Classifying spaces with virtually cyclic stabilizers for linear groups

We show that every discrete subgroup of $\mathrm{G}\mathrm{L}(n,\mathbb{R})$ admits a finite dimensional classifying space with virtually cyclic stabilizers. Applying our methods to $\mathrm{S}\mathrm{L}(3,\mathbb{Z})$, we obtain a four dimensional classifying space with virtually cyclic stabilizers and a decomposition of the algebraic $K$-theory of its group ring.