Classification of Hyperbolic Dehn fillings II: Quadratic case

Abstract

This paper is subsequent to [5]. In this paper, we extend the classification of hyperbolic Dehn fillings with sufficiently large coefficients by addressing the remaining case not covered in [5]. Specifically, by considering the case in which the two cusp shapes lie in the same quadratic field, we obtain the complete classification under a mild assumption satisfied by most manifolds. The content of this paper is not limited to the classification of hyperbolic Dehn fillings. Along the way, we also classify key types of automorphisms of the holonomy variety of a two-cusped hyperbolic $3$-manifold and uncover an intriguing hidden structure in the complex volume of certain manifolds. Concrete examples illustrating these phenomena were discovered by S. Oh, and we elaborate on them in this paper. All the results presented here appear to be effective. In the third paper of this series, joint with S. Oh, we will provide examples confirming the optimality of our results.