ZL-amenability constants of finite groups with two character degrees

TL;DR

This paper computes the exact amenability constants for the centers of $\, ext{l}^1(G)$ for specific classes of finite groups with two character degrees, providing explicit formulas and addressing a previously posed question.

Contribution

It derives a general formula for the amenability constant of the center of $\, ext{l}^1(G)$ for all finite groups with two character degrees, including dihedral, extraspecial, and Frobenius groups.

Findings

Calculated exact amenability constants for specified group classes.

Provided a universal formula applicable to all finite groups with two character degrees.

Answered a previously open question negatively.

Abstract

We calculate the exact amenability constant of the centre of $\ell^1(G)$ when $G$ is one of the following classes of finite group: dihedral; extraspecial; or Frobenius with abelian complement and kernel. This is done using a formula which applies to all finite groups with two character degrees. In passing, we answer in the negative a question raised in work of the third author with Azimifard and Spronk (J. Funct. Anal. 2009).