  Let $G$ be a group that admits a cocompact classifying space for proper
actions $X$. We derive a formula for the Bredon cohomological dimension for
proper actions of $G$ in terms of the relative cohomology with compact support
of certain pairs of subcomplexes of $X$. We use this formula to compute the
Bredon cohomological dimension for proper actions of fundamental groups of
non-positively curved simple complexes of finite groups. As an application we
show that if a virtually torsion-free group acts properly and chamber
transitively on a building, its virtual cohomological dimension coincides with
its Bredon cohomological dimension. This covers the case of Coxeter groups and
graph products of finite groups.
