Random walks and expansion in $ {\mathrm{SL}}_{d}(\mathbb{Z}/{p}^{n}\mathbb{Z})$

Jean Bourgain; Alex Gamburd

doi:10.1016/j.crma.2008.04.006

Group Theory

Random walks and expansion in

{SL}_{d} (Z / p^{n} Z)

[Marches au hasard et l'expansion en

{SL}_{d} (Z / p^{n} Z)

]

Présenté par : Jean Bourgain

Jean Bourgain ¹ ; Alex Gamburd ¹

¹ School of Mathematics, Institute for Advanced Study, Princeton, NJ 08540, USA

Comptes Rendus. Mathématique, Volume 346 (2008) no. 11-12, pp. 619-623.

Résumé
Abstract

Soit $S = {g_{1}, \dots, g_{k}}$ un sous-ensemble de ${SL}_{d} (Z)$ engendrant un sous-groupe de ${SL}_{d} (R)$ Zariski dense. On considère les graphes de Cayley $G ({SL}_{d} (Z / p^{n} Z), π_{p^{n}} (S)) = G_{n}$ , òu l'on varie n. Alors ${G_{n}}$ forment une famille d'expanseurs.

Let $S = {g_{1}, \dots, g_{k}}$ be a set of elements of ${SL}_{d} (Z)$ generating a Zariski dense subgroup of ${SL}_{d} (R)$ and let p be a sufficiently large prime. Consider the family of Cayley graphs $G ({SL}_{d} (Z / p^{n} Z), π_{p^{n}} (S)) = G_{n}$ , where we vary n. Then ${G_{n}}$ forms an expander family.

Accepté le : 2008-04-15
Publié le : 2008-05-20

DOI : 10.1016/j.crma.2008.04.006

Affiliations des auteurs :

Jean Bourgain ¹ ; Alex Gamburd ¹

¹ School of Mathematics, Institute for Advanced Study, Princeton, NJ 08540, USA

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     author = {Jean Bourgain and Alex Gamburd},
     title = {Random walks and expansion in $ {\mathrm{SL}}_{d}(\mathbb{Z}/{p}^{n}\mathbb{Z})$},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {619--623},
     publisher = {Elsevier},
     volume = {346},
     number = {11-12},
     year = {2008},
     doi = {10.1016/j.crma.2008.04.006},
     language = {en},
}

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Jean Bourgain; Alex Gamburd. 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. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2008.04.006/

Bibliographie
Cité par

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[3] J. Bourgain; A. Gamburd Uniform expansion bounds for Cayley graphs of ${SL}_{2} (F_{p})$ , Ann. of Math., Volume 167 (2008), pp. 625-642

[4] J. Bourgain, A. Gamburd, Expansion and random walks in ${SL}_{d} (Z / p^{n} Z)$ : I, preprint

[5] J. Bourgain, A. Gamburd, Expansion and random walks in ${SL}_{d} (Z / p^{n} Z)$ : II, preprint

[6] J. Bourgain; A. Gamburd; P. Sarnak 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] Y. Guivarc'h 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] H. Helfgott Growth and generation in ${SL}_{2} (Z / p Z)$ , Ann. of Math., Volume 167 (2008), pp. 601-623

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

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

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