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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?

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up vote 10 down vote accepted

These questions have been studied (perhaps except the third one). See Section 5.8.7 in my book and the references there.

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Thank you very much for this nice reference, however, since I'm not interested in the standard generating set (except for the case in which is the slowest), I'm unable to extract useful information concerning my questions... –  Dan Sălăjan May 22 '13 at 23:41
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A uniform lower bound for the growth function of $F$ (independent of the generating set) is found here: unige.ch/math/folks/delaharpe/articles/… –  Mark Sapir May 23 '13 at 0:43
    
Thanks! This answers probably the first two questions. I expect though the minimal rate to go well over 2, in fact doesn't look easy to beat the standard generators. –  Dan Sălăjan May 23 '13 at 1:36
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