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
 A: For some famous open problems, there are known "obstructions" to resolving them.  For example in number theory we have Siegel zeros and the parity problem for sieves, and in computational complexity theory we have naturalization and relativization/algebrization barriers.  The obstructions are well known and ideas for surmounting them are scarce, so if a proposed solution appears, experts can often zoom in quickly on where the new idea for dealing with the obstruction must lie.  It's harder to be fooled in such cases.
For other open problems, the conceptual territory surrounding the problem is less well understood and mapped out.  Instead of a mountain range with only a few well-known passes, the obstacles form a forest with myriad trees among which it is easy to get lost.  I know nothing about amenable groups, but if what you say is correct, I would hypothesize that the conjecture falls into the latter category.  There are lots of different approaches one could try, and instead of running into the same difficulty over and over again, each new attempt makes a foray into uncharted territory where there is not much past experience to help detect the pitfalls.
