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