EDIT(August 2013): I accepted Mark's answer as being the state of art- there are two relevant references, one in the answer and one in the comments. The minimal growth rate of $F$ remains unknown with no conjectural answer. END OF EDIT


EDIT: Mark Sapir pointed a reference (in the comments) giving a lower bound of $2^{1/4}$ for the minimal rate. Is this the state of art? The third question remains unanswered. If the answer is NO then the lower bound jumps suddenly to $\frac{\sqrt{5}+3}{2}$ by known results. END OF EDIT


What is it known about the minimal growth rate of the Thompson's group $F$? Is there an easy lower bound? Is there a generating set growing slower than the standard one?



Growth of Thompson's group $F$
These questions have been studied (perhaps except the third one). See Section 5.8.7 in [my book](http://www.math.vanderbilt.edu/~msapir/book/book11513.pdf)  and the references there. 


