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Expansion in $\mathrm{SL}_d(\cal O_K/I)$, $I$ square-free. (English) Zbl 1269.20044
Let $S$ be a fixed symmetric finite subset of $\mathrm{SL}_d(\cal O_K)$ that generates a Zariski dense subgroup of $\mathrm{SL}_d(\cal O_K)$ when considered as an algebraic group over $\Bbb Q$ by restriction of scalars. The author proves that the Cayley graphs of $\mathrm{SL}_d(\cal O_K/I)$ with respect to the projections of $S$ is an expander family when $I$ ranges over square-free ideals of $\cal O_K$ if $d=2$ and $K$ is an arbitrary number field or if $d=3$ and $K=\Bbb Q$.

##### MSC:
 20G30 Linear algebraic groups over global fields and their integers 20F05 Generators, relations, and presentations of groups 20F65 Geometric group theory 05C25 Graphs and abstract algebra 11B30 Arithmetic combinatorics; higher degree uniformity 11B75 Combinatorial number theory
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##### References:
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