nLab virtual cohomological dimension

Idea

In group cohomology theory, if $G$ is a group which has torsion-free subgroups of finite index, then all such subgroups have the same cohomological dimension; this common dimension is called the virtual cohomological dimension of $G$ and denoted $vcd(G)$.

For example, $SL_n(\mathbb{Z})$ has infinite cohomological dimension, and yet

$$vcd(SL_n(\mathbb{Z})) = \binom{n}{2}.$$

Related concepts

• Farrell-Tate cohomology

notion of dimension

• homotopy dimension

• cohomology dimension (virtual)

• covering dimension

• Heyting dimension

• dimension of a manifold

• dimension of a cell complex

• dimension of a separable metric space

• small inductive dimension of a regular space

• large inductive dimension of a normal space