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)

Presented by: Jean Bourgain

Jean Bourgain ¹ ; Alex Gamburd ¹

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

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

Abstract (Original language)
Résumé

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.

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.

Accepted: 2008-04-15
Published online: 2008-05-20

DOI: 10.1016/j.crma.2008.04.006

Author's affiliations:

Jean Bourgain ¹ ; Alex Gamburd ¹

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

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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

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