Subgroup Growth in pro-p Groups

Abstract

Let G be any group. We write a n (G) for the number of subgroups of G of index n. If G is a profinite group, we consider only open subgroups. 1 If G is finitely generated, either as an abstract or as a profinite group, then a n (G) is finite. The study of the series a n (G), a subject that is known as subgroup growth,was begun by Hurwitz in the 19th century, with geometric motivation, and has become an active area of research in recent years. Let us mention, e.g., that a pro-p Groups G is p-adic analytic if and only if its subgroup growth is polynomial, i.e., there exist constants C and s such that a n (G) ≤ Cns for all n (see Section4 below and [2]). This characterisation of p-adic analytic groups was an instrumental stage in the characterisation of all groups of polynomial subgroup growth (known for short as PSG groups). Other applications of subgroup growth are found in [12] and [4].