Transience of algebraic varieties in linear groups - applications to generic Zariski density
[Transience des variétés algébriques dans les groupes linéaires - applications à la généricité de la notion de densité Zariski]
Annales de l'Institut Fourier, Tome 63 (2013) no. 5, pp. 2049-2080.

Nous étudions la transience des variétés algébriques dans les groupes linéaires. En particulier, nous montrons qu’une marche aléatoire sur un sous-groupe non élémentaire de SL 2 () évite toute sous-variété algébrique propre avec une probabilité convergeant vers 1 de façon exponentielle. Nous étudions aussi le cas où la marche aléatoire vit dans un sous-groupe Zariski dense du groupe des points réels d’un groupe algébrique semi-simple, défini et déployé sur .

Nous utilisons ces résultats pour montrer qu’un sous-groupe aléatoire (en un sens à préciser) d’un groupe algébrique est Zariski dense.

We study the transience of algebraic varieties in linear groups. In particular, we show that a “non elementary” random walk in SL 2 () escapes exponentially fast from every proper algebraic subvariety. We also treat the case where the random walk takes place in the real points of a semisimple split algebraic group and show such a result for a wide family of random walks.

As an application, we prove that generic subgroups (in some sense) of linear groups are Zariski dense.

DOI : 10.5802/aif.2822
Classification : 20P05, 20G20, 60B15
Keywords: transience, algebraic varieties, Zariski density, random matrix products, random walks, probability of return
Mot clés : propriétés génériques des groupes linéaires, marches aléatoires sur les groupes, produits de matrices aléatoires, sous-variétés des groupes algébriques linéaires
Aoun, Richard 1

1 Université Paris Sud 11 Laboratoire de Mathématiques Bâtiment 425 91405 Orsay (France) Département de Mathématiques Faculté des Sciences de l’Université Saint-Joseph Campus des Sciences et Technologies B.P. 11-514 Riad El Solh Beyrouth 1107 205 (Liban)
@article{AIF_2013__63_5_2049_0,
     author = {Aoun, Richard},
     title = {Transience of algebraic varieties in linear groups - applications to generic {Zariski} density},
     journal = {Annales de l'Institut Fourier},
     pages = {2049--2080},
     publisher = {Association des Annales de l{\textquoteright}institut Fourier},
     volume = {63},
     number = {5},
     year = {2013},
     doi = {10.5802/aif.2822},
     zbl = {06284540},
     mrnumber = {3203113},
     language = {en},
     url = {http://archive.numdam.org/articles/10.5802/aif.2822/}
}
TY  - JOUR
AU  - Aoun, Richard
TI  - Transience of algebraic varieties in linear groups - applications to generic Zariski density
JO  - Annales de l'Institut Fourier
PY  - 2013
SP  - 2049
EP  - 2080
VL  - 63
IS  - 5
PB  - Association des Annales de l’institut Fourier
UR  - http://archive.numdam.org/articles/10.5802/aif.2822/
DO  - 10.5802/aif.2822
LA  - en
ID  - AIF_2013__63_5_2049_0
ER  - 
%0 Journal Article
%A Aoun, Richard
%T Transience of algebraic varieties in linear groups - applications to generic Zariski density
%J Annales de l'Institut Fourier
%D 2013
%P 2049-2080
%V 63
%N 5
%I Association des Annales de l’institut Fourier
%U http://archive.numdam.org/articles/10.5802/aif.2822/
%R 10.5802/aif.2822
%G en
%F AIF_2013__63_5_2049_0
Aoun, Richard. Transience of algebraic varieties in linear groups - applications to generic Zariski density. Annales de l'Institut Fourier, Tome 63 (2013) no. 5, pp. 2049-2080. doi : 10.5802/aif.2822. http://archive.numdam.org/articles/10.5802/aif.2822/

[1] Aoun, R. Random subgroups of linear groups are free., Duke Math. J., Volume 160 (2011) no. 1, pp. 117-173 | DOI | MR | Zbl

[2] de lya Arp, P.; Grigorchuk, R. I.; Chekerini-Sil’berstaĭn, T. Amenability and paradoxical decompositions for pseudogroups and discrete metric spaces, Tr. Mat. Inst. Steklova, Volume 224 (1999) no. Algebra. Topol. Differ. Uravn. i ikh Prilozh., pp. 68-111 | MR | Zbl

[3] Bekka, M.E.B. Amenable unitary representations of locally compact groups, Invent. Math., Volume 100 (1990) no. 2, pp. 383-401 | DOI | MR | Zbl

[4] Benoist, Y. Propriétés asymptotiques des groupes linéaires, Geom. Funct. Anal., Volume 7 (1997) no. 1, pp. 1-47 | DOI | MR | Zbl

[5] Benoist, Y. Propriétés asymptotiques des groupes linéaires. II, Analysis on homogeneous spaces and representation theory of Lie groups, Okayama–Kyoto (1997) (Adv. Stud. Pure Math.), Volume 26, Math. Soc. Japan, Tokyo, 2000, pp. 33-48 | MR | Zbl

[6] Benoist, Y. Convexes divisibles. I, Algebraic groups and arithmetic, Tata Inst. Fund. Res., Mumbai, 2004, pp. 339-374 | MR | Zbl

[7] Borel, A. Linear algebraic groups, Graduate Texts in Mathematics, 126, Springer-Verlag, New York, 1991 | MR | Zbl

[8] Borel, A.; Tits, J. Groupes réductifs, Inst. Hautes Études Sci. Publ. Math. (1965) no. 27, pp. 55-150 | DOI | Numdam | MR | Zbl

[9] Borel, A.; Tits, J. Éléments unipotents et sous-groupes paraboliques de groupes réductifs. I, Invent. Math., Volume 12 (1971), pp. 95-104 | DOI | MR | Zbl

[10] Bougerol, P.; Lacroix, J. Products of random matrices with applications to Schrödinger operators, Progress in Probability and Statistics, 8, Birkhäuser Boston Inc., Boston, MA, 1985 | MR | Zbl

[11] Bourgain, J.; Gamburd, A. Uniform expansion bounds for Cayley graphs of SL 2 (𝔽 p ), Ann. of Math. (2), Volume 167 (2008) no. 2, pp. 625-642 | DOI | MR | Zbl

[12] Bourgain, J.; Gamburd, A. Expansion and random walks in SL d (/p n ) II - with an appendix by J. Bourgain, J. Eur. Math. Soc. (JEMS), Volume 5 (2009), pp. 1057-1103 | DOI | MR | Zbl

[13] Breuillard, E. A Strong Tits Alternative, 2008 (preprint)

[14] Breuillard, E.; Gamburd, A. Strong uniform expansion in SL(2,p), 2010 (to appear in GAFA) | MR | Zbl

[15] Eymard, P. Moyennes invariantes et représentations unitaires, Lecture Notes in Mathematics, Vol. 300, Springer-Verlag, Berlin, 1972 | MR | Zbl

[16] Fulton, W.; Harris, J. Representation theory, Graduate Texts in Mathematics, 129, Springer-Verlag, New York, 1991 (A first course, Readings in Mathematics) | MR | Zbl

[17] Furstenberg, H. Noncommuting random products, Trans. Amer. Math. Soc., Volume 108 (1963), pp. 377-428 | DOI | MR | Zbl

[18] Goldsheid, I. Ya.; Margulis, G. A. Lyapunov exponents of a product of random matrices, Russian Math. Surveys, Volume 44 (1989) no. 5, pp. 11-71 | DOI | MR | Zbl

[19] Guivarc’h, Y. Produits de matrices aléatoires et applications aux propriétés géométriques des sous-groupes du groupe linéaire, Ergodic Theory Dynam. Systems, Volume 10 (1990) no. 3, pp. 483-512 | MR | Zbl

[20] Guivarc’h, Y.; Raugi, A. Frontière de Furstenberg, propriétés de contraction et théorèmes de convergence, Z. Wahrsch. Verw. Gebiete, Volume 69 (1985) no. 2, pp. 187-242 | DOI | MR | Zbl

[21] Helgason, S. Differential geometry, Lie groups, and symmetric spaces, Graduate Studies in Mathematics, 34, American Mathematical Society, Providence, RI, 2001 (Corrected reprint of the 1978 original) | MR | Zbl

[22] Humphreys, J.E. Linear algebraic groups, Springer-Verlag, New York, 1975 (Graduate Texts in Mathematics, No. 21) | MR | Zbl

[23] Kesten, H. Symmetric random walks on groups, Trans. Amer. Math. Soc., Volume 92 (1959), pp. 336-354 | DOI | MR | Zbl

[24] Kingman, J. F. C. Subadditive ergodic theory, Ann. Probability, Volume 1 (1973), pp. 883-909 | DOI | MR | Zbl

[25] Kowalski, E. The large sieve and its applications, Cambridge Tracts in Mathematics, 175, Cambridge University Press, Cambridge, 2008 (Arithmetic geometry, random walks and discrete groups) | MR | Zbl

[26] Le Page, E. Théorèmes limites pour les produits de matrices aléatoires, Probability measures on groups (Oberwolfach, 1981) (Lecture Notes in Math.), Volume 928, Springer, Berlin, 1982, pp. 258-303 | MR | Zbl

[27] Mostow, G. D. Strong rigidity of locally symmetric spaces, Princeton University Press, Princeton, N.J., 1973 (Annals of Mathematics Studies, No. 78) | MR | Zbl

[28] Rivin, I. Walks on groups, counting reducible matrices, polynomials, and surface and free group automorphisms, Duke Math. J., Volume 142 (2008) no. 2, pp. 353-379 | DOI | MR | Zbl

[29] Rivin, I. Zariski density and genericity, Int. Math. Res. Not. IMRN (2010) no. 19, pp. 3649-3657 | DOI | MR | Zbl

[30] Tits, J. Représentations linéaires irréductibles d’un groupe réductif sur un corps quelconque, J. Reine Angew. Math., Volume 247 (1971), pp. 196-220 | MR | Zbl

[31] Tits, J. Free subgroups in linear groups, J. Algebra, Volume 20 (1972), pp. 250-270 | DOI | MR | Zbl

[32] Varjú, P. Expansion in SL d (O K /I), I square-free, arXiv:1001.3664

[33] Vinberg, È. B.; Kac, V. G. Quasi-homogeneous cones, Mat. Zametki, Volume 1 (1967), pp. 347-354 | MR | Zbl

Cité par Sources :