Groups of Intermediate Subgroup Growth and a Problem of Grothendieck

Let f be a function such that for every ε > 0, nlog n ≤ f (n) ≤ nεn holds if n is sufficiently large. Suppose that log f (n)/ log n is nondecreasing. Using sequences of finite alternating groups, for every such f we construct a 4-generator group 0 such that sn(0), the number of subgroups of index at most n in 0, grows like f (n). This essentially completes the investigation of the “spectrum” of possible subgroup growth types and settles several questions posed by Lubotzky, Mann, and Segal. As a by-product we obtain continuously many nonisomorphic 4-generator residually finite groups with isomorphic profinite completions. Our construction also sheds some light on a problem of Grothendieck [Gr]; we obtain an abundance of pairs of finitely generated residually finite groups 00 < 0, such that the natural map ı̂ : 0̂0 → 0̂ between profinite completions is an isomorphism, but 00 6∼= 0. 0. Introduction Let 0 be a finitely generated group. Denote by sn(0) the number of subgroups of index at most n in 0. The series {sn(0)} has been the subject of intense investigation in the past two decades (see [L1], [L2], [MS] and the references therein). Indeed, the behaviour of sn(0) is the main topic of the forthcoming book of Lubotzky and Segal [LS]. Given a function f , we say that 0 has (subgroup) growth type f if there exist positive constants a and b such that (1) sn(0) ≤ f (n)a for all n; (2) sn(0) ≥ f (n)b for infinitely many n. Moreover, if (2) holds for all sufficiently large n, we say that 0 has strict growth type f . Note that having strict growth type f is an equivalence relation. A classical result of M. Hall [H] implies that Fr , the free group on r generators DUKE MATHEMATICAL JOURNAL Vol. 121, No. 1, c © 2004 Received 21 October 2002. Revision received 28 February 2003. 2000 Mathematics Subject Classification. Primary 20E07; Secondary 20E18. Author’s research supported by Hungarian National Science Foundation (OTKA) grant numbers T 034878 and