On the degree of groups of polynomial subgroup growth

Abstract:

Let $G$ be a finitely generated residually finite group and let $a_n(G)$ denote the number of index $n$ subgroups of $G$. If $a_n(G) \le n^{\alpha }$ for some $\alpha$ and for all $n$, then $G$ is said to have polynomial subgroup growth (PSG, for short). The degree of $G$ is then defined by $\operatorname {deg}(G) = \limsup {{\log a_n(G)} \over {\log n}}$. Very little seems to be known about the relation between $\operatorname {deg}(G)$ 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 $H \le G$ is a finite index subgroup, then $\operatorname {deg}(G) \le \operatorname {deg}(H)+1$. A large part of the paper is devoted to the structure of groups of small degree. We show that $a_n(G)$ 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.