On the degree of groups of polynomial subgroup growth

Let G be a finitely generated residually finite group and let an(G) denote the number of index n subgroups of G. If a(n)(G) less than or equal to n(al pha) for some alpha and for all n, then G is said to have polynomial subgro up growth (PSG, for short). The degree of G is then defined by deg(G) = lim sup log a(n) (G)/log n. Very little seems to be known about the relation between deg(G) and the alg ebraic structure of G. We derive a formula for computing the degree of cert ain metabelian groups, which serves as a main tool in this paper. Addressin g a problem posed by Lubotzky, we also show that if H less than or equal to G is a finite index subgroup, then deg(G) less than or equal to deg(H) + 1 . A large part of the paper is devoted to the structure of groups of small de gree. We show that an(G) is bounded above by a linear function of n if and only if G is virtually cyclic. We then determine all groups of degree less than 3/2, and reveal some connections with plane crystallographic groups. I t follows from our results that the degree of a finitely generated group ca nnot lie in the open interval (1; 3/2). Our methods are largely number-theoretic, and density theorems a la Chebota rev play essential role in the proofs. Most of the results also rely implic itly on the Classification of Finite Simple Groups.