Let be a fixed symmetric finite subset of that generates a Zariski dense subgroup of when we consider it as an algebraic group over by restriction of scalars. We prove that the Cayley graphs of with respect to the projections of is an expander family if ranges over square-free ideals of if and is an arbitrary numberfield, or if and .
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Péter P. Varjú, Expansion in , square-free. J. Eur. Math. Soc. 14 (2012), no. 1, pp. 273–305DOI 10.4171/JEMS/302