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. 



Finite models for torsion-free lattices
In fact, more is true and you do not need separate arguments for rank 1 and higher rank. 


The following is Theorem 13.1(i) in the book of Ballmann, Gromov and Schroeder "Manifolds of nonpositive curvature":


Suppose that $(M,g)$ is a complete real-analytic Riemannian manifold of nonpositive curvature and finite volume. Then $M$ is tame: It is diffeomorphic to the interior of a compact manifold with boundary $M'$. 


Moreover, the proof shows that $M'$ can be realized as a submanifold (with boundary) of $M$. 


Applying this to the locally-symmetric space $(M,g)=X/\Gamma$, where $\Gamma$ is a torsion-free lattice in the isometry group of a nonpositively curved symmetric space $X$, after triangulating $M'$, we obtain a finite model for $\Gamma$. 


