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

Bourgain, Jean; Gamburd, Alex

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

Bourgain, Jean ¹ ; Gamburd, Alex ¹

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

Comptes Rendus. Mathématique, Tome 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 :

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, Tome 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/

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