Nilpotency of Bocksteins, Kropholler's hierarchy and a conjecture of Moore

TL;DR

This paper investigates Moore's conjecture on projective modules over groups, proving it for specific classes of groups, including those in Kropholler's hierarchy, and explores closure properties and counterexamples related to Serre's theorem.

Contribution

It extends Moore's conjecture verification to groups within Kropholler's hierarchy and establishes new closure properties for group pairs satisfying the conjecture.

Findings

Proves Moore's conjecture for groups satisfying Serre's theorem analog.

Establishes the conjecture for all groups in Kropholler's hierarchy LHF.

Identifies cases where Serre's theorem analog fails but the conjecture holds.

Abstract

A conjecture of Moore claims that if G is a group and H a finite index subgroup of G such that G - H has no elements of prime order (e.g. G is torsion free), then a G-module which is projective over H is projective over G. The conjecture is known for finite groups. In that case, it is a direct consequence of Chouinard's theorem which is based on a fundamental result of Serre on the vanishing of products of Bockstein operators. It was observed by Benson, using a construction of Baumslag, Dyer and Heller, that the analog of Serre's Theorem for infinite groups is not true in general. We prove that the conjecture is true for groups which satisfy the analog of Serre's theorem. Using a result of Benson and Goodearl, we prove that the conjecture holds for all groups inside Kropholler's hierarchy LHF, extending a result of Aljadeff, Cornick, Ginosar, and Kropholler. We show two closure properties for the class of pairs of groups (G,H) which satisfy the conjecture, the one is closure under morphisms, and the other is a closure operation which comes from Kropholler's construction. We use this in order to exhibit cases in which the analog of Serre's theorem does not hold, and yet the conjecture is true. We will show that in fact there are pairs of groups (G,H) in which H is a perfect normal subgroup of prime index in G, and the conjecture is true for (G,H). Moreover, we will show that it is enough to prove the conjecture for groups of this kind only.