On amenability constants of Fourier algebras: new bounds and new examples

Let $G$ be a locally compact group. If $G$ is finite, then the amenability constant of its Fourier algebra, denoted by $\mathrm{AM}\left(A\right(G\left)\right)$, admits an explicit formula [Johnson, J. Lond. Math. Soc. 1994]; if $G$ is infinite, then no such formula for $\mathrm{AM}\left(A\right(G\left)\right)$ is known, although lower and upper bounds were established by Runde [Proc. Am. Math. Soc. 2006]. Using non-abelian Fourier analysis, we obtain a sharper upper bound for $\mathrm{AM}\left(A\right(G\left)\right)$ when $G$ is discrete. Combining this with previous work of the first author [Choi, Int. Math. Res. Not. 2023], we exhibit new examples of discrete groups and compact groups where $\mathrm{AM}\left(A\right(G\left)\right)$ can be calculated explicitly; previously this was only known for groups that are products of finite groups with “degenerate” cases. Our new examples also provide additional evidence to support the conjecture that Runde's lower bound for the amenability constant is in fact an equality.