A menagerie of SU(2)-cyclic 3-manifolds

We classify \(SU(2)\)-cyclic and \(SU(2)\)-abelian 3-manifolds, for which every representation of the fundamental group into \(SU(2)\) has cyclic or abelian image respectively, among geometric 3-manifolds which are not hyperbolic. As an application, we give examples of hyperbolic 3-manifolds which do not admit degree-1 maps to any Seifert fibered manifold other than \(S^3\) or a lens space. We also produce infinitely many one-cusped hyperbolic manifolds with at least four \(SU(2)\)-cyclic Dehn fillings, one more than the number of cyclic fillings allowed by the cyclic surgery theorem.