Rigidity in hyperbolic Dehn filling

TL;DR

This paper investigates the rigidity properties of core geodesics in hyperbolic Dehn fillings, demonstrating that large coefficient fillings are uniquely determined by holonomy products, with implications and alternative proofs provided.

Contribution

It establishes a new rigidity result linking Dehn filling parameters to core geodesic holonomies in hyperbolic 3-manifolds.

Findings

Large coefficient Dehn fillings are uniquely determined by core geodesic holonomies.

Non-symmetric cusp shapes lead to rigidity in hyperbolic Dehn fillings.

An alternative geometric proof of a key corollary is provided by I. Agol.

Abstract

This paper concerns with a rigidity of core geodesics in hyperbolic Dehn fillings. For instance, for an $n$-cusped hyperbolic $3$-manifold $M$ having non-symmetric cusp shapes, we show any Dehn filling of $M$ with sufficiently large coefficient is uniquely determined by the product of the holonomies of its core geodesics. We also explore various implications of the main results. An appendix by I. Agol provides an alternative geometric proof of one of the corollaries of our main arguments.