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

TL;DR

This paper investigates how hyperbolic cone-manifolds deform when their singularity lengths stay bounded, showing that collapsing sequences lead to specific geometric structures like Seifert fibered or Sol manifolds.

Contribution

It proves that collapsing sequences of hyperbolic cone-manifolds with bounded singularity lengths result in Seifert fibered or Sol manifolds, addressing a question by Thurston.

Findings

Collapse implies Seifert fibered or Sol manifolds

Bounded singularity lengths constrain geometric limits

Application to Thurston's question and holonomy sequences

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}%, p_{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.