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29 December, 2021

This is the one year anniversary of this blogpost, announcing an arXiv posting of an article which had not yet been submitted. The arXiv posting was born from a (justified) fear that if I didn’t get some work I had done in 2020 written up and posted to the arXiv, it would just sit on my TO DO list for 2021 and beyond…

In that post, I grumbled that:

In more easy-going times, professionally speaking, I would have preferred to sit on this and use the summer of 2021 to push the techniques in the paper to their limits. But, well, these days one has to keep the wolf from the door.

As it turned out, during March and April 2021, some new ideas emerged as a result of the teaching I had been doing, alluded to here. (This was not so much a case of finding inspiration as seeking escape/relief!) I’ll say more about the new material below.

In addition, I realised during the summer of 2021 that some technical issues which I had overlooked or brushed under the carpet, concerning the unitary dual of the Cartesian square of certain groups, needed to be treated more carefully: this is related to the phenomenon/issue, discussed in various places on Kevin Buzzard’s blog, that as professional mathematicians we get increasingly comfortable – but also too cavalier – about assuming that two things which “surely must be isomorphic in a canonical way” are actually equal.

So, after a couple of months dealing with other demands, and several spells of tearing my hair out over various “proofs by reference to earlier results, or results in other books”, I finally managed to put together a proper revised version of the preprint, posted to arXiv just before Christmas.

An explicit minorant for the amenability constant of the Fourier algebra

For a locally compact group G, let AM(A(G)) denote the amenability constant of the Fourier algebra of G. We show that AM(A(G)) ≥ 3/2 for every non-abelian G, extending a result of Johnson (JLMS, 1994) who obtained this for finite non-abelian groups. This lower bound, which is known to be best possible, improves on results by previous authors and answers a question raised by Runde (PAMS, 2006).

To do this we study a minorant for AM(G), related to the anti-diagonal in G×G, which was implicitly used in Runde’s work but hitherto not studied in depth. Our main novelty is an explicit formula for this minorant when G is a countable virtually abelian group, in terms of the Plancherel measure for G. As further applications, we characterize those non-abelian groups where the minorant attains its minimal value, and present some examples to support the conjecture that the minorant always coincides with the amenability constant.

Some remarks on the new material

Although I took the opportunity to do a lot of minor rewrites to the exposition, the most significant change in v2 compared to v1, mathematically speaking, is that the first version merely stopped at showing that AD(Γ) ≥ 3/2 for every non-abelian Γ, while the second version includes a new section that characterizes those Γ for which equality occurs: they are exactly those groups in which the centre has index 4. Such groups also have the property that the derived subgroup has order 2, and hence in various ways they can be thought of as the non-abelian groups that are closest to being abelian.

Originally I worked this out for the finite case (see this blogpost for a self-contained exposition), since this is the setting where we know that the minorant AD actually coincides with the amenability constant of the Fourier algebra. However, since I am hopeful that the two invariants agree for infinite groups as well, it seemed worthwhile to work out as much as possible for AD, for future use/interest.

The proof of the general case is technically more involved than the finite case, since we can no longer use arguments based on counting conjugacy classes. One theme, which didn’t quite survive into the final write-up, but did play an important heuristic role, is that certain statements about the centre and particular centralizers are countably determined, i.e. if they are true for all countable subgroups of Γ then they are true for Γ itself. This was important because the main formula in the paper for AD(Γ) was only proved under a countability assumption; it seems likely that the assumption could be removed, but one would have to reinvent or recheck a lot of material on direct integrals of group representations in order to ensure that the relevant technical machinery still remains valid.

The new version of the paper also includes a new section with some questions raised by this work (the most obvious, and the most important, being whether AD is always equal to the amenability constant of the Fourier algebra). One question which might be worth mentioning here for those interested in finite group theory: if G is a finite group and AD(G) ≤ 2, does it follow that maxdeg(G) ≤2? (This condition on degrees is clearly sufficient to ensure AD(G) ≤ 2; and dihedral groups show that we can get as close to 2 from below as we wish.) A related question, which I didn’t include explicitly in the paper: what is the infimum of AD(G) over all finite groups G which have maxdeg ≥ 3?

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