Let $G$ be a group acting on $X$, $X$ a cellular complex and $cd(G)$ the cohomological dimension of $G$.

2 things:

(1) I'm looking for a reference (or proof!) of this:

Suppose $X$ is acyclic. Then $cd(G) \leq max_{\sigma} \space cd(Stab(\sigma) + dim \space \sigma$, where $\sigma$ runs over the cells of $X$.

(2) If $X$ isn't acyclic, can anything be said about $cd(G)$?