Sylvester domains and pro-$p$ groups

Abstract

Let $G$ be a finitely generated torsion-free pro-$p$ group containing an open free-by-${\mathbb{Z}}_p$ pro-$p$ subgroup. We show that the completed group algebra ${\mathbb{F}}_p[[G]]$ is a Sylvester domain. Moreover, the inner rank ${\mathrm{irk}}_{{\mathbb{F}}_p[[G]]}(A)$ of a matrix $A$ over ${\mathbb{F}}_p[[G]]$ can be calculated by approximation by ranks corresponding to finite quotients of $G$. As a consequence, we obtain a particular case of the mod $p$ Lück approximation for abstract finitely generated subgroups of free-by-${\mathbb{Z}}_p$ pro-$p$ groups.