Expansion in perfect groups

By Alireza Salehi Golsefidy, P. P. Varj'u. 2011

116 citations recorded by the source corpus.

Let Γ 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 πq 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 πq(Γ) with respect to the generating sets πq(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 Γ is perfect, i.e. it has no nontrivial Abelian quotients.

Subfields: Algebra and Number Theory, Discrete Mathematics and Combinatorics

View Paper: https://www.semanticscholar.org/paper/0f12c4aadb3653fd9c902587b7994cc4cc156e7e

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In Algebra and Number Theory

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Citations in the corpus

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