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

TL;DR

This paper classifies certain non-hyperbolic 3-manifolds based on their $SU(2)$ representation properties and explores implications for hyperbolic manifolds and Dehn fillings, revealing new examples and constraints.

Contribution

It provides a classification of $SU(2)$-cyclic and $SU(2)$-abelian 3-manifolds among non-hyperbolic geometries and constructs hyperbolic examples with unusual Dehn filling properties.

Findings

Classified $SU(2)$-cyclic and $SU(2)$-abelian 3-manifolds among geometric non-hyperbolic cases.

Constructed hyperbolic 3-manifolds with more $SU(2)$-cyclic Dehn fillings than predicted by the cyclic surgery theorem.

Provided examples of hyperbolic manifolds with degree-1 maps only to $S^3$ or lens spaces.

Abstract

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.