Gorenstein homological dimension and some invariants of groups

For any group $G$, the Gorenstein homological dimension ${\mathrm{G}\mathrm{h}\mathrm{d}}_{R}G$ is defined to be the Gorenstein flat dimension of the coefficient ring $R$, which is considered as an $RG$-module with trivial group action. We prove that ${\mathrm{G}\mathrm{h}\mathrm{d}}_{R}G<\mathrm{\infty }$ if and only if the Gorenstein flat dimension of any $RG$-module is finite, if and only if there exists an $R$-pure $RG$-monic $R\to A$ with $A$ being $R$-flat and ${\mathrm{G}\mathrm{h}\mathrm{d}}_{R}G={\mathrm{f}\mathrm{d}}_{RG}A$, where $R$ is a commutative ring with finite Gorenstein weak global dimension. As applications, properties of $\mathrm{G}\mathrm{h}\mathrm{d}$ on subgroup, quotient group, extension of groups as well as Weyl group are investigated. Moreover, we compare the relations between some invariants such as $\mathrm{s}\mathrm{f}\mathrm{l}\mathrm{i}RG$, $\mathrm{s}\mathrm{i}\mathrm{l}\mathrm{f}RG$, $\mathrm{s}\mathrm{p}\mathrm{l}\mathrm{i}RG$, $\mathrm{s}\mathrm{i}\mathrm{l}\mathrm{p}RG$, and Gorenstein projective, Gorenstein flat and PGF dimensions of $RG$-modules; a sufficient condition for Gorenstein projective-flat problem over group rings is given.