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?



What is the growth of the rank of a power of a finite simple group?
One has
$$1 \leq s\_n - \frac{\log(n)}{\log|S|} \leq 2r$$
based on an elementary argument in Remark 1.1 in [Moshe Jarden and Alexander Lubotzky, *Random normal subgroups of free pro-finite groups*, J. Group Theory 2 (1999) 213-224], where $r$ denotes the minimal number of generators of $S$. By the classification of finite simple groups, we know that $r = 2$.


