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In this paper, we study finiteness criteria for the Gorenstein homologicaldimension of groups over a commutative ring of finite weak Gorenstein globaldimension and provide good estimations for the Gorenstein weak global dimensionof group rings. As a result, we obtain Gorenstein analogues of well knownproperties in classical homological algebra over some large families ofinfinite groups. Moreover, we prove that over a commutative ring of finite weakGorenstein global dimension, the Gorenstein cohomological dimension of a group$G$ bounds the Gorenstein homological dimension of $G$. Finally, we compare thegeneralized cohomological dimension and the generalized homological dimensionof a group, defined by Ikenaga.

Finiteness criteria for Gorenstein homological dimension and some invariants of groups