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

#### Authors

Yemon Choi

Keywords: Amenability constant; character degrees; just non-abelian groups

DOI: 10.22108/ijgt.2016.9562

MSC 2010: 20C15 (primary); 43A20, 43A62 (secondary)

#### Status

Appeared as Int. J. Group Th. 5 (2016) no. 4, 27–46

Preprint version (final accepted version) available at arXiv 1410.5134

#### Reviews

[ Math Review (summary) | Zentralblatt (summary ]

#### 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.

After this article was accepted for publication, I was informed by F. Ladisch that for $G={\rm SL}(2,5)$ and $N=\{\pm I\}$, calculations with GAP show that ${\rm ass}(G/N) \gt {\rm ass}(G)$. Thus Question 1 has a positive answer.