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

Péter P. Varjú

Type: Article

Publication Date: 2011-11-16

Citations: 35

DOI: https://doi.org/10.4171/jems/302

Abstract

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 .

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