Relative hyperbolicity, classifying spaces, and lower algebraic -theory

For Γ a relatively hyperbolic group, we construct a model for the universal space among Γ -spaces with isotropy on the family V C 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 ( H n ) . We use the information we obtain to explicitly compute the lower algebraic K -theory of the Coxeter group Γ 3 (a non-uniform lattice in O + ( 3 , 1 ) ). Part of this computation involves calculating certain Waldhausen Nil-groups for Z [ D 2 ] , Z [ D 3 ] .