Classification of hyperbolic Dehn fillings I

TL;DR

This paper classifies large Dehn fillings of a 2-cusped hyperbolic 3-manifold using an invariant derived from holonomy derivatives, linking it to the complex volume of the fillings.

Contribution

It introduces a classification method for large Dehn fillings based on a holonomy derivative invariant, connecting it to the complex volume.

Findings

Dehn fillings with large coefficients can be classified using the invariant.

Equal invariants imply equal complex volumes for large Dehn fillings.

The invariant is a powerful tool for understanding the geometry of hyperbolic 3-manifolds.

Abstract

Let $M$ be a $2$-cusped hyperbolic $3$-manifold. By the work of Thurston, the product of the derivatives of the holonomies of core geodesics of each Dehn filling of $M$ is an invariant of it. In this paper, we classify Dehn fillings of $M$ with sufficiently large coefficients using this invariant. Further, for any given two Dehn fillings of $M$ (with sufficiently larger coefficients), if their aforementioned invariants are the same, it is shown their complex volumes are the same as well.