Alexander Lubotzky, L\'aszl\'o Pyber, Aner Shalev:

        Discrete groups of slow subgroup growth
               In memory of S.A. Amitsur


It is known that the subgroup growth of finitely generated linear groups
is either polynomial or at least  $n^{{\log n} \over {\log \log n}}$. In 
this paper we prove the existence of a finitely generated group whose
subgroup growth is of type $n^{{\log n} \over {(\log \log n)^2}}$. This
is the slowest non-polynomial subgroup growth obtained so far for 
finitely generated groups. The subgroup growth type $n^{\log n}$ is also 
realized. The proofs involve analysis of the subgroup structure of 
finite alternating groups and finite simple groups in general. For
example, we show that there is an absolute constant $c$ such that, if
$T$ is any finite simple group, then $T$ has at most $n^{c \log n}$ 
subgroups of index $n$.