Trou spectral dans les groupes simples

TL;DR

This paper establishes a spectral gap property for Cayley graphs derived from reductions modulo q of certain subgroups of SL_d(Z) with simple Zariski closure, contributing to understanding expansion properties in algebraic groups.

Contribution

It proves the spectral gap property for Cayley graphs associated with specific subgroups of SL_d(Z), extending known results to new algebraic group contexts.

Findings

Spectral gap property is proven for the family of Cayley graphs.

Results apply to subgroups with simple Zariski closure.

Advances understanding of expansion in algebraic groups.

Abstract

Nous montrons la propri\'et\'e du trou spectral pour la famille des graphes de Cayley obtenus par r\'eduction modulo $q$ d'un sous-groupe de $\mathrm{SL}_d(\mathbb{Z})$ dont l'adh\'erence de Zariski est un $\mathbb{Q}$-groupe simple. -- We show a spectral gap property for the family of Cayley graphs obtained by reduction modulo $q$ of a subgroup of $\mathrm{SL}_d(\mathbb{Z})$ whose Zariski closure is a simple $\mathbb{Q}$-group.