Group Theory
Random walks and expansion in SLd(Z/pnZ)
Comptes Rendus. Mathématique, Volume 346 (2008) no. 11-12, pp. 619-623.

Let S={g1,,gk} be a set of elements of SLd(Z) generating a Zariski dense subgroup of SLd(R) and let p be a sufficiently large prime. Consider the family of Cayley graphs G(SLd(Z/pnZ),πpn(S))=Gn, where we vary n. Then {Gn} forms an expander family.

Soit S={g1,,gk} un sous-ensemble de SLd(Z) engendrant un sous-groupe de SLd(R) Zariski dense. On considère les graphes de Cayley G(SLd(Z/pnZ),πpn(S))=Gn, òu l'on varie n. Alors {Gn} forment une famille d'expanseurs.

Accepted:
Published online:
DOI: 10.1016/j.crma.2008.04.006
Bourgain, Jean 1; Gamburd, Alex 1

1 School of Mathematics, Institute for Advanced Study, Princeton, NJ 08540, USA
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Bourgain, Jean; Gamburd, Alex. Random walks and expansion in $ {\mathrm{SL}}_{d}(\mathbb{Z}/{p}^{n}\mathbb{Z})$. Comptes Rendus. Mathématique, Volume 346 (2008) no. 11-12, pp. 619-623. doi : 10.1016/j.crma.2008.04.006. http://www.numdam.org/articles/10.1016/j.crma.2008.04.006/

[1] Bougerol, P.; Lacroix, J. Products of Random Matrices with Applications to Schrödinger Operators, Progress in Probability and Statistics, vol. 8, Birkhäuser, 1985

[2] J. Bourgain, The sum–product theorem Zq with q arbitrary, preprint

[3] Bourgain, J.; Gamburd, A. Uniform expansion bounds for Cayley graphs of SL2(Fp), Ann. of Math., Volume 167 (2008), pp. 625-642

[4] J. Bourgain, A. Gamburd, Expansion and random walks in SLd(Z/pnZ): I, preprint

[5] J. Bourgain, A. Gamburd, Expansion and random walks in SLd(Z/pnZ): II, preprint

[6] Bourgain, J.; Gamburd, A.; Sarnak, P. Sieving and expanders, C. R. Math. Acad. Sci. Paris, Ser. I, Volume 343 (2005), pp. 155-159

[7] J. Bourgain, A. Gamburd, P. Sarnak, Affine linear sieve, expanders, and sum–product, preprint

[8] 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), pp. 483-512

[9] Helfgott, H. Growth and generation in SL2(Z/pZ), Ann. of Math., Volume 167 (2008), pp. 601-623

[10] Long, D.D.; Lubotzky, A.; Reid, A.W. Heegaard genus and property ‘tau’ for hyperbolic 3-manifolds, J. Topol., Volume 1 (2008) no. 1, pp. 152-158

[11] Sarnak, P.; Xue, X. Bounds for multiplicities of automorphic representations, Duke Math. J., Volume 64 (1991), pp. 207-227

[12] T. Tao, Product sets estimates for non-commutative groups, Combinatorica, in press

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