On the Gorenstein and cohomological dimension of groups

Abstract

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