Let $G$ be a real, connected, semisimple Lie group and $\Gamma < G$ a torsion-free lattice. Then does there exist a finite $CW$-model for $B\Gamma$?

I know this to be true in many instances (e.g. when $\Gamma$ is uniform, when $G=\mathrm{SO}_0(n,1)$ or when $\mathrm{rank}_{\mathbb R}(G) \geq 2$ and $\Gamma$ is irreducible). In these instances, there always exist certain "canonical" finite CW-models for $B\Gamma$.

However, I am unaware of the situation for *general* such lattices, hence the question.

virtuallyhas a finite CW-model, i.e that there exists a finite-index subgroup $\Gamma' < \Gamma$ possessing a finite CW-model. $\endgroup$"I think your argument shows only that each $\Gamma$ virtually..."No, I was careful on that point: I project the lattices on factors, so I embed $\Gamma$ into a larger torsion-free lattice which is a direct product (I said "overgroup"). $\endgroup$16more comments