Subgroup growth and sieve methods

Abstract

We study the subgroup growth of profinite groups. We obtain a structure theorem for profinite groups of polynomial subgroup growth (PSG groups, for short), which essentially reduces their characterization to the case where the group is a cartesian product of finite simple groups. Analysing the growth behaviour of such cartesian products, we construct, for any real number $\alpha \ge 1$, a PSG profinite group whose degree is exactly $\alpha$. Applications to the behaviour of the abscissa of convergence of the associated zeta function $\sum a_n(G)n^{-s}$ are drawn. We also show that there is no gap between polynomial and non-polynomial subgroup growth by constructing non-PSG groups whose subgroup growth is arbitrarily slow. Our arguments rely heavily on the use of sieve methods in number theory. In particular, a Bombieri-type short intervals theorem and the so-called Fundamental lemma in sieve theory play an essential role in this paper.

1991 Mathematics Subject Classification: 20E07, 11N36.