Gorenstein homological dimension of group extensions

We study the Gorenstein homological dimension $Gh{d}_{k}G$ of groups $G$ which are of type $F{P}_{\mathrm{\infty }}$ over a commutative ring $k$. Our main result shows that, over an arbitrary ring $R$, every finitely presented Gorenstein flat $R$-module is projectively coresolved Gorenstein flat. Consequently, for a group $G$ of type $F{P}_{\mathrm{\infty }}$ over $k$ with sfli$k<\mathrm{\infty }$, the three natural dimensions for $G$, namely Gorenstein homological, Gorenstein cohomological and projectively coresolved Gorenstein flat, all coincide. Building on this collapse, we establish a Gorenstein homological analogue of Fel${}^{\mathrm{\prime }}$dman’s theorem, a formula for iterated $m$-fold self-extensions of a group $N$ of type $F{P}_{\mathrm{\infty }}$ over $\mathbb{Z}$, and a field-detection theorem, showing that the Gorenstein homological dimension of a group of type $F{P}_{\mathrm{\infty }}$ over a principal ideal domain is realized after passing to a suitable field.