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A gap theorem for the ZL-amenability constant of a finite group

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<mark>Journal publication date</mark>12/2016
<mark>Journal</mark>International Journal of Group Theory
Issue number4
Number of pages20
Pages (from-to)27-46
Publication StatusPublished
Early online date19/05/15
<mark>Original language</mark>English


It was shown in [A. Azimifard, E. Samei, N. Spronk, JFA 2009; arxiv 0805.3685] that the ZL-amenability constant of a finite group is always at least 1, with equality if and only if the group is abelian. It was also shown in the same paper that for any finite non-abelian group this invariant is at least 301/300, but the proof relies crucially on a deep result of D. A. Rider on norms of central idempotents in group algebras. Here we show that if G is finite and non-abelian then its ZL-amenability constant is at least 7/4, which is known to be best possible. We avoid use of Rider's result, by analyzing the cases where G is just non-abelian, using calculations from [M. Alaghmandan, Y. Choi, E. Samei, CMB 2014; arxiv 1302.1929], and establishing a new estimate for groups with trivial centre.