Cohomological Dimension - Examples

Examples

In the first group of examples, let the ring R of coefficients be Z.

• A free group has cohomological dimension one. As shown by John Stallings (for finitely generated group) and Richard Swan (in full generality), this property characterizes free groups.
• The fundamental group of a compact, connected, orientable Riemann surface other than the sphere has cohomological dimension two.
• More generally, the fundamental group of a compact, connected, orientable aspherical manifold of dimension n has cohomological dimension n. In particular, the fundamental group of a closed orientable hyperbolic n-manifold has cohomological dimension n.
• Nontrivial finite groups have infinite cohomological dimension over Z. More generally, the same is true for groups with nontrivial torsion.

Now let us consider the case of a general ring R.

• A group G has cohomological dimension 0 if and only if its group ring RG is semisimple. Thus a finite group has cohomological dimension 0 if and only if its order (or, equivalently, the orders of its elements) is invertible in R.
• Generalizing the Stallings–Swan theorem for R = Z, Dunwoody proved that a group has cohomological dimension at most one over an arbitrary ring R if and only if it is the fundamental group of a connected graph of finite groups whose orders are invertible in R.