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, p. 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 : https://doi.org/10.5802/aif.2822
Classification:  20P05,  20G20,  60B15
Mots clés: propriétés génériques des groupes linéaires, marches aléatoires sur les groupes, produits de matrices aléatoires, sous-variétés des groupes algébriques linéaires
@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},
     publisher = {Association des Annales de l'institut Fourier},
     volume = {63},
     number = {5},
     year = {2013},
     pages = {2049-2080},
     doi = {10.5802/aif.2822},
     mrnumber = {3203113},
     zbl = {06284540},
     language = {en},
     url = {http://http://www.numdam.org/item/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://www.numdam.org/item/AIF_2013__63_5_2049_0/

[1] Aoun, R. Random subgroups of linear groups are free., Duke Math. J., Tome 160 (2011) no. 1, pp. 117-173 | Article | MR 2838353 | Zbl 1239.20051

[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, Tome 224 (1999) no. Algebra. Topol. Differ. Uravn. i ikh Prilozh., pp. 68-111 | MR 1721355 | Zbl 0968.43002

[3] Bekka, M.E.B. Amenable unitary representations of locally compact groups, Invent. Math., Tome 100 (1990) no. 2, pp. 383-401 | Article | MR 1047140 | Zbl 0702.22010

[4] Benoist, Y. Propriétés asymptotiques des groupes linéaires, Geom. Funct. Anal., Tome 7 (1997) no. 1, pp. 1-47 | Article | MR 1437472 | Zbl 0947.22003

[5] Benoist, Y. Propriétés asymptotiques des groupes linéaires. II, Analysis on homogeneous spaces and representation theory of Lie groups, Okayama–Kyoto (1997), Math. Soc. Japan, Tokyo (Adv. Stud. Pure Math.) Tome 26 (2000), pp. 33-48 | MR 1770716 | Zbl 0960.22012

[6] Benoist, Y. Convexes divisibles. I, Algebraic groups and arithmetic, Tata Inst. Fund. Res., Mumbai (2004), pp. 339-374 | MR 2094116 | Zbl 1084.37026

[7] Borel, A. Linear algebraic groups, Springer-Verlag, New York, Graduate Texts in Mathematics, Tome 126 (1991) | MR 1102012 | Zbl 0726.20030

[8] Borel, A.; Tits, J. Groupes réductifs, Inst. Hautes Études Sci. Publ. Math. (1965) no. 27, pp. 55-150 | Article | Numdam | MR 207712 | Zbl 0145.17402

[9] Borel, A.; Tits, J. Éléments unipotents et sous-groupes paraboliques de groupes réductifs. I, Invent. Math., Tome 12 (1971), pp. 95-104 | Article | MR 294349 | Zbl 0238.20055

[10] Bougerol, P.; Lacroix, J. Products of random matrices with applications to Schrödinger operators, Birkhäuser Boston Inc., Boston, MA, Progress in Probability and Statistics, Tome 8 (1985) | MR 886674 | Zbl 0572.60001

[11] Bourgain, J.; Gamburd, A. Uniform expansion bounds for Cayley graphs of SL 2 (𝔽 p ), Ann. of Math. (2), Tome 167 (2008) no. 2, pp. 625-642 | Article | MR 2415383 | Zbl 1216.20042

[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), Tome 5 (2009), pp. 1057-1103 | Article | MR 2538500 | Zbl 1193.20060

[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 2746951 | Zbl 1253.20051

[15] Eymard, P. Moyennes invariantes et représentations unitaires, Springer-Verlag, Berlin, Lecture Notes in Mathematics, Vol. 300 (1972) | MR 447969 | Zbl 0249.43004

[16] Fulton, W.; Harris, J. Representation theory, Springer-Verlag, New York, Graduate Texts in Mathematics, Tome 129 (1991) (A first course, Readings in Mathematics) | MR 1153249 | Zbl 0744.22001

[17] Furstenberg, H. Noncommuting random products, Trans. Amer. Math. Soc., Tome 108 (1963), pp. 377-428 | Article | MR 163345 | Zbl 0203.19102

[18] Goldsheid, I. Ya.; Margulis, G. A. Lyapunov exponents of a product of random matrices, Russian Math. Surveys, Tome 44 (1989) no. 5, pp. 11-71 | Article | MR 1040268 | Zbl 0705.60012

[19] Guivarc’H, Y. Produits de matrices aléatoires et applications aux propriétés géométriques des sous-groupes du groupe linéaire, Ergodic Theory Dynam. Systems, Tome 10 (1990) no. 3, pp. 483-512 | MR 1074315 | Zbl 0715.60008

[20] Guivarc’H, Y.; Raugi, A. Frontière de Furstenberg, propriétés de contraction et théorèmes de convergence, Z. Wahrsch. Verw. Gebiete, Tome 69 (1985) no. 2, pp. 187-242 | Article | MR 779457 | Zbl 0558.60009

[21] Helgason, S. Differential geometry, Lie groups, and symmetric spaces, American Mathematical Society, Providence, RI, Graduate Studies in Mathematics, Tome 34 (2001) (Corrected reprint of the 1978 original) | MR 1834454 | Zbl 0993.53002

[22] Humphreys, J.E. Linear algebraic groups, Springer-Verlag, New York (1975) (Graduate Texts in Mathematics, No. 21) | MR 396773 | Zbl 0471.20029

[23] Kesten, H. Symmetric random walks on groups, Trans. Amer. Math. Soc., Tome 92 (1959), pp. 336-354 | Article | MR 109367 | Zbl 0092.33503

[24] Kingman, J. F. C. Subadditive ergodic theory, Ann. Probability, Tome 1 (1973), pp. 883-909 | Article | MR 356192 | Zbl 0311.60018

[25] Kowalski, E. The large sieve and its applications, Cambridge University Press, Cambridge, Cambridge Tracts in Mathematics, Tome 175 (2008) (Arithmetic geometry, random walks and discrete groups) | MR 2426239 | Zbl 1177.11080

[26] Le Page, E. Théorèmes limites pour les produits de matrices aléatoires, Probability measures on groups (Oberwolfach, 1981), Springer, Berlin (Lecture Notes in Math.) Tome 928 (1982), pp. 258-303 | MR 669072 | Zbl 0506.60019

[27] Mostow, G. D. Strong rigidity of locally symmetric spaces, Princeton University Press, Princeton, N.J. (1973) (Annals of Mathematics Studies, No. 78) | MR 385004 | Zbl 0265.53039

[28] Rivin, I. Walks on groups, counting reducible matrices, polynomials, and surface and free group automorphisms, Duke Math. J., Tome 142 (2008) no. 2, pp. 353-379 | Article | MR 2401624 | Zbl 1207.20068

[29] Rivin, I. Zariski density and genericity, Int. Math. Res. Not. IMRN (2010) no. 19, pp. 3649-3657 | Article | MR 2725508 | Zbl 1207.20045

[30] Tits, J. Représentations linéaires irréductibles d’un groupe réductif sur un corps quelconque, J. Reine Angew. Math., Tome 247 (1971), pp. 196-220 | MR 277536 | Zbl 0227.20015

[31] Tits, J. Free subgroups in linear groups, J. Algebra, Tome 20 (1972), pp. 250-270 | Article | MR 286898 | Zbl 0236.20032

[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, Tome 1 (1967), pp. 347-354 | MR 208470 | Zbl 0163.16902