Discrete groups of slow subgroup growth

It is known that the subgroup growth of finitely generated linear groups is either polynomial or at least {Mathematical expression}. In this paper we prove the existence of a finitely generated group whose subgroup growth is of type {Mathematical expression}. 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 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.