Relative Hyperbolicity, Classifying Spaces, and Lower Algebraic K-theory

For Γ a relatively hyperbolic group, we construct a model for the universal space among Γ-spaces with isotropy on the family VC of virtually cyclic subgroups of Γ. We provide a recipe for identifying the maximal infinite virtually cyclic subgroups of Coxeter groups which are lattices in O+(n, 1) = Isom(Hn). We use the information we obtain to explicitly compute the lower algebraic K-theory of the Coxeter group Γ3 (a non-uniform lattice in O+(3, 1)). Part of this computation involves calculating certain Waldhausen Nil-groups for Z[D2], Z[D3].