A characterisation of uniform pro-p groups

TL;DR

This paper investigates the properties and characterisation of uniform pro-p groups, which are fundamental in p-adic Lie theory, proposing a conjecture relating their algebraic and topological dimensions.

Contribution

It proposes a conjecture characterising uniform pro-p groups via their minimal number of generators and p-adic dimension, providing evidence for specific cases.

Findings

Proved the conjecture for soluble groups.

Established the conjecture when p > dim(G).

Linked algebraic properties to p-adic manifold dimension.

Abstract

Let p be a prime. Uniform pro-p groups play a central role in the theory of p-adic Lie groups. Indeed, a topological group admits the structure of a p-adic Lie group if and only if it contains an open pro-p subgroup which is uniform. Furthermore, uniform pro-p groups naturally correspond to powerful Lie lattices over the p-adic integers and thus constitute a cornerstone of p-adic Lie theory. In the present paper we propose and supply evidence for the following conjecture, aimed at characterising uniform pro-p groups. Suppose that p > 2 and let G be a torsion-free pro-p group of finite rank. Then G is uniform if and only if its minimal number of generators is equal to the dimension of G as a p-adic manifold, i.e., d(G) = dim(G). In particular, we prove that the assertion is true whenever G is soluble or p > dim(G).