Expansion in $S{L}_d({\mathcal{O}}_K/I)$, $I$ square-free

Abstract

Let $S$ be a fixed symmetric finite subset of $S{L}_d({\mathcal{O}}_K)$ that generates a Zariski dense subgroup of $S{L}_d({\mathcal{O}}_K)$ when we consider it as an algebraic group over $\mathbb{Q}$ by restriction of scalars. We prove that the Cayley graphs of $S{L}_d({\mathcal{O}}_K/I)$ with respect to the projections of $S$ is an expander family if $I$ ranges over square-free ideals of ${\mathcal{O}}_K$ if $d=2$ and $K$ is an arbitrary numberfield, or if $d=3$ and $K=\mathbb{Q}$.