The torsion-free rank of homology in towers of soluble pro-p groups

Abstract

We show that for every finitely presented pro-$p$ nilpotent-by-abelian-by-finite group $G$ there is an upper bound on ${\dim }_{{\mathbb{Q}}_p}(H_1(M,{\mathbb{Z}}_p){\otimes }_{{\mathbb{Z}}_p}{\mathbb{Q}}_p)$, as $M$ runs through all pro-$p$ subgroups of finite index in $G$.

Final accepted version. To appear in the Israel Journal of Mathematics