On the hierarchy of cohomological dimensions of groups

Using Auslander’s G-dimension, we assign a numerical invariant to any group ΓΓ. It provides a refinement of the cohomological dimension and fits well into the well-known hierarchy of dimensions assigned already to ΓΓ. We study this dimension and show its power in reflecting the properties of the underlying group. We also discuss its connections to relative and Tate cohomology of groups.