# What makes the amenability of Thompsons group $F$ such a tricky problem?

An open problem that seems to get a lot of attention every once in a while is the amenability of Thompsons group $F$.
The problem seems to generate both proofs and disproofs at a fairly high rate, compared to many other open problems.
What is more, it seems that a big part of these are actually serious attempts by serious mathematicians, rather than the "usual" elementary attempts one sees for the more famous problems.

For examples, see for instance the MO question Is Thompson's Group F amenable? as well as (what as far as I can tell is the newest attempt, but I may have missed some) http://arxiv.org/abs/1408.2188.

Is there something inherent to this problem which causes this, i.e. some aspect that makes so many serious mathematicians convince themselves that they have a solution, and so many other serious mathematicians to take so long to find the errors?

Note that I am specifically not asking about what the errors were in previous attempts, unless there is some general type of error that tends to come up in many of them

• @QiaochuYuan But mathematicians thinking they have solved a (quite wellknown) open problem for long enough to actually announce the proof is a fairly rare occurrence, so given the large number of such problems, it seems that it should not take that many to be an unlikely outlier. – Tobias Kildetoft Nov 13 '14 at 9:04
• – Todd Trimble Nov 13 '14 at 13:01
• @QiaochuYuan By way of comparison, one doesn't see at least two people announce proofs that all free group factors are distinct while two more announce proofs that they are all isomorphic – Yemon Choi Nov 13 '14 at 13:32
• One reason might be that there are some numerical quantities that imply amenability of a group and some of the experimental approximations of these quantities for F seem to hang right near the border. – Benjamin Steinberg Nov 13 '14 at 14:01
• Many famous open problems probably have some consensus as to which way the answer lies. My feeling is the experts are 50-50 on this one. – Benjamin Steinberg Nov 13 '14 at 14:54