Amenability of actions/groups

Stream: Is there code for X?

[Stefano Rocca]

I am about to submit a PR for Følner filters and some of their properties, and I was wondering whether predicates like IsAmenableMulAction or IsAmenableGroup, or code on amenable groups in general, already exist or are currently being worked on.

I am asking because, in my code, an amenable action of $G$ on a measure space $X$ amounts to the existence of a finitely additive probability measure that is invariant under the action. Thus I am carrying around

  ∃ m : Set X → ℝ≥0∞,
    m .univ = 1 ∧
    (∀ s t, MeasurableSet s → MeasurableSet t → Disjoint s t → m (s ∪ t) = m s + m t) ∧
    ∀ (g : G) (s : Set X), m (g • s) = m s

If you think it would be useful, I can implement that as well.

[Anatole Dedecker]

We don't have this, and I don't think anyone is working on it.

IIRC there was a previous attempt at amenable groups, but one issue is that out of the $10^{10^{10}}$ (according to Brown-Ozawa) characterization of amenable groups, one needs to choose the one that generalizes most conveniently to the most general setup. I think the consensus is that the right characterization is the fixed point characterization, so one has to do some nontrivial work to get back to Følner filters.

The situation of amenable actions is even worse, because there are (at least) two orthogonal notions of "G acts amenably on the measure space X": for one of them (which is what you suggest), a (discrete) group G is amenable iff G acts amenably on G, while for the other one (called Zimmer amenability) G is amenable iff it acts amenably on a point. And then you have the topological analog of these, and so on...

[Geoffrey Irving]

Is there really only one characterisation?

[Anatole Dedecker]

Not quite of course, but I've only ever seen the following three definitions given for a topological group $G$ to be amenable:

1. Every affine action of $G$ on a compact convex in a (Hausdorff) LCTVS admits a fixed point,
2. Whenever $G$ acts continuously on a compact space $X$, there exists a $G$-invariant regular probability measure on $X$,
3. There exists a $G$-invariant positive and unital linear form $m$ on the space $UC_b(G_r)$, where $G_r$ denotes the group $G$ equipped with its right uniform structure and $UC_b(X)$ is the space of bounded complex-valued uniformly continuous functions on the uniform space $X$.

Note that 3 sounds very similar to the usual mean characterization (in fact $m$ is often called a mean as well), but the key point is that its definition does not make reference to a Haar measure, which allows it to make sense for topological groups.

The nice thing is that the equivalences between these are quite elementary to prove (you only really need facts about barycenters of measures), which is not the case when you have to work with the $10^{10^{10}}$ versions in the locally compact setting.

[Stefano Rocca]

The situation of amenable actions is even worse

I believe there is a fixed point version also for actions of $G$ on $X$, so that when you set $X=pt$ you recover the fixed point definition of amenability for $G$. Not sure about the hypotheses on $X$ tho.

In any case, I do see the issue of finding a general definition. I will PR as it is for now , thanks!

[Anatole Dedecker]

Yes, my point was that there are multiple genuinely different notions of amenable actions, but indeed each of them also has a bunch of definitions...