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nLab

virtual cohomological dimension

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# Contents

## 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

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

Last revised on March 21, 2021 at 03:30:41.
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