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Justin Moore's proof turned out to have an error


I just attended Justin Moore's talk on this today. Since I am neither a group theorist nor a combinatorist, and is not familiar with ultrafilters I cannot judge the correctness of his proof. But more elementary - if Thompson's group F is amenable, what would this signify? Can we deduce some non-elementary results from this statement? I was not able to ask this in person, but it seems this would imply the amenability of some other related groups as well. Any comments, answer, etc are welcome.

I suspect this is too low for mathoverflow (as I am not a researcher in this field) so I ask in here.

To cite the background, the wikipedia link on Thompson group F is here. Justin Moore's proof is here. Some related mathoverflow question can be found at here, here, and there. I have read about 60 pages' paper not related to Justin Moore's proof on F in the past, and raise the question under this context. Also, often that the statement itself is not significant (like Fermat's last theorem), but what mathematical structure the proof involved is important. So any comments, answers, etc on that direction is also welcome. I personally felt very surprised and happy to know ramsey theory was lurking in the background of the proof.

A summary of Justin's proof is this (borrowed from others):

  1. The definition of amenability of a group G demands a. a positive measure m defined on all subsets of G b. m(G) = 1 c. m is finitely additive d. m is invariant under left mult by elements of G.

  2. The motivation for the definition is that amenable groups acting on other spaces allow for measures on the space with the properties above.

  3. Often a space can be found with a G action so that if a measure with the properties above can be found then the group is amenable.

  4. For F, such a space is the space T of all finite binary trees.

  5. The spaces of trees T has a binary operation. (Trees A and B are made into a tree A^B whose left subtree is A and right subtree is B).

  6. The space M of measures on T having all the properties but (d) inherits a binary operation from the binary operation ^ on T.

  7. An idempotent in M will have (d) and prove the amenability of F.

  8. There is a famous lemma that finds idempotents in spaces that are compact and have an associative operation.

  9. The space M is (I think) compact, but the operation is not associative.

  10. Moore's work is 30% proving (7) above and 70% generalizing the result (8) to the non-associative setting of the binary operation on M.

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    @MustafaGokhanBenli: ah ok, thanks.2012-09-19

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