Group class operations and homological conditions

TL;DR

This paper explores how specific group operations can generate many examples of groups satisfying certain homological conditions, including virtually Gorenstein group algebras and Moore's conjecture.

Contribution

It applies Kropholler's and Talelli's operations to demonstrate the abundance of groups meeting these homological conditions.

Findings

Many groups with virtually Gorenstein group algebras

Groups satisfying Moore's conjecture are abundant

Homological conditions are preserved under specific group operations

Abstract

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.