On Profinite Groups of Type FP_∞

2014/12/05 by Ged Corob Cook, Cook, Ged Corob
Mathematics · #FOS: Mathematics #Group Theory (math.GR) #math.GR

paper · pdf · doi:10.48550/arxiv.1412.1876

arxiv created 2014/12/05 · arxiv updated 2014/12/08

Abstract

Suppose R is a profinite ring. We construct a large class of profinite groups \widehat\scriptstyle\bf L'\scriptstyle\bf H_R\mathfrakF, including all soluble profinite groups and profinite groups of finite cohomological dimension over R. We show that, if G ∈ \widehat\scriptstyle\bf L'\scriptstyle\bf H_R\mathfrakF is of type FP_∞ over R, then there is some n such that H_Rⁿ(G,R [[ G ]]) ≠ 0, and deduce that torsion-free soluble pro-p groups of type FP_∞ over ℤ_p have finite rank, thus answering the torsion-free case of a conjecture of Kropholler.

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