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

In this short note, we characterise some Gorenstein versions of the concept of a group being of type $\mathrm{\Phi }$ 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 $\text{LH}\mathcal{F}$ class.