Keep your ABC; I’ll take an F (updated)
Update 2nd October 2012
Moore has withdrawn his claim: see the comments at 1209.2063v4. Thus, the amenability or otherwise of F remains open!
The original post now follows beneath.
While the rest of the mathematical blogosphere is probably more excited about the announced proof of the ABC conjecture — and who can blame them, or you? — this bear of little brain was rather more enthused by other developments today.
Thompson’s group F is amenable.
(Thompson himself apparently raised this question back in the 1970s; see this MathOveflow post for some details.)
It has long been known that F is not elementary amenable and has exponential growth, so that in some sense it cannot be “amenable for bvious reasons”.
A preprint is apparently forthcoming; obviously, Moore’s purported proof will require a good deal of scrutiny. (That’s even before one goes into the unfortunate history surrounding previous announcements by various authors regarding the amenability or non-amenability of F…) Still, if everything holds up, I for one will be very interested to see the details; not least because Moore has previously obtained lower bounds for the growth of Følner sequences for F.
Update 2012-09-11: the preprint is now on the arXiv at 1209.2063.