Discrete groups of slow subgroup growth
Discrete groups of slow subgroup growth
A. Lubotzky, L. Pyber, A. 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.