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 an(G) ≤ nα for some α and for all n, then G is said to have polynomial subgroup growth (PSG, for short). The degree of G is then defined by deg(G) = limsup log an(G) log n . Very little seems to be known about the relation between deg(G) and the algebraic structure of G. We derive a formula for computing the degree of certain metabelian groups, which serves as a main tool in this paper. Addressing a problem posed by Lubotzky, we also show that if H ≤ G is a finite index subgroup, then deg(G) ≤ deg(H) + 1. A large part of the paper is devoted to the structure of groups of small degree. 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. It follows from our results that the degree of a finitely generated group cannot lie in the open interval (1, 3/2). Our methods are largely number-theoretic, and density theorems à la Chebotarev play essential role in the proofs. Most of the results also rely implicitly on the Classification of Finite Simple Groups.