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arxiv: 1001.3664 · v1 · pith:V36OWMK3new · submitted 2010-01-20 · 🧮 math.GR

Expansion in SL_d(O_K/I), I square-free

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keywords square-freealgebraicarbitrarycayleyconsiderdenseexpanderexpansion
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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.

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