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 of $G$ over $\mathbb{F}_p$ is a Sylvester domain. Moreover the inner rank of a matrix $A$ over this completed group algebra can be calculated by approximation by ranks corresponding to finite quotients of $G$, that is, if $G=G_1>G_2>\ldots$ is a chain of normal open subgroups of $G$ with trivial intersection and $A_i$ is the matrix over $\mathbb{F}_p[G/G_i]$ obtained from the matrix $A$ by applying the natural homomorphism induced from $G \to G/G_i$, then the inner rank of $A$ equals $\lim{i\to \infty} \frac{\operatorname{rk}_{\mathbb{F}_p} (A_i)}{|G:G_i|}$. As a consequence, we obtain a particular case of the mod $p$ L\"uck approximation for abstract finitely generated subgroups of free-by-$\mathbb{Z}_p$ pro-$p$ groups.