Some special cases of Moore's conjecture on projective modules and finite index subgroups

Ehud Meir (Technion)

Wednesday 22nd October, 2008 14:00-15:00 Mathematics Building, Rm 516


(joint work with Eli Aljadeff) Let G be a group, and let H be a finite index subgroup. It was conjectured by Moore that if H intersects every nontrivial subgroup of G nontrivially (that is- H satisfies Moore’s condition in G), then every G-module M which is H-projective is also G-projective. This conjecture was proved to be true in several cases. In case the group G is finite, it is known to be true by a result of Chouinard, which follows from a result of Serre on the nilpotency of a certain element in the cohomology ring H*(G,Z). It is also known to be true (by a theorem of Aljadeff) in case H^ satisfies the Moore’s condition in G^ (where H^, G^ stands for the profinite completions). By a theorem of Aljadeff, Cornick, Ginosar and Kropholler, it is known to hold for a group G in Kropholler’s hierarchy LHF, in case the module M is also finitely generated. In this talk we shall discuss known cases in which the conjecture is true, We will show why the conjecture holds for every group in LHF, without the finite generation assumption, and we shall show how, using a result of Benson and Goodearl, we can build examples for a group G and a finite index normal acyclic subgroup H for which the conjecture holds. This would provide us examples in which the cohomology element mentioned above is not nilpotent, and yet the conjecture holds.

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