Pro-p groups of rank 3 and the question of Iwasawa

Abstract

For a prime p > 3 we determine all pro-p groups that satisfy d(G) = d(H) = 3 for all open subgroups H of G.

Pro-p groups with constant generating number on open subgroups

“…This generalises the partial results in [10], which were obtained by quite different methods. Thematically our work is linked to other recent results on generating numbers of pro-p groups; e.g.…”

“…Note that E n is non-empty for every n because, clearly, it contains the free abelian pro-p group Z n p of rank n. In [11], Yamagishi remarked that no other examples are known to him when n ≥ 3. Recently, for p > 3, the second author determined all p-adic analytic pro-p groups that belong to the class E 3 ; see [10]. His approach relied on the classification of 3-dimensional soluble Z p -Lie lattices provided in [4].…”

“…Let p be a prime. Motivated by a question of Kenkichi Iwasawa, which we discuss below, we consider profinite groups G satisfying the condition This generalises the partial results in [10], which were obtained by quite different methods. Thematically our work is linked to other recent results on generating numbers of pro-p groups; e.g.…”

A CHARACTERIZATION OF UNIFORM PRO-p GROUPS

“…In [11, Theorem 1.1] we classified finitely generated pro-p groups with constant generating number on open subgroups, that is, pro-p groups G with the property d(H) = d(G) for every open subgroup H ≤ G; see also [19]. (1) the abelian group Z d p , for d ≥ 0; (2) the metabelian group y ⋉ A, for d ≥ 2, where y ∼ = Z p , A ∼ = Z d−1 p and y acts on A as scalar multiplication by λ, with λ = 1 + p s for some s ≥ 1, if p > 2, and λ = ±(1 + 2 s ) for some s ≥ 2, if p = 2;…”

“…In [11,Theorem 1.1] we classified finitely generated pro-p groups with constant generating number on open subgroups, that is, pro-p groups G with the property d(H) = d(G) for every open subgroup H ≤ G; see also [19]. Note that, if G is a hereditarily uniform pro-p group, then d(H) = d(G) for all open subgroups H ≤ G. Hence, Corollary 1.12 can also be regarded as a consequence of Theorem 3.1.…”