Suppose $R$ is a profinite ring. We construct a large class of profinite groups $\widehat{ { \scriptstyle\bf L}'{ \scriptstyle\bf H}_R}\mathfrak{ F}$, including all soluble profinite groups and profinite groups of finite cohomological dimension over $R$. We show that, if $G \in \widehat{ { \scriptstyle\bf L}'{ \scriptstyle\bf H}_R}\mathfrak{ F}$ is of type $\operatorname{ FP}_\infty$ over $R$, then there is some $n$ such that $H_R^n(G,R [[ G ]]) \neq 0$, and deduce that torsion-free soluble pro-$p$ groups of type $\operatorname{ FP}_\infty$ over $\mathbb{ Z}_p$ have finite rank, thus answering the torsion-free case of a conjecture of Kropholler.
Categories: math.GR
