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

Grégoire Montcouquiol and Hartmut Weiß

Geometry & Topology 17 (2013) 369–412

In the former articles [arXiv:0903.4743 and this volume pp 329-367], it was independently proven by the authors that the space of hyperbolic cone–3–manifolds with cone angles less than 2π 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, ie 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.

Cone-manifolds, surface group representations, hyperbolic geometry
Mathematical Subject Classification 2010
Primary: 57M50, 58D27
Secondary: 53C35
Received: 3 May 2011
Revised: 2 May 2012
Accepted: 6 September 2012
Published: 11 March 2013
Proposed: Jean-Pierre Otal
Seconded: Simon Donaldson, David Gabai
Grégoire Montcouquiol
Univ. Paris-Sud, Laboratoire de Mathématiques
Orsay F-91405
Hartmut Weiß
LMU München
Mathematisches Institut
Theresienstr. 39
D-80333 München