan Linial, and Avi Wigderson, Expander graphs and their applications, Bull. Amer. Math. Soc. (N.S.) 43 (2006), no. 4, 439–561. MR 2247919, DOI 10.1090/S0273-0979-06-01126-8
E. Hrushovski, P. H. Kropholler, A. Lubotzky, and A. Shalev, Powers in finitely generated groups, Trans. Amer. Math. Soc. 348 (1996), no. 1, 291–304. MR 1316851, DOI 10.1090/S0002-9947-96-01456-0
F. Jouve, E. Kowalski and D. Zywina, Splitting fields of characteristic polynomials of random elements in arithmetic groups, Israel J. of Math., to appear, arXiv:1008.3662.
E. Kowalski, The large sieve and its applications, Cambridge Tracts in Mathematics, vol. 175, Cambridge University Press, Cambridge, 2008. Arithmetic geometry, random walks and discrete groups. MR 2426239, DOI 10.1017/CBO9780511542947
Alexander Lubotzky, Discrete groups, expanding graphs and invariant measures, Modern Birkhäuser Classics, Birkhäuser Verlag, Basel, 2010. With an appendix by Jonathan D. Rogawski; Reprint of the 1994 edition. MR 2569682, DOI 10.1007/978-3-0346-0332-4
A. Lubotzky, Expander Graphs in Pure and Applied Mathematics. Bull. Amer. Math. Soc. 49 (2012), 113–162.
Alexander Lubotzky and Avinoam Mann, On groups of polynomial subgroup growth, Invent. Math. 104 (1991), no. 3, 521–533. MR 1106747, DOI 10.1007/BF01245088
A. Lubotzky and C. Meiri, Sieve methods in group theory II: The Mapping Class Group, Geometriae Dedicata, to appear, arXiv:1104.2450 .
A. Lubotzky and C. Meiri, Sieve methods in group theory III: $\mathrm {aut}(F_n)$, arXiv:1106.4637v1.
Alexander Lubotzky, Shahar Mozes, and M. S. Raghunathan, The word and Riemannian metrics on lattices of semisimple groups, Inst. Hautes Études Sci. Publ. Math. 91 (2000), 5–53 (2001). MR 1828742, DOI 10.1007/BF02698740
Alexander Lubotzky and Dan Segal, Subgroup growth, Progress in Mathematics, vol. 212, Birkhäuser Verlag, Basel, 2003. MR 1978431, DOI 10.1007/978-3-0348-8965-0
Serge Lang and André Weil, Number of points of varieties in finite fields, Amer. J. Math. 76 (1954), 819–827. MR 65218, DOI 10.2307/2372655
Joseph Maher, Random walks on the mapping class group, Duke Math. J. 156 (2011), no. 3, 429–468. MR 2772067, DOI 10.1215/00127094-2010-216
A.I. Mal$’$cev, Homomorphisms onto finite groups. Ivanov. Gos. Ped. Inst. Uchen. Zap. Fiz-Mat. Nauki 8 (1958), 49–60.
J.S. Milne, Algebraic number Theory. Online: www.jmilne.org/math/CourseNotes/.
Madhav V. Nori, On subgroups of $\textrm {GL}_n(\textbf {F}_p)$, Invent. Math. 88 (1987), no. 2, 257–275. MR 880952, DOI 10.1007/BF01388909
L. Pyber and E. Szabó, Growth in finite simple groups of Lie type of bounded rank, arXiv:1005.1858.
Igor Rivin, Walks on groups, counting reducible matrices, polynomials, and surface and free group automorphisms, Duke Math. J. 142 (2008), no. 2, 353–379. MR 2401624, DOI 10.1215/00127094-2008-009
A. Salehi-Golsefidy and P. Varju. Expansion in perfect groups, arXiv:1108.4900.
Robert Steinberg, Automorphisms of finite linear groups, Canadian J. Math. 12 (1960), 606–615. MR 121427, DOI 10.4153/CJM-1960-054-6
P. Varju, Expansion in $SL_d(O_K/I)$, $I$ square-free. arXiv:1001.3664v1.
Boris Weisfeiler, Strong approximation for Zariski-dense subgroups of semisimple algebraic groups, Ann. of Math. (2) 120 (1984), no. 2, 271–315. MR 763908, DOI 10.2307/2006943