Deformations of Hyperbolic Cone-Structures: Study of the Collapsing case

Abstract

This work is devoted to the study of deformations of hyperbolic cone structures under the assumption that the lengths of the singularity remain uniformly bounded over the deformation. Given a sequence $(M_i$ of pointed hyperbolic cone-manifolds with topological type $(M,\Sigma )$, where $M$ is a closed, orientable and irreducible 3-manifold and $\Sigma$ an embedded link in $M$. If the sequence $M_i$ collapses and assuming that the lengths of the singularity remain uniformly bounded, we prove that $M$ is either a Seifert fibered or a $Sol$ manifold. We apply this result to a question stated by Thurston and to the study of convergent sequences of holonomies.