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

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

2012-03-07

arxiv.org/abs/1203.1591v1

Mathematics

Geometric Topology

Scientific paper

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.

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