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

TL;DR

This paper improves bounds on the amenability constant of Fourier algebras for certain groups, provides explicit calculations for new examples, and supports a conjecture relating to these bounds.

Contribution

It introduces sharper upper bounds for the amenability constant of Fourier algebras on discrete groups and offers new explicit examples, advancing understanding of this constant.

Findings

Sharper upper bounds for ${ m AM}({ m A}(G))$ when $G$ is discrete.

Explicit calculation of ${ m AM}({ m A}(G))$ for new group examples.

Evidence supporting the conjecture that Runde's lower bound is exact.

Abstract

Let $G$ be a locally compact group. If $G$ is finite then the amenability constant of its Fourier algebra, denoted by ${\rm AM}({\rm A}(G))$, admits an explicit formula [Johnson, JLMS 1994]; if $G$ is infinite then no such formula for ${\rm AM}({\rm A}(G))$ 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 ${\rm AM}({\rm A}(G))$ 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 ${\rm AM}({\rm A}(G))$ 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.