Geodesic
Sieve methods in group theory I: Powers in linear groups
Lubotzky, Alexander1 ; Meiri, Chen1, 2
1 Einstein Institute of Mathematics, Hebrew University, Jerusalem 90914, Israel
2 Institute for Advanced Study, Princeton, New Jersey 08540
Journal of the American Mathematical Society, Tome 25 (2012) no. 4, pp. 1119-1148

Voir la notice de l'article provenant de la source American Mathematical Society

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.
DOI : 10.1090/S0894-0347-2012-00736-X

Lubotzky, Alexander 1 ; Meiri, Chen 1, 2

1 Einstein Institute of Mathematics, Hebrew University, Jerusalem 90914, Israel
2 Institute for Advanced Study, Princeton, New Jersey 08540 
@article{10_1090_S0894_0347_2012_00736_X,
     author = {Lubotzky, Alexander and Meiri, Chen},
     title = {Sieve methods in group theory {I:} {Powers} in linear groups},
     journal = {Journal of the American Mathematical Society},
     pages = {1119--1148},
     publisher = {mathdoc},
     volume = {25},
     number = {4},
     year = {2012},
     doi = {10.1090/S0894-0347-2012-00736-X},
     url = {http://geodesic.mathdoc.fr/articles/10.1090/S0894-0347-2012-00736-X/}
}
TY  - JOUR
AU  - Lubotzky, Alexander
AU  - Meiri, Chen
TI  - Sieve methods in group theory I: Powers in linear groups
JO  - Journal of the American Mathematical Society
PY  - 2012
SP  - 1119
EP  - 1148
VL  - 25
IS  - 4
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/articles/10.1090/S0894-0347-2012-00736-X/
DO  - 10.1090/S0894-0347-2012-00736-X
ID  - 10_1090_S0894_0347_2012_00736_X
ER  - 
%0 Journal Article
%A Lubotzky, Alexander
%A Meiri, Chen
%T Sieve methods in group theory I: Powers in linear groups
%J Journal of the American Mathematical Society
%D 2012
%P 1119-1148
%V 25
%N 4
%I mathdoc
%U http://geodesic.mathdoc.fr/articles/10.1090/S0894-0347-2012-00736-X/
%R 10.1090/S0894-0347-2012-00736-X
%F 10_1090_S0894_0347_2012_00736_X
Lubotzky, Alexander; Meiri, Chen. Sieve methods in group theory I: Powers in linear groups. Journal of the American Mathematical Society, Tome 25 (2012) no. 4, pp. 1119-1148. doi: 10.1090/S0894-0347-2012-00736-X

[1] Bourgain, Jean, Gamburd, Alex Uniform expansion bounds for Cayley graphs of 𝑆𝐿₂(𝔽_{𝕡}) Ann. of Math. (2) 2008 625 642

[2] Bourgain, Jean, Gamburd, Alex Expansion and random walks in 𝑆𝐿_{𝑑}(ℤ/𝕡ⁿℤ). I J. Eur. Math. Soc. (JEMS) 2008 987 1011

[3] Bourgain, Jean, Gamburd, Alex Expansion and random walks in 𝑆𝐿_{𝑑}(ℤ/𝕡ⁿℤ). II J. Eur. Math. Soc. (JEMS) 2009 1057 1103

[4] Bourgain, Jean, Gamburd, Alex, Sarnak, Peter Affine linear sieve, expanders, and sum-product Invent. Math. 2010 559 644

[5] Breuillard, Emmanuel, Green, Ben, Tao, Terence Approximate subgroups of linear groups Geom. Funct. Anal. 2011 774 819

[6] Carter, Roger W. Simple groups of Lie type 1972

[7] Chatzidakis, Zoã©, Van Den Dries, Lou, Macintyre, Angus Definable sets over finite fields J. Reine Angew. Math. 1992 107 135

[8] Davenport, Harold Multiplicative number theory 1980

[9] Fried, Michael D., Haran, Dan, Jarden, Moshe Effective counting of the points of definable sets over finite fields Israel J. Math. 1994 103 133

[10] Friedlander, John, Iwaniec, Henryk Opera de cribro 2010

[11] Golod, E. S. On nil-algebras and finitely approximable 𝑝-groups Izv. Akad. Nauk SSSR Ser. Mat. 1964 273 276

[12] Golod, E. S., Å Afareviä, I. R. On the class field tower Izv. Akad. Nauk SSSR Ser. Mat. 1964 261 272

[13] Helfgott, H. A. Growth and generation in 𝑆𝐿₂(ℤ/𝕡ℤ) Ann. of Math. (2) 2008 601 623

[14] Hoory, Shlomo, Linial, Nathan, Wigderson, Avi Expander graphs and their applications Bull. Amer. Math. Soc. (N.S.) 2006 439 561

[15] Hrushovski, E., Kropholler, P. H., Lubotzky, A., Shalev, A. Powers in finitely generated groups Trans. Amer. Math. Soc. 1996 291 304

[16] Kowalski, E. The large sieve and its applications 2008

[17] Lubotzky, Alexander Discrete groups, expanding graphs and invariant measures 2010

[18] Lubotzky, Alexander, Mann, Avinoam On groups of polynomial subgroup growth Invent. Math. 1991 521 533

[19] Lubotzky, Alexander, Mozes, Shahar, Raghunathan, M. S. The word and Riemannian metrics on lattices of semisimple groups Inst. Hautes Études Sci. Publ. Math. 2000

[20] Lubotzky, Alexander, Segal, Dan Subgroup growth 2003

[21] Lang, Serge, Weil, Andrã© Number of points of varieties in finite fields Amer. J. Math. 1954 819 827

[22] Maher, Joseph Random walks on the mapping class group Duke Math. J. 2011 429 468

[23] Nori, Madhav V. On subgroups of 𝐺𝐿_{𝑛}(𝐹_{𝑝}) Invent. Math. 1987 257 275

[24] Rivin, Igor Walks on groups, counting reducible matrices, polynomials, and surface and free group automorphisms Duke Math. J. 2008 353 379

[25] Steinberg, Robert Automorphisms of finite linear groups Canadian J. Math. 1960 606 615

[26] Weisfeiler, Boris Strong approximation for Zariski-dense subgroups of semisimple algebraic groups Ann. of Math. (2) 1984 271 315

Cité par Sources :