On the Gorenstein cohomological dimension of group extensions

Abstract

The Gorenstein cohomological dimension is defined for any group G and coincides with the virtual cohomological dimension vcdG, whenever the latter is defined and finite. Unlike the virtual cohomological dimension, the Gorenstein cohomological dimension behaves well with respect to extensions, finite graphs of groups and ascending unions. In this paper, we study the Gorenstein cohomological dimension GcdkG of groups G, which are of type FP∞ over a commutative ring k. We show that if 1⟶N⟶G⟶Q⟶1 is an extension of groups with N of type FP∞ over a field F and GcdFQ<∞, then GcdFG=GcdFN+GcdFQ. We also show that for any group G of type FP∞ over Z with GcdZG<∞, there exists a field F such that GcdFG=GcdZG. This implies, in particular, that if G is a group of type FP∞ over Z and G is an extension of N by itself, then GcdZG=2GcdZN.