Expansion in SL_d(Z/qZ), q arbitrary

TL;DR

This paper proves that Cayley graphs derived from Zariski-dense subgroups of SL_d(Z) project to expanders over Z/qZ for arbitrary q, extending the understanding of expansion properties in algebraic groups.

Contribution

It establishes that the Cayley graphs of projections of Zariski-dense subgroups of SL_d(Z) are expanders for all q, generalizing previous results to arbitrary moduli.

Findings

Cayley graphs of pi_q(G) are expanders for all q

Expansion holds for Zariski-dense subgroups of SL_d(Z)

Generalizes expansion results to arbitrary q

Abstract

Let S be a fixed finite symmetric subset of SL_d(Z), and assume that it generates a Zariski-dense subgroup G. We show that the Cayley graphs of pi_q(G) with respect to the generating set pi_q(S) form a family of expanders, where pi_q is the projection map Z->Z/qZ.