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

For ΓΓ a relatively hyperbolic group, we construct a model for the universal space among ΓΓ-spaces with isotropy on the family VCVC of virtually cyclic subgroups of ΓΓ. 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 KK-theory of the Coxeter group Γ3Γ3 (a non-uniform lattice in O+(3,1)O+(3,1)). Part of this computation involves calculating certain Waldhausen Nil-groups for Z[D2]Z[D2], Z[D3]Z[D3].