On Profinite Groups of Type $\operatorname{FP}_\infty$

TL;DR

This paper constructs a broad class of profinite groups and proves that torsion-free soluble pro-p groups of type FP_infinity over Z_p have finite rank, addressing a conjecture by Kropholler.

Contribution

It introduces a new class of profinite groups and establishes a key property of torsion-free soluble pro-p groups of type FP_infinity.

Findings

Existence of non-zero cohomology in some degree for groups of type FP_infinity

Torsion-free soluble pro-p groups of type FP_infinity have finite rank

Addresses a conjecture of Kropholler regarding these groups

Abstract

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.