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

Authors

Yemon Choi

Metadata

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) ]

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.

Updates/comments

The problem resolved in this paper had been bugging me since 2009, and to some extent motivated the work that led to the Alaghmandan–C.–Samei paper that is mentioned above. Thanks are due to both Mahmood and Ebrahim for discussions and for their interest. Thanks also go to various people on MathOverflow for provding various references and examples in character theory.

Note added in proof

The following was added at the end of Section 6 (i.e. before the Appendix)

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.

(For more details and context, see this MathOverflow answer.)


Yemon Choi