Sieve methods in group theory I: Powers in linear groups
Authors:
Alexander Lubotzky and Chen Meiri
Journal:
J. Amer. Math. Soc. 25 (2012), 11191148
MSC (2010):
Primary 20Pxx
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 is a finitely generated nonvirtually solvable linear group of characteristic zero, then the set of proper powers in is exponentially small. This is a farreaching generalization of a result of Hrushovski, Kropholler, Lubotzky, and Shalev.
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 [BG1]
 J. Bourgain and A. Gamburd, Uniform expansion bounds for Cayley graphs of . Ann. of Math. (2) 167 (2008), no. 2, 625642. MR 2415383 (2010b:20070)
 [BG2]
 J. Bourgain and A. Gamburd, Expansion and random walks in . I. J. Eur. Math. Soc. (JEMS) 10 (2008), no. 4, 9871011. MR 2443926 (2010a:05093)
 [BG3]
 J. Bourgain and A. Gamburd, Expansion and random walks in . With an appendix by Bourgain. J. Eur. Math. Soc. (JEMS) 11 (2009), no. 5, 10571103. MR 2538500 (2011a:60021)
 [BGS1]
 J. Bourgain, A. Gamburd and P. Sarnak, Affine linear sieve, expanders, and sumproduct, Invent. Math. 179 (2010), no. 3, 559644. MR 2587341 (2011d:11018)
 [BGS2]
 J. Bourgain, A. Gamburd and P. Sarnak, Generalization of Selberg's Theorem and Affine Sieve, arXiv:0912.5021
 [BGT]
 E. Breuillard, B. Green and T. Tao, Approximate subgroups of linear groups, Geom. Funct. Anal. 21 (2011), no. 4, 774819. MR 2827010
 [Ca]
 R.W. Carter, Simple groups of Lie type. Pure and Applied Mathematics, Vol. 28. John Wiley & Sons, LondonNew YorkSydney, 1972. viii+331 pp. MR 0407163 (53:10946)
 [CvdDM]
 Z. Chatzidakis, L. van den Dries and A. Macintyre, Definable sets over finite fields. J. Reine Angew. Math. 427 (1992), 107135. MR 1162433 (94c:03049)
 [Da]
 H. Davenport, Multiplicative number theory, Second edition, Revised by Hugh L. Montgomery. Graduate Texts in Mathematics, 74. SpringerVerlag, New YorkBerlin, 1980. xiii+177 pp. MR 606931 (82m:10001)
 [FHJ]
 M.D. Fried, D. Haran, and M. Jarden, Effective counting of the points of definable sets over finite fields. Israel J. Math. 85 (1994), no. 13, 103133. MR 1264342 (95k:12016)
 [FI]
 J. Friedlander and H. Iwaniec, Opera de cribro. American Mathematical Society Colloquium Publications, 57. American Mathematical Society, Providence, RI, 2010. xx+527 pp. MR 2647984 (2011d:11227)
 [Go]
 E.S. Golod, On nilalgebras and finitely approximable pgroups. (Russian) Izv. Akad. Nauk SSSR Ser. Mat. 28 (1964), 273276. MR 0161878 (28:5082)
 [GS]
 E.S. Golod and I.R. Shafarevich, On the class field tower. (Russian) Izv. Akad. Nauk SSSR Ser. Mat. 28 (1964), 261272. MR 0161852 (28:5056)
 [He]
 H.A. Helfgott, Growth and generation in , Ann. of Math. (2) 167 (2008), no. 2, 601623. MR 2415382 (2009i:20094)
 [HLW]
 S. Hoory, N. Linial and A. Wigderson, Expander graphs and their applications, Bull. Amer. Math. Soc. (N.S.) 43 (2006), no. 4, 439561. MR 2247919 (2007h:68055)
 [HKLS]
 E. Hrushovski, P.H. Kropholler, A. Lubotzky and A. Shalev, Powers in finitely generated groups, Trans. Amer. Math. Soc. 348 (1996), no. 1, 291304. MR 1316851 (96f:20061)
 [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, Arithmetic geometry, random walks and discrete groups. Cambridge Tracts in Mathematics, 175. Cambridge University Press, Cambridge, 2008. xxii+293 pp. MR 2426239 (2009f:11123)
 [Lu1]
 A. Lubotzky, Discrete groups, expanding graphs and invariant measures, with an appendix by Jonathan D. Rogawski. Reprint of the 1994 edition. Modern Birkhäuser Classics. Birkhäuser Verlag, Basel, 2010. iii+192 pp. MR 2569682 (2010i:22011)
 [Lu2]
 A. Lubotzky, Expander Graphs in Pure and Applied Mathematics. Bull. Amer. Math. Soc. 49 (2012), 113162.
 [LuMa]
 A. Lubotzky and A. Mann, On groups of polynomial subgroup growth. Invent. Math. 104 (1991), no. 3, 521533. MR 1106747 (92d:20038)
 [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]
 A. Lubotzky, S. Mozes and M.S. Raghunathan, The word and Riemannian metrics on lattices of semisimple groups. Inst. Hautes Études Sci. Publ. Math. No. 91 (2000), 553 (2001). MR 1828742 (2002e:22011)
 [LuSe]
 A. Lubotzky and D. Segal, Subgroup growth. Progress in Mathematics, 212. Birkhäuser Verlag, Basel, 2003. MR 1978431 (2004k:20055)
 [LW]
 S. Lang and A. Weil, Number of points of varieties in finite fields. Amer. J. Math. 76, (1954). 819827. MR 0065218 (16:398d)
 [Mah]
 J. Maher, Random walks on the mapping class group, Duke Math. J. 156 (2011), no. 3, 429468. MR 2772067
 [Mal]
 A.I. Malcev, Homomorphisms onto finite groups. Ivanov. Gos. Ped. Inst. Uchen. Zap. FizMat. Nauki 8 (1958), 4960.
 [Mi]
 J.S. Milne, Algebraic number Theory. Online: www.jmilne.org/math/CourseNotes/.
 [No]
 M.V. Nori, On subgroups of . Invent. Math. 88 (1987), no. 2, 257275. MR 880952 (88d:20068)
 [PS]
 L. Pyber and E. Szabó, Growth in finite simple groups of Lie type of bounded rank, arXiv:1005.1858.
 [Ri]
 I. Rivin, Walks on groups, counting reducible matrices, polynomials, and surface and free group automorphisms, Duke Math. J. 142 (2008), no. 2, 353379. MR 2401624 (2009m:20077)
 [SGV]
 A. SalehiGolsefidy and P. Varju. Expansion in perfect groups, arXiv:1108.4900.
 [St]
 R. Steinberg, Automorphisms of finite linear groups. Canad. J. Math. 12 (1960), 606615. MR 0121427 (22:12165)
 [Va]
 P. Varju, Expansion in , squarefree. arXiv:1001.3664v1.
 [We]
 B. Weisfeiler, Strong approximation for Zariskidense subgroups of semisimple algebraic groups. Ann. of Math. (2) 120 (1984), no. 2, 271315. MR 763908 (86m:20053)
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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:
http://dx.doi.org/10.1090/S08940347201200736X
PII:
S 08940347(2012)00736X
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.
