Geodesic
Expansion in $SL_d(\mathcal{O}_K/I)$, $I$ square-free
Journal of the European Mathematical Society, Tome 14 (2012) no. 1, pp. 273-305 Cet article a éte moissonné depuis la source EMS Press

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Let S be a fixed symmetric finite subset of SLd​(OK​) that generates a Zariski dense subgroup of SLd​(OK​) when we consider it as an algebraic group over Q by restriction of scalars. We prove that the Cayley graphs of SLd​(OK​/I) with respect to the projections of S is an expander family if I ranges over square-free ideals of OK​ if d=2 and K is an arbitrary numberfield, or if d=3 and K=Q.
DOI : 10.4171/jems/302
Classification : 60-XX, 05-XX, 20-XX, 00-XX
Keywords: Expanders, property tau, Cayley graphs, random walks on groups, affine sieve 
@article{JEMS_2012_14_1_a7,
     author = {P\'eter P. Varj\'u},
     title = {Expansion in $SL_d(\mathcal{O}_K/I)$, $I$ square-free},
     journal = {Journal of the European Mathematical Society},
     pages = {273--305},
     year = {2012},
     volume = {14},
     number = {1},
     doi = {10.4171/jems/302},
     url = {http://geodesic.mathdoc.fr/articles/10.4171/jems/302/}
}
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Péter P. Varjú. Expansion in $SL_d(\mathcal{O}_K/I)$, $I$ square-free. Journal of the European Mathematical Society, Tome 14 (2012) no. 1, pp. 273-305. doi: 10.4171/jems/302

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