Homological growth of nilpotent-by-abelian pro-p groups

Abstract: We show that the torsion-free rank of $H_i(M, \mathbb{Z}p)$ has finite upper bound for $i \leq m$, where $M$ runs through the pro-$p$ subgroups of finite index in a pro-$p$ group $G$ that is (nilpotent of class $c$)-by-abelian such that $ G/N'$ is of type $FP{2cm}$.