On the Gorenstein and cohomological dimension of groups

Abstract:

Here we relate the Gorenstein dimension of a group $G$, $\mathrm {Gcd}_{R}G$, over $\mathbb Z$ and $\mathbb Q$ to the cohomological dimension of $G$, $\mathrm {cd}_{R}G$, over $\mathbb Z$ and $\mathbb Q$, and show that if $G$ is in ${\scriptstyle \bf {LH}}\mathfrak F$, a large class of groups defined by Kropholler, then $\mathrm {cd}_{\mathbb Q}G=\mathrm {Gcd}_{\mathbb Q}G$ and if $G$ is torsion free, then $\mathrm {Gcd}_{\mathbb Z}G= \mathrm {cd}_{\mathbb Z}G$. We also show that for any group $G$, $\mathrm {Gcd}_{\mathbb Q}G\leq \mathrm {Gcd}_{\mathbb Z}G$.