Characteristic subsurfaces, character varieties and Dehn fillings

Abstract

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 $\alpha$ and $\beta$ in several situations. These include cases where $M\left(\beta \right)$ is reducible and where $M\left(\alpha \right)$ has finite ${\pi }_1$, or $M\left(\alpha \right)$ is very small, or $M\left(\alpha \right)$ admits a ${\pi }_1$–injective immersed torus.