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Expansion in perfect groups |
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| Authors: | Alireza Salehi Golsefidy Péter P Varjú |
| Institution: | 1. Department of Mathematics, University of California, San Diego, CA, 92122, USA 2. Department of Mathematics, Princeton University, Princeton, NJ, 08544, USA 3. Analysis and Stochastics Research Group of the Hungarian Academy of Sciences, University of Szeged, Szeged, Hungary |
| Abstract: | Let ?? be a subgroup of ${{\rm GL}_d(\mathbb{Z}1/q_0])}$ generated by a finite symmetric set S. For an integer q, denote by ?? q the projection map ${\mathbb{Z}1/q_0] \to \mathbb{Z}1/q_0]/q \mathbb{Z}1/q_0]}$ . We prove that the Cayley graphs of ?? q (??) with respect to the generating sets ?? q (S) form a family of expanders when q ranges over square-free integers with large prime divisors if and only if the connected component of the Zariski-closure of ?? is perfect, i.e. it has no nontrivial Abelian quotients. |
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