  We investigate strictly developable simple complexes of groups with arbitrary
local groups, or equivalently, group actions admitting a strict fundamental
domain. We introduce a new method for computing the cohomology of such groups.
We also generalise Bestvina's construction to obtain a polyhedral complex
equivariantly homotopy equivalent to the standard development of the lowest
possible dimension.
  As applications, for a group acting chamber transitively on a building of
type $(W,S)$, we show that its Bredon cohomological dimension is equal to the
virtual cohomological dimension of $W$ and give a realisation of the building
of the lowest possible dimension.
  We introduce the notion of a reflection-like action, and use it to give a new
family of counterexamples to the strong form of Brown's conjecture on the
equality of virtual cohomological dimension and Bredon cohomological dimension
for proper actions.
  We show that the fundamental group $G$ of a simple complex of groups acts on
a tree with stabilisers generating a family of subgroups $\mathcal{F}$ if and
only if its Bredon cohomological dimension with respect to $\mathcal{F}$ is at
most one. This confirms a folklore conjecture under the assumption that a model
for the classifying space $E_{\mathcal{F}}G$ of $G$ for the family
$\mathcal{F}$ has a strict fundamental domain.
  In order to handle complexes of groups arising from arbitrary group actions,
we define a number of combinatorial invariants such as the block poset, which
may be of independent interest. We also derive a general formula for Bredon
cohomological dimension for a group $G$ admitting a cocompact model for
$E_{\mathcal{F}}G$. As a consequence of both, we obtain a simple formula for
proper cohomological dimension of $\mathrm{CAT}(0)$ groups whose actions admit
a strict fundamental domain.
