Comptes Rendus
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.

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

1 School of Mathematics, Institute for Advanced Study, Princeton, NJ 08540, USA
     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},
AU  - Jean Bourgain
AU  - Alex Gamburd
TI  - Random walks and expansion in $ {\mathrm{SL}}_{d}(\mathbb{Z}/{p}^{n}\mathbb{Z})$
JO  - Comptes Rendus. Mathématique
PY  - 2008
SP  - 619
EP  - 623
VL  - 346
IS  - 11-12
PB  - Elsevier
DO  - 10.1016/j.crma.2008.04.006
LA  - en
ID  - CRMATH_2008__346_11-12_619_0
ER  - 
%0 Journal Article
%A Jean Bourgain
%A Alex Gamburd
%T Random walks and expansion in $ {\mathrm{SL}}_{d}(\mathbb{Z}/{p}^{n}\mathbb{Z})$
%J Comptes Rendus. Mathématique
%D 2008
%P 619-623
%V 346
%N 11-12
%I Elsevier
%R 10.1016/j.crma.2008.04.006
%G en
%F CRMATH_2008__346_11-12_619_0
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.

[1] P. Bougerol; J. Lacroix 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] J. Bourgain; A. Gamburd 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] 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 SL2(Z/pZ), 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

[11] P. Sarnak; X. Xue 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

Cited by Sources:

Comments - Policy

Articles of potential interest

New results on expanders

Jean Bourgain; Alex Gamburd

C. R. Math (2006)

Sieving and expanders

Jean Bourgain; Alex Gamburd; Peter Sarnak

C. R. Math (2006)

Sum–product theorems and exponential sum bounds in residue classes for general modulus

Jean Bourgain

C. R. Math (2007)