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Expander graphs and amenable quotients. (English) Zbl 0989.20028

Hejhal, Dennis A. (ed.) et al., Emerging applications of number theory. Based on the proceedings of the IMA summer program, Minneapolis, MN, USA, July 15-26, 1996. New York, NY: Springer. IMA Vol. Math. Appl. 109, 571-581 (1999).
Let \(\Gamma=\langle S\rangle\) be a finitely generated group and \(N_i\) a sequence of normal subgroups such that the Cayley graphs \(X\langle\Gamma/N_i,S\rangle\) form an expander family. The paper investigates under what conditions on \(F\), where \(F\) is a subset of \(\Gamma\), is \(X\langle\Gamma/N_i,F\rangle\) an expander family as well. It is shown, using the invariant mean on the profinite completion of \(\Gamma\) with respect to the \(N_i\)’s, that such an \(F\) can be found as a subset of \(H\), provided \(H\) is a co-amenable, but not necessarily normal, subgroup of \(\Gamma\).
For the entire collection see [Zbl 0919.00047].

MSC:

20F05 Generators, relations, and presentations of groups
05C25 Graphs and abstract algebra (groups, rings, fields, etc.)
20H05 Unimodular groups, congruence subgroups (group-theoretic aspects)
22E40 Discrete subgroups of Lie groups
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