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

TL;DR

This paper establishes a new lower bound of 7/4 for the ZL-amenability constant of finite non-abelian groups, improving previous bounds and avoiding complex prior proofs.

Contribution

It proves the optimal lower bound of 7/4 for non-abelian finite groups' ZL-amenability constant without relying on deep existing theorems.

Findings

Lower bound of 7/4 for non-abelian groups' ZL-amenability constant

Avoids use of Rider's deep result in the proof

Provides new estimates for groups with trivial centre

Abstract

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.