Sylvester domains and pro-$p$ groups

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}\llbracket G\rrbracket is a Sylvester domain. Moreover, the inner rank \mathrm{irk}_{\mathbb{F}_p\llbracket G\rrbracket}(A) of a matrix A over \mathbb{F}_{p}\llbracket G\rrbracket 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.