Expansion in $S{L}_d(O_K/I)$, $I$ square-free

Abstract

Let S be a fixed symmetric finite subset of SL_d(O_K) that generates a Zariski dense subgroup of SL_d(O_K) when we consider it as an algebraic group over Q by restriction of scalars. We prove that the Cayley graphs of SL_d(O_K/I) with respect to the projections of S is an expander family if I ranges over square-free ideals of O_K if d=2 and K is an arbitrary numberfield, or if d=3 and K=Q.

References in corpus (2)

• The sum-product phenomenon in arbitrary rings
• Expansion properties of finite simple groups

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• Super-approximation, I: p-adic semisimple case
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• Sieve methods in group theory I: Powers in Linear groups
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