Constructing irreducible representations of discrete groups

The decomposition of unitary representations of a discrete group obtained by induction from a subgroup involves commensurators. In particular Mackey has shown that quasi-regular representations are irreducible if and only if the corresponding subgroups are self-commensurizing. The purpose of this work is to describe general constructions of pairs of groups Gamma(0)<Gamma with Gamma(0) its own commensurator in Gamma. These constructions are then applied to groups of isometries of hyperbolic spaces and to lattices in algebraic groups.