Cohomological dimension of a group

Summary

The cohomological dimension of a group $G$ is the largest degree in which group cohomology of $G$ can be nonzero. It equals the shortest length of a projective resolution of $\Z$ over the group ring $\Z[G]$. Free groups are exactly the groups of dimension at most $1$, by the Stallings–Swan theorem, and $\Z^n$ has dimension $n$. A group of finite cohomological dimension is torsion-free.