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Expansion in perfect groups. (English) Zbl 1284.20044

Summary: Let \(\Gamma\) be a subgroup of \(\mathrm{GL}_d(\mathbb Z[1/q_0])\) generated by a finite symmetric set \(S\). For an integer \(q\), denote by \(\pi_q\) 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 \(\pi_q(\Gamma)\) with respect to the generating sets \(\pi_q(S)\) 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 \(\Gamma\) is perfect, i.e. it has no nontrivial Abelian quotients.

MSC:

20F65 Geometric group theory
20F05 Generators, relations, and presentations of groups
20G20 Linear algebraic groups over the reals, the complexes, the quaternions
05C25 Graphs and abstract algebra (groups, rings, fields, etc.)
11N36 Applications of sieve methods
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