  For a discrete group $\Gamma$ satisfying some finiteness conditions we give a
Bredon projective resolution of the trivial module in terms of projective
covers of the chain complex associated to certain posets of subgroups. We use
this to give new results on dimensions of $E\gamma$ and to reprove that for
virtually solvable groups, $\underline{\cd}\Gamma=\vcd\Gamma$. We also deduce a
formula to compute the Euler class of $E\gamma$ for $\Gamma$ virtually solvable
of type $\FP_\infty$ and use it to compute orbifold Euler characteristics.
