# Expansion in $SL_{d}(O_{K}/I)$, $I$ square-free

### Péter P. Varjú

Princeton University, USA

## Abstract

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

## Cite this article

Péter P. Varjú, Expansion in $SL_{d}(O_{K}/I)$, $I$ square-free. J. Eur. Math. Soc. 14 (2012), no. 1, pp. 273–305

DOI 10.4171/JEMS/302