Characteristic subsurfaces, character varieties and Dehn fillings

Let M be a one cusped hyperbolic 3-manifold. A slope on the boundary of the compact core of M is called exceptional if the corresponding Dehn filling produces a non-hyperbolic manifold. We give new upper bounds for the distance between two exceptional slopes α and β in several situations. These include cases where M(β) is reducible and where M(α) has finite π₁, or M(α) is very small, or M(α) admits a π₁-injective immersed torus.