Subgroup posets, Bredon cohomology and equivariant Euler characteristics

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.