Deformation of Hyperbolic Cone-Structures: Study of the non-Colapsing case

TL;DR

This paper investigates how hyperbolic cone-structures deform when the singularity lengths stay bounded, showing either collapse to certain geometries or convergence to a hyperbolic space with finite volume, with implications for Thurston's conjecture.

Contribution

It establishes conditions under which sequences of hyperbolic cone-manifolds either collapse or converge, advancing understanding of their deformation space and addressing Thurston's conjecture.

Findings

Sequences either collapse to Seifert fibered or Sol manifolds or converge to a hyperbolic space.

Bounded singularity lengths imply specific geometric behaviors.

Results apply to small links and Thurston's conjecture.

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. Assuming that the lengths of the singularity remain uniformly bounded, we prove that either the sequence M_{i} collapses and M is Seifert fibered or a Sol manifold, or the sequence M_{i} does not collapse and in this case a subsequence of (M_{i},p_{i}) converges to a complete Alexandrov space of dimension 3 endowed with a hyperbolic metric of finite volume on the complement of a finite union of quasi-geodesics. We apply this result to a conjecture of Thurston and to the case where {\Sigma} is a small link in M.