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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 $K=\Bbb Q$.

20G30Linear algebraic groups over global fields and their integers
20F05Generators, relations, and presentations of groups
20F65Geometric group theory
05C25Graphs and abstract algebra
11B30Arithmetic combinatorics; higher degree uniformity
11B75Combinatorial number theory
Full Text: DOI arXiv
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[5] Bourgain, J., Gamburd, A.: Uniform expansion bounds for Cayley graphs of SL2(Fp), Ann. of Math. 167, 625-642 (2008) · Zbl 1216.20042 · doi:10.4007/annals.2008.167.625 · http://annals.math.princeton.edu/annals/2008/167-2/p07.xhtml
[6] Bourgain, J., Gamburd, A.: Expansion and random walks in SLd (Z/pnZ): I. J. Eur. Math. Soc. 10, 987-1011 (2008) · Zbl 1193.20059 · doi:10.4171/JEMS/137 · http://www.ems-ph.org/journals/show_pdf.php?issn=1435-9855&vol=10&iss=4&rank=5
[7] Bourgain, J., Gamburd, A.: Expansion and random walks in SLd (Z/pnZ): II. With an