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Zbl pre05995804
Varjú, Péter P.
Expansion in $SL_d(\cal{O}_K/I)$, $I$ square-free. (English)
[J] J. Eur. Math. Soc. (JEMS) 14, No. 1, 273-305 (2012). ISSN 1435-9855; ISSN 1435-9863/e

Let $S$ be a fixed symmetric finite subset of $SL_d(\cal{O}_K)$ that generates a Zariski dense subgroup of $SL_d(\cal{O}_K)$ when considered as an algebraic group over $\Bbb Q$ by restriction of scalars. The author proves that the Cayley graphs of $SL_d(\cal{O}_K/I)$ with respect to the projections of $S$ is an expander family when $I$ ranges over square-free ideals of $\cal{O}_K$ if $d=2$ and $K$ is an arbitrary number field or if $d=3$ and $K=\Bbb Q$.
[L. N. Vaserstein (University Park)]

MSC 2000:: *20-99 Group theory and generalizations
05C99 Graph theory

Keywords: expanders; property tau; Cayley graphs; random walks on groups; affine sieve

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