Sylvester domains and pro-p groups

Let G be a finitely generated torsion-free pro-p group containing an open free-by-Zₚ pro-p subgroup. We show that the completed group algebra of G over Fₚ 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=G₁>G₂>… is a chain of normal open subgroups of G with trivial intersection and Aᵢ is the matrix over Fₚ[G/Gᵢ] obtained from the matrix A by applying the natural homomorphism induced from G → G/Gᵢ, then the inner rank of A equals limi→ ∞ rkFₚ (Aᵢ)|G:Gᵢ|. As a consequence, we obtain a particular case of the mod p L\"uck approximation for abstract finitely generated subgroups of free-by-Zₚ pro-p groups.