Which asymptotic bounds (upper and lower) are known for $s_n$  the minimal number of generators of $S^n$ where $S$ is a nonabelian finite simple group?
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1$\begingroup$ You should probably check the work of J. Wiegold $\endgroup$ – Geoff Robinson Mar 1 '15 at 17:01

$\begingroup$ @GeoffRobinson which one? do you have a reference? $\endgroup$ – Pablo Mar 1 '15 at 17:02

1$\begingroup$ He wrote a few papers on "Growth sequences of Groups" $\endgroup$ – Geoff Robinson Mar 1 '15 at 18:42

2$\begingroup$ it's logarithmic, check Thevenaz's elementary argument: arxiv.org/abs/math/9703201 $\endgroup$ – YCor Mar 1 '15 at 19:04

3$\begingroup$ Essentially a duplicate of this question:mathoverflow.net/questions/187736/… $\endgroup$ – Ian Agol Mar 1 '15 at 21:59

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One has $$1 \leq s_n  \frac{\log(n)}{\logS} \leq 2r$$ based on an elementary argument in Remark 1.1 in [Moshe Jarden and Alexander Lubotzky, Random normal subgroups of free profinite groups, J. Group Theory 2 (1999) 213224], where $r$ denotes the minimal number of generators of $S$. By the classification of finite simple groups, we know that $r = 2$.

$\begingroup$ One may find this article here: tau.ac.il/~jarden/Articles/paper68.pdf $\endgroup$ – Ian Agol Mar 1 '15 at 23:30