Gorenstein versions of type $\Phi$ groups and some Gcd-cd, Ghd-hd coincidences

TL;DR

This paper characterizes Gorenstein variants of type Φ groups and explores conditions under which classical and Gorenstein cohomological dimensions coincide for certain torsion-free groups.

Contribution

It introduces Gorenstein versions of type Φ groups and generalizes results on cohomological dimension coincidences within Kropholler's class.

Findings

Characterization of Gorenstein type Φ groups

Generalization of cohomological dimension coincidence results

Insights into Gorenstein cohomology in group theory

Abstract

In this short note, we characterise some Gorenstein versions of the concept of a group being of type $\Phi$ as introduced by Olympia Talelli. And, we also generalize a different Talelli result regarding the coincidence of the classical and the Gorenstein cohomological dimension of torsion-free groups in Kropholler's $\LH\mathscr{F}$ class.