While I did not participate in most of the checking of Shavgulidze's
argument, I can offer the following partial account of the situation. I
am told the paper was correct except for a lemma (or sequence of them)
claiming that a sequence of auxiliary measures had certain properties.
These were Borel measures on the $n$-simplex (one for each $n$). I
believe it was shown that the original proposed auxiliary sequence of
measures did not have one of the two properties. Shavgulidze
proposed other sequences of measures. The most recent attempt that I am
aware of (which was presented during his 2010 trip to the US mentioned
by Mark Sapir in the above comment) involved the direct construction of
Folner sets for the action of $F$ on the finite subsets of dyadic
rationals (see the next paragraphs). The details were somewhat sparse
and the definitions involved many unspecified numerical parameters, but
it appeared to be the case that these sets could not be Folner in the
necessary sense (see below for a clarification of "necessary sense").
This is because they would likely
both contradict the iterated exponential lower bound on the Folner
function which I have demonstrated and because they appear to violate
the qualitative properties which I have demonstrated that Folner sets of
trees must have (see the pre-print on my webpage; the qualitative
condition appears in lemma 5.7, noting that marginal implies measure
0 with respect to any invariant measure).
Meanwhile I was able to provide a direct elementary proof that the
existence of such a sequence having these properties implied the amenability
of $F$. In fact the proof gives an explicit procedure for constructing
(weighted) Folner sets from the sequence of measures satisfying the hypotheses
mentioned above. A note containing the details was circulated to a few people around the
time of Shavgulidze's visit to Vanderbilt.
While I am reluctant to speak for anyone else (including the author), it
appears to me that after the dust had settled (which took a considerable amount of time),
the problem with the proof seems to have at least some of its roots
in the following observation (which I now include for the sake of prosperity). $F$
acts on the finite subsets of the dyadic rations (let's call this set
$\mathcal{D}$) by taking the set-wise image (here I am utilizing the
piecewise linear function model of $F$). Now let $\mathcal{T}$ denote
the finite subset of $[0,1]$ which contain $0$ and $1$ and are such that
any consecutive pair is of the form $p/2^q,(p+1)/2^q$ (for natural numbers $p,q$). $F$ only acts
partially on $\mathcal{T}$: the action $T \cdot f$ is defined if $f'$
is defined on the complement of $T$ in $[0,1]$ (there may be other cases
when $T \cdot f$ is in $\mathcal{T}$, but let's restrict the domain of
the action as above). The full action of $F$ on $\mathcal{D}$ is
amenable. The point here is that the action of the standard generators
on the sets $\{0,1-2^{-n},1\}$ is the same for large enough $n$
and thus we can build
Folner sets as in a $\mathbb{Z}$ action. The amenability of partial
action of $F$ on $\mathcal{T}$ is, on the other hand, equivalent to the
amenability of $F$ (this is well known, but see the preprint above to see
this spelled out in the present jargon).
Now here is the catch, if we also require that the invariant
measure/Folner sets for the action of $F$ on $\mathcal{D}$ to
concentrate on sets of mesh less than $1/16$, then one again arrives at
an equivalent formulation of the amenability of $F$. The author was
aware of the need for the mesh condition, but (in the most recent example)
arranged it only in a
modification after the fact (which interferes with invariance).
Incidentally the hypotheses on the sequence of measures mentioned above
are a condition requiring that the measures concentrate on sets of
arbitrarily small mesh as $n$ tends to infinity and a condition which
is an analog of translation invariance.
I apologize if this borders on ``too much information.''
[Added 1/28/2011]
Shavgulidze's 1/14/2011 posting to the ArXiv is essentially a more detailed version of what he was saying in notes, seminars, and private communication in January 2010 during his visit to the US mentioned in Mark Sapir's post above. The present note is still sufficiently vague and full of sufficiently many errors (many typographical in nature) that it is hard (or easy, if you like) to say explicitly which line of the proof is incorrect. It is possible, however to point to places where crucial details are missing and where there are certainly going to be errors (specifically the problems will be on page 11, if not elsewhere as well). The comments from my answer above still apply equally well to the present version. It appears that the present version (or any perturbation of it) still would violate the lower bound on the growth of the Folner function which I have established. The present version still totally ignores that the combinatorial statements on page 11 themselves readily imply the amenability of F, without the involvement of any analytical concepts.
[Added 2/3/2011]
Details on what is incorrect with Shavgulidze's proof of the amenablity of $F$ can be found here.
[Added 10/3/2012]
Well, well, well: now I'm in the position of having announced a proof that $F$ is amenable only to have an error be found. The error was finally found by Azer Akhemedov after being overlooked for roughly 4 weeks by myself and 9 or more people who had checked the proof and found no problems. The basic strategy of the proof still may be valid: it began by considering an extension of the free binary system $(\mathbb{T},*)$ on one generator to the finitely additive probability measures on this system:
$$\mu * \nu (E) = \int \int \chi_E(s * t) d \nu (t) d \mu (s).$$
It was shown (correctly) that any idempotent measure is $F$-invariant (there is a natural way of identifying $\mathbb{T}$ with the positive elements of $F$).
The difficulty came in constructing the idempotent measure.
A version of the Kakutani Fixed Point Theorem was used to construct approximations $K_{\mathcal{B},k,n}$ to the set of idempotent measures.
The error occurs in attempting to intersect these compact families of measures.
In the proof, it was claimed that the parameter $k$ could be stablized along the an ultrafilter
(Lemma 4.13 in the most recent version), allowing one to take a directed intersection of nonempty compact sets.
This lemma is likely false and at least is not proved as claimed.
One may still be able to argue that a relevant intersection of these approximations is nonempty and hence that there is an idempotent.
This seems to require new ideas though.
