JournalsjemsVol. 14, No. 1pp. 273–305

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

  • Péter P. Varjú

    Princeton University, USA
Expansion in $SL_d(\mathcal{O}_K/I)$, $I$ square-free cover
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Let SS be a fixed symmetric finite subset of SLd(OK)SL_d(\mathcal{O}_K) that generates a Zariski dense subgroup of SLd(OK)SL_d(\mathcal{O}_K) when we consider it as an algebraic group over Q\mathbb Q by restriction of scalars. We prove that the Cayley graphs of SLd(OK/I)SL_d(\mathcal{O}_K/I) with respect to the projections of SS is an expander family if II ranges over square-free ideals of OK\mathcal{O}_K if d=2d=2 and KK is an arbitrary numberfield, or if d=3d=3 and K=QK=\mathbb Q.

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

DOI 10.4171/JEMS/302