On the collapsing along deformations of hyperbolic cone 3 -manifolds

Abstract

This article focuses on deformations of hyperbolic cone structures under the assumption that the length of the singularity remains uniformly bounded during the deformation. Let $M$ be a closed, orientable, and irreducible $3$-manifold, and let $\Sigma$ be an embedded link in $M$. For a collapsing sequence of hyperbolic cone structures with topological type $(M,\Sigma )$ and with uniformly bounded lengths of singularities, we prove that $M$ is either Seifert fibered or a $\mathrm{Sol}$ manifold.