Since it is not …

For a finite group G, let Irr(G) denote the set of irreducible characters of G (working over the complex field) and define AD(G) to be

This invariant of G first appeared in a 1994 paper of B.E. Johnson under a different guise. Johnson was studying the amenability properties (in the Banach algebraic sense) of Fourier algebras of compact groups, and he showed en route that for a finite group G, the amenability constant of its Fourier algebra coincides with AD(G).

From basic character theory, we know that AD(G)≥1, with equality if and only if φ(e)=1 for every φ in Irr(G), i.e. if and only if G is abelian. It is also observed in Johnson’s paper that if G is a non-abelian finite group, then AD(G)≥ 3/2. (See this earlier blogpost for a proof.)

For almost two years, on and off, I have been toying with — and occasionally obsessing over as displacement activity — the following problem.

Question 1. What are the possible values of AD(G) in the interval ?

The reason for the threshold 2 is that we can produce certain values easily: namely, by considering dihedral groups, whose irreducible characters all have degree 1 or 2, one can obtain the values . (One could also obtain the same values by considering binary dihedral groups, as shown in this more recent blogpost.) Moreover, one can show easily that if G is non-abelian and every irreducible character of G has degree ≤2, then only these values are obtained.

So an answer to Question 1 would follow from a positive answer to the following question, which I asked on MathOverflow back in 2021:

Question 2. Suppose that AD(G)≤ 2. Does it follow that every irreducible character of G has degree ≤2?

The following result is not quite as strong as I had been hoping, but it does suffice to show that Question 2 has a positive answer.

Theorem 3.

Let G be a finite group and suppose it has an irreducible character of degree ≥3. Then

where [G,G] denotes the derived subgroup (a.k.a. the commutator subgroup) of G.

Corollary 4. Let G be a finite group, and suppose that AD(G)≤2. Then

.

The techniques used are surprisingly generic, in the sense that although the finiteness of G is used in an essential way to perform an infinite descent/minimal criminal argument, I don’t need Sylow’s theorems, nor any of the harder results concerning how the set of character degrees influences the structure of a finite group.

I have finally got around to writing up the details; hopefully a preprint will materialize in early 2024.

Some acknowledgments and room for improvement

I would like to thank Geoff Robinson for taking an interest in this problem back in 2021 and sharing some of his own thoughts/notes; even if the techniques that he showed me don’t really play a role in my eventual proof of the theorem, it is fair to say that some of the comments and ideas helped in developing my own attack on the problem.

Geoff conjectured that in fact, the lower bound in Theorem 2 could be improved to 9/4, which would be another gap result. But so far I am unable to even achieve a lower bound 2+δ for some small positive δ.