Analytic pro-p groups of small dimensions

TL;DR

This paper proves that small-dimensional torsion-free p-adic analytic pro-p groups are saturable, enabling classification and deeper understanding of their structure using Lie theory.

Contribution

It establishes a converse to Lazard's result for small dimensions, showing torsion-free p-adic analytic pro-p groups of dimension less than p are saturable.

Findings

Proves all torsion-free p-adic analytic pro-p groups of dimension < p are saturable.

Provides an effective classification of 3-dimensional soluble torsion-free p-adic analytic pro-p groups for p > 3.

Uses Lie theory to analyze and classify small-dimensional pro-p groups.

Abstract

According to Lazard, every p-adic Lie group contains an open pro-p subgroup which is saturable. This can be regarded as the starting point of p-adic Lie theory, as one can naturally associate to every saturable pro-p group G a Lie lattice L(G) over the p-adic integers. Essential features of saturable pro-p groups include that they are torsion-free and p-adic analytic. In the present paper we prove a converse result in small dimensions: every torsion-free p-adic analytic pro-p group of dimension less than p is saturable. This leads to useful consequences and interesting questions. For instance, we give an effective classification of 3-dimensional soluble torsion-free p-adic analytic pro-p groups for p > 3. Our approach via Lie theory is comparable with the use of Lazard's correspondence in the classification of finite p-groups of small order.