Well, the Shavgulidze preprint [henceforth [ETS]] and Brin’s notes [henceforth [MGB]] have been sitting in the intray this week – “real life” intervened in the shape of chores, forms, lost property and other such tedious necessities. So it’s only in the last couple of days that I’ve sat down to read the calculations and definitions…
Having now skimmed over [MGB] and taken a cursory look at [ETS], the current blogging plan starts as follows:
- A few basics on normed spaces.
Although everything can be done on an as-and-when-needed basis, several of the basic preliminaries make more sense to me (given my mathematical background) as special instances of a general outlook. Moreover, this discussion should lead into…
- Amenability for right topological groups?*
As remarked in [MGB], the argument in [ETS] is surprising or odd-looking for group theorists, because it works with a definition in terms of a left-invariant mean on a function space, rather than families of almost invariant subsets. Note that the former definition can have technical advantages: sometimes it allows quicker/slicker proofs of certain results (for instance, see Prop. 5 in these expository notes by Terence Tao).
So, I thought it might be worth a post looking at amenability from the function-space perspective, and saying something about how we take into account a `compatible’ topology on the group when one is given to us.
- Un soupçon de `F’
Here, since I’m really an outsider and mere spectator when it comes to geometric group theory, I’m not sure I’ll be able to say much about Thompson’s group `F’, except to point to other sources of various degrees of technicality. But I will at least try to give an indication of why the question of its (non-)amenability has occupied people on and off for the last 40 years or so.
After this, there seems to be a choice of emphasis or ordering. I want to spend a post saying something about Wiener measure, although I don’t know if this will just be confined to those properties used in [ETS], or whether I’ll try to throw in some of the more general context.
If I get this far and still have enough energy/enthusiasm for the project, then I may try to write a post presenting (an approximate version of) my take on the detailed calculations in [ETS]. My current intuition is that the messy estimates on iterated integrals of unpleasant looking formulas must have some probabilistic interpretation, which might not make the proof any shorter but might make it seem less opaque for Bears of Little Brain such as myself. Detailed checking can be found in [MGB], and some of those involved are posting updates here.
[*] This comes with a question mark, because I’m not sure at time of writing if there exists an established definition in the generality I’ve claimed. (Amenability for locally compact groups has many well-known and much studied equivalent definitions; amenability for general topological groups has a less studied but well-established definition. But for right topological groups, I don’t know if the obvious candidate has been borne out by experience.)
 Unfortunately, the rather solid set of notes I had when I first learned such things is now in someone else’s house, or possibly a nearby ditch, on the other side of the Atlantic, so the post may end up being rather hand-wavy and nebulous, depending on how much time I have to check the details in other sources.