Let G be a group, H a finite index subgroup. Moore's conjecture says that under a certain condition on G and H (which we call the Moore's condition), a G-module M which is projective over H is projective over G. In other words- if we know that a module is ``almost projective'', then it is projective.
In this talk we will survey cases in which the conjecture is known to be true. This includes the case in which the group G is finite and the case in which the group G has finite hohomological dimension. As a generalization of these two cases, we shall present Kropholler's hierarchy LHF, and discuss the conjecture for groups in this hierarchy.
In the case of finite groups and in the case of finite cohomological dimension groups, the conjecture is proved by the same finiteness argument. This argument is straightforward in the finite cohomological dimension case, and is a result of a theorem of Serre in case the group is finite. We will show that inside Kropholler's hierarchy the conjecture holds even though this finiteness condition might fail to hold.
We will also discuss some other cases in which the conjecture is known to be true (e.g. Thompson's group F)