Complex twist flows on surface group representations and the local shape of the deformation space of hyperbolic cone-3-manifolds

TL;DR

This paper explores the local deformation space of hyperbolic cone-3-manifolds when the singular locus can split, revealing a parametrization involving cone angles and edge lengths linked to pair-of-pants decompositions.

Contribution

It extends previous work by analyzing the deformation space without fixing the singular locus, connecting splittings to pair-of-pants decompositions and new parameters.

Findings

Deformation space parametrized by cone angles and edge lengths.

Splittings correspond to pair-of-pants decompositions.

Under certain conditions, the local shape is explicitly described.

Abstract

In two former papers, the authors independently proved that the space of hyperbolic cone-3-manifolds with cone angles less than 2{\pi} and fixed singular locus is locally parametrized by the cone angles. In this sequel, we investigate the local shape of the deformation space when the singular locus is no longer fixed, i.e. when the singular vertices can be split. We show that the different possible splittings correspond to specific pair-of-pants decompositions of the smooth parts of the links of the singular vertices, and that under suitable assumptions the corresponding subspace of deformations is parametrized by the cone angles of the original edges and the lengths of the new ones.