Separable-stable representations of a compression body

Let M   be a hyperbolizable, non-trivial compression body without toroidal boundary components. In this paper, we characterize which discrete and faithful representations of π1(M)π1(M) into PSL(2,C)PSL(2,C) are separable-stable. The set of separable-stable representations forms a domain of discontinuity for the action of Out(π1(M))Out(π1(M)) on the PSL(2,C)PSL(2,C)-character variety of π1(M)π1(M).