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 $\mathrm{A}\mathrm{M}(\mathrm{A}(G))$, admits an explicit formula [Johnson, JLMS 1994]; if $G$ is infinite then no such formula for $\mathrm{A}\mathrm{M}(\mathrm{A}(G))$ is known, although lower and upper bounds were established by Runde [PAMS 2006]. Using non-abelian Fourier analysis, we obtain a sharper upper bound for $\mathrm{A}\mathrm{M}(\mathrm{A}(G))$ when $G$ is discrete. Combining this with previous work of the first author [Choi, IMRN 2023], we exhibit new examples of discrete groups and compact groups where $\mathrm{A}\mathrm{M}(\mathrm{A}(G))$ 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.