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 Fₚ G is a Sylvester domain. Moreover, the inner rank irkF_p G(A) of a matrix A over Fₚ 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- Zₚ pro- p groups.