Cohomological dimension of a group

Definition
The cohomological dimension of a group $$G$$ is defined as the projective dimension of its integral group ring $$\mathbb{Z}G$$, i.e., it is the minimum possible length of a projective resolution of $$\mathbb{Z}$$ as a $$\mathbb{Z}G$$-module.

If there is no projective resolution of finite length, the cohomological dimension is defined to be $$\infty$$. A group whose cohomological dimension is finite is termed a group of finite cohomological dimension.

Related notions

 * Geometric dimension of a group: This is the minimum possible dimension of an Eilenberg-MacLane space $$K(G,1)$$ (i.e., a classifying space for the group viewed as a discrete group). The geometric dimension is equal to the cohomological dimension except possibly when the cohomological dimension is $$2$$ and the geometric dimension is $$3$$, though no examples are known even for this case.

In particular, a group of finite cohomological dimension is the same as a group of finite geometric dimension.