On the degree of groups of polynomial subgroup growth

Abstract

LetGGbe a finitely generated residually finite group and letan(G)a_n(G)denote the number of indexnnsubgroups ofGG. Ifan(G)≤nαa_n(G) \le n^{\alpha }for someα\alphaand for allnn, thenGGis said to have polynomial subgroup growth (PSG, for short). The degree ofGGis then defined bydeg⁡(G)=lim suplog⁡an(G)log⁡n\operatorname {deg}(G) = \limsup {{\log a_n(G)} \over {\log n}}. Very little seems to be known about the relation betweendeg⁡(G)\operatorname {deg}(G)and the algebraic structure ofGG. 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 ifH≤GH \le Gis a finite index subgroup, thendeg⁡(G)≤deg⁡(H)+1\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 thatan(G)a_n(G)is bounded above by a linear function ofnnif and only ifGGis virtually cyclic. We then determine all groups of degree less than3/23/2, 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)(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.