On character varieties, sets of discrete characters, and nonzero degree maps

A {\it knot manifold} is a compact, connected, irreducible, orientable $3$-manifold whose boundary is an incompressible torus. We first investigate virtual epimorphisms between the fundamental groups of small knot manifolds and prove minimality results for small knot manifolds with respect to nonzero degree maps. These results are applied later in the paper where we fix a small knot manifold $M$ and investigate various sets of characters of representations $\rho: \pi_1(M) \to {\rm PSL}_2(\Bbb{C})$ whose images are discrete. We show that the topology of these sets is intimately related to the algebraic structure of the ${\rm PSL}_2(\Bbb{C})$-character variety of $M$ as well as dominations of manifolds by $M$ and its Dehn fillings. We apply our results to the following question of Shicheng Wang: {\it Are nonzero degree maps between infinitely many distinct Dehn fillings of two hyperbolic knot manifolds $M$ and $N$ induced by a nonzero degree map $M \to N$?} We show that the answer is yes generically. Using this we show that if a small $\mathcal{H}$-minimal hyperbolic knot manifold admits non-homeomorphic $\mathcal{H}$-minimal Dehn fillings, it admits infinitely many such fillings. We also construct the first infinite families of small, closed, connected, orientable manifolds which are minimal in the sense that they do not admit nonzero degree maps, other than homotopy equivalences, to any aspherical manifold.