a gap theorem for the zl-amenability constant of a finite group

it was shown in [a. azimifard, e. samei, n. spronk, jfa 256 (2009)] 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 [a. azimifard, e. samei, n. spronk, op cit.] 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 reslt, by analyzing the cases where $g$ is just non-abelian, using calculations from [m. alaghmandan, y. choi, e. samei, cmb 57 (2014)], and establishing a new estimate for groups with trivial centre.