Geometry & Topology

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

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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.

Article information

Geom. Topol., Volume 17, Number 1 (2013), 369-412.

Received: 3 May 2011
Revised: 2 May 2012
Accepted: 6 September 2012
First available in Project Euclid: 20 December 2017

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 57M50: Geometric structures on low-dimensional manifolds 58D27: Moduli problems for differential geometric structures
Secondary: 53C35: Symmetric spaces [See also 32M15, 57T15]

Cone-manifolds surface group representations hyperbolic geometry


Montcouquiol, Grégoire; Weiß, Hartmut. Complex twist flows on surface group representations and the local shape of the deformation space of hyperbolic cone–$3$–manifolds. Geom. Topol. 17 (2013), no. 1, 369--412. doi:10.2140/gt.2013.17.369.

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