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

Abstract

Let $m\ge 3$ 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 $\left(\genfrac{}{}{0ex}{}{m}{2}\right)$).