Journal of the European Mathematical Society

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Volume 14, Issue 1, 2012, pp. 273–305
DOI: 10.4171/JEMS/302

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

Péter P. Varjú[1]

(1) Department of Mathematics, Princeton University, Fine Hall, Washington Road, NJ 08544-1000, PRINCETON, UNITED STATES

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 $K=\mathbb Q$.

Keywords: Expanders, property tau, Cayley graphs, random walks on groups, affine sieve

Varjú Péter: Expansion in $SL_d(\mathcal{O}_K/I)$, $I$ square-free. J. Eur. Math. Soc. 14 (2012), 273-305. doi: 10.4171/JEMS/302