Gorenstein dimension and proper actions

Abstract

We conjecture that a group G admits a finite‐dimensional classifying space for proper actions if and only if the Gorenstein projective dimension of G is finite. We verify the one‐dimensional case of this conjecture. Some evidence are given for the hypothesis that the Gorenstein projective ℤG‐modules are precisely Benson's class of cofibrant modules.

Finiteness conditions in the stable module category

“…In fact, if we define the Gorenstein cohomological dimension Gcd G of G by letting Gcd G = Gpd ZG Z, then for any ZG-module M there is an inequality Gpd ZG M ≤ Gcd G + 1 (cf. [1], Proposition 2.4(c)). As shown in [loc.cit., Theorem 2.5], the Gorenstein cohomological dimension Gcd G of G is the supremum of those integers n, for which there exist ZG-modules M and P , with M Z-free and P projective, such that Ext n ZG (M, P ) = 0.…”

A note on complete resolutions

“…In [3] it was proved that if G is an LHF-group and M a ZG-module that admits a projective resolution by finitely generated projective modules, then M is a Gorenstein projective ZG-module if and only if it is a cofibrant module.…”

“…It turns out that G admits a complete resolution in the strong sense if and only if the generalized cohomological dimension cdG is finite [3], where cdG = sup{n ∈ N| ∃M Z-free, ∃F ZG-free : Ext n ZG (M, F ) = 0} was defined by Ikenaga in his study of generalized Farrell-Tate cohomology in [10].…”

“…Note that cdG = Gcd Z G [3], the Gorenstein projective dimension of G, which is defined via resolutions of the trivial ZG-module Z by Gorenstein projective modules and is related to the G-dimension defined by Auslander in [1] (see also [2] and [6]). …”

On some questions regarding projectivity criteria

“…( [18,3]) The class HF is the smallest class of groups containing the class of finite groups and which contains a group G whenever G admits a finite dimensional contractible G-C-complex whose stabilizers are already in HF. The class LHF is the class of groups such that all of its finitely generated subgroups are in HF.…”