Relative hyperbolicity, classifying spaces, and lower algebraic K-theory

For $\Gamma$ a relatively hyperbolic group, we construct a model for the universal space among $\Gamma$-spaces with isotropy on the family $VC$ of virtually cyclic subgroups of $\Gamma$. We provide a recipe for identifying the maximal infinite virtually cyclic subgroups of Coxeter groups which are lattices in $O^{+}(n,1)=\mathrm{Isom}\left(H^n\right)$. We use the information we obtain to explicitly compute the lower algebraic $K$-theory of the Coxeter group ${\Gamma }_3$ (a non-uniform lattice in $O^{+}(3,1)$). Part of this computation involves calculating certain Waldhausen Nil-groups for $Z\left[D_2\right]$, $Z\left[D_3\right]$.