Some Characterisations of p-adic Analytic Groups

TL;DR

This paper characterizes p-adic analytic pro-p groups through necessary and sufficient conditions, linking properties like finite rank and chain conditions, and resolves related conjectures for countably based groups.

Contribution

It provides new characterizations of p-adic analytic groups and confirms conjectures for noetherian and countably based pro-p groups.

Findings

A noetherian pro-p group with finite chain length has finite rank.

A noetherian pro-p group has finite rank iff it satisfies the weak descending chain condition.

Countably based pro-p groups with all non-open closed subgroups of finite rank are p-adic analytic.

Abstract

We give three necessary and sufficient conditions for a pro-p group to be p-adic analytic. We show that a noetherian pro-p group having finite chain length has a finite rank and conversely. We further deduce that a noetherian pro-p group has a finite rank precisely when it satisfies the weak descending chain condition. Using these results, we resolve a conjecture posed by Lubotzky and Mann in the affirmative within the class of noetherian groups which are countably based. Using these results, we answer a related conjecture about pro-p groups for the case of countably based pro-p groups. Namely, we prove that if every closed but non-open subgroup of a countably based pro-p group has finite rank, then the group is p-adic analytic and conversely.