Uncountably many non-commensurable finitely presented pro-$p$ groups

TL;DR

The paper demonstrates the existence of uncountably many non-commensurable finitely presented pro-p groups with a fixed number of generators, expanding understanding of the diversity of such groups.

Contribution

It proves the existence of uncountably many non-commensurable finitely presented pro-p groups with a given minimal number of generators and relations, a significant advancement in group theory.

Findings

Uncountably many non-commensurable metabelian uniform pro-p groups of dimension m exist.

Uncountably many non-commensurable finitely presented pro-p groups with minimal generators m.

These groups have minimal number of relations {m }.

Abstract

Let $m\geq 3$ be a positive integer. We prove that there are uncountably many non-commensurable metabelian uniform pro-$p$ groups of dimension $m$. Consequently, there are uncountably many non-commensurable finitely presented pro-$p$ groups with minimal number of generators $m$ (and minimal number of relations $ {m \choose 2}$).