Sieve methods in group theory I: Powers in linear groups

Authors:
Alexander Lubotzky and Chen Meiri

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

MSC (2010):
Primary 20Pxx

Published electronically:
April 11, 2012

MathSciNet review:
2947947

Full-text PDF

Abstract | References | Similar Articles | Additional Information

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 is a finitely generated non-virtually solvable linear group of characteristic zero, then the set of proper powers in is exponentially small. This is a far-reaching generalization of a result of Hrushovski, Kropholler, Lubotzky, and Shalev.

**[BG1]**Jean Bourgain and Alex Gamburd,*Uniform expansion bounds for Cayley graphs of 𝑆𝐿₂(𝔽_{𝕡})*, Ann. of Math. (2)**167**(2008), no. 2, 625–642. MR**2415383**, 10.4007/annals.2008.167.625**[BG2]**Jean Bourgain and Alex Gamburd,*Expansion and random walks in 𝑆𝐿_{𝑑}(ℤ/𝕡ⁿℤ). I*, J. Eur. Math. Soc. (JEMS)**10**(2008), no. 4, 987–1011. MR**2443926**, 10.4171/JEMS/137**[BG3]**Jean Bourgain and Alex Gamburd,*Expansion and random walks in 𝑆𝐿_{𝑑}(ℤ/𝕡ⁿℤ). II*, J. Eur. Math. Soc. (JEMS)**11**(2009), no. 5, 1057–1103. With an appendix by Bourgain. MR**2538500**, 10.4171/JEMS/175**[BGS1]**Jean Bourgain, Alex Gamburd, and Peter Sarnak,*Affine linear sieve, expanders, and sum-product*, Invent. Math.**179**(2010), no. 3, 559–644. MR**2587341**, 10.1007/s00222-009-0225-3**[BGS2]**J. Bourgain, A. Gamburd and P. Sarnak,*Generalization of Selberg's Theorem and Affine Sieve*, arXiv:0912.5021**[BGT]**Emmanuel Breuillard, Ben Green, and Terence Tao,*Approximate subgroups of linear groups*, Geom. Funct. Anal.**21**(2011), no. 4, 774–819. MR**2827010**, 10.1007/s00039-011-0122-y**[Ca]**Roger W. Carter,*Simple groups of Lie type*, John Wiley & Sons, London-New York-Sydney, 1972. Pure and Applied Mathematics, Vol. 28. MR**0407163****[CvdDM]**Zoé Chatzidakis, Lou van den Dries, and Angus Macintyre,*Definable sets over finite fields*, J. Reine Angew. Math.**427**(1992), 107–135. MR**1162433****[Da]**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****[FHJ]**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**, 10.1007/BF02758639**[FI]**John Friedlander and Henryk Iwaniec,*Opera de cribro*, American Mathematical Society Colloquium Publications, vol. 57, American Mathematical Society, Providence, RI, 2010. MR**2647984****[Go]**E. S. Golod,*On nil-algebras and finitely approximable 𝑝-groups*, Izv. Akad. Nauk SSSR Ser. Mat.**28**(1964), 273–276 (Russian). MR**0161878****[GS]**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****[He]**H. A. Helfgott,*Growth and generation in 𝑆𝐿₂(ℤ/𝕡ℤ)*, Ann. of Math. (2)**167**(2008), no. 2, 601–623. MR**2415382**, 10.4007/annals.2008.167.601**[HLW]**Shlomo Hoory, Nathan Linial, and Avi Wigderson,*Expander graphs and their applications*, Bull. Amer. Math. Soc. (N.S.)**43**(2006), no. 4, 439–561 (electronic). MR**2247919**, 10.1090/S0273-0979-06-01126-8**[HKLS]**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**, 10.1090/S0002-9947-96-01456-0**[JKZ]**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.**[Ko]**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****[Lu1]**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****[Lu2]**A. Lubotzky,*Expander Graphs in Pure and Applied Mathematics.*Bull. Amer. Math. Soc. 49 (2012), 113-162.**[LuMa]**Alexander Lubotzky and Avinoam Mann,*On groups of polynomial subgroup growth*, Invent. Math.**104**(1991), no. 3, 521–533. MR**1106747**, 10.1007/BF01245088**[LuMe1]**A. Lubotzky and C. Meiri,*Sieve methods in group theory II: The Mapping Class Group*, Geometriae Dedicata, to appear, arXiv:1104.2450 .**[LuMe2]**A. Lubotzky and C. Meiri,*Sieve methods in group theory III*: , arXiv:1106.4637v1.**[LMR]**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****[LuSe]**Alexander Lubotzky and Dan Segal,*Subgroup growth*, Progress in Mathematics, vol. 212, Birkhäuser Verlag, Basel, 2003. MR**1978431****[LW]**Serge Lang and André Weil,*Number of points of varieties in finite fields*, Amer. J. Math.**76**(1954), 819–827. MR**0065218****[Mah]**Joseph Maher,*Random walks on the mapping class group*, Duke Math. J.**156**(2011), no. 3, 429–468. MR**2772067**, 10.1215/00127094-2010-216**[Mal]**A.I. Malcev,*Homomorphisms onto finite groups.*Ivanov. Gos. Ped. Inst. Uchen. Zap. Fiz-Mat. Nauki 8 (1958), 49-60.**[Mi]**J.S. Milne,*Algebraic number Theory.*Online: www.jmilne.org/math/CourseNotes/.**[No]**Madhav V. Nori,*On subgroups of 𝐺𝐿_{𝑛}(𝐹_{𝑝})*, Invent. Math.**88**(1987), no. 2, 257–275. MR**880952**, 10.1007/BF01388909**[PS]**L. Pyber and E. Szabó,*Growth in finite simple groups of Lie type of bounded rank*, arXiv:1005.1858.**[Ri]**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**, 10.1215/00127094-2008-009**[SGV]**A. Salehi-Golsefidy and P. Varju.*Expansion in perfect groups*, arXiv:1108.4900.**[St]**Robert Steinberg,*Automorphisms of finite linear groups*, Canad. J. Math.**12**(1960), 606–615. MR**0121427****[Va]**P. Varju,*Expansion in , square-free*. arXiv:1001.3664v1.**[We]**Boris Weisfeiler,*Strong approximation for Zariski-dense subgroups of semisimple algebraic groups*, Ann. of Math. (2)**120**(1984), no. 2, 271–315. MR**763908**, 10.2307/2006943

Retrieve articles in *Journal of the American Mathematical Society*
with MSC (2010):
20Pxx

Retrieve articles in all journals with MSC (2010): 20Pxx

Additional Information

**Alexander Lubotzky**

Affiliation:
Einstein Institute of Mathematics, Hebrew University, Jerusalem 90914, Israel

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

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

Keywords:
Sieve,
property-$𝜏$,
powers,
linear groups,
finite groups of Lie type

Received by editor(s):
July 19, 2011

Received by editor(s) in revised form:
January 20, 2012

Published electronically:
April 11, 2012

Article copyright:
© Copyright 2012
American Mathematical Society

The copyright for this article reverts to public domain 28 years after publication.