# Expansion in $SL_d(\mathcal{O}_K/I)$, $I$ square-free

### Péter P. Varjú

Princeton University, USA

## Abstract

Let $S$ be a fixed symmetric finite subset of $SL_d(\mathcal{O}_K)$ that generates a Zariski dense subgroup of $SL_d(\mathcal{O}_K)$ when we consider it as an algebraic group over $\mathbb Q$ by restriction of scalars. We prove that the Cayley graphs of $SL_d(\mathcal{O}_K/I)$ 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 $K=\mathbb Q$.

## Cite this article

Péter P. Varjú, Expansion in $SL_d(\mathcal{O}_K/I)$, $I$ square-free. J. Eur. Math. Soc. 14 (2012), no. 1, pp. 273–305

DOI 10.4171/JEMS/302