## Sieve methods in group theory I: Powers in linear groups

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- by Alexander Lubotzky and Chen Meiri PDF
- J. Amer. Math. Soc.
**25**(2012), 1119-1148 Request permission

## 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

## Additional Information

**Alexander Lubotzky**- Affiliation: Einstein Institute of Mathematics, Hebrew University, Jerusalem 90914, Israel
- MR Author ID: 116480
- Email: alexlub@math.huji.ac.il
**Chen Meiri**- Affiliation: Einstein Institute of Mathematics, Hebrew University, Jerusalem 90914, Israel
- Address at time of publication: Institute for Advanced Study, Princeton, New Jersey 08540
- Email: chen7meiri@gmail.com
- Received by editor(s): July 19, 2011
- Received by editor(s) in revised form: January 20, 2012
- Published electronically: April 11, 2012
- © Copyright 2012
American Mathematical Society

The copyright for this article reverts to public domain 28 years after publication. - Journal: J. Amer. Math. Soc.
**25**(2012), 1119-1148 - MSC (2010): Primary 20Pxx
- DOI: https://doi.org/10.1090/S0894-0347-2012-00736-X
- MathSciNet review: 2947947