Group class operations and homological conditions

Kropholler's operation ${\scriptstyle{{\bf LH}}}$ and Talelli's operation $\Phi$ can be often used to formally enlarge the class of available examples of groups that satisfy certain homological conditions. In this paper, we employ this enlargement technique regarding two specific homological conditions. We thereby demonstrate the abundance of groups that (a) have virtually Gorenstein group algebras, as defined by Beligiannis and Reiten, and (b) satisfy Moore's conjecture on the relation between projectivity and relative projectivity, that was studied by Chouinard, Aljadeff, Cornick, Ginosar, Kropholler and Meir.