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Complex twist flows on surface group representations and the local shape of the deformation space of hyperbolic cone-3-manifolds. (English) Zbl 1267.57021
Hyperbolic cone-3-manifolds are hyperbolic manifolds with singular locus an embedded graph and a conical metric around the singularity. They can be constructed by gluing hyperbolic simplices by isometries along faces, so that the singular locus is contained in the 1-skeleton.
Local rigidity of three-dimensional cone-manifolds with (fixed) cone angles less than \(2\pi\) has been established by several authors, provided that the topology of the singular graph is preserved: namely the space of deformations is locally parameterized by the cone angles. Among others we cite C. D. Hodgson and S. P. Kerckhoff [J. Differ. Geom. 48, No. 1, 1–59 (1998; Zbl 0919.57009)], R. Mazzeo and G. Montcouquiol [ibid. 87, No. 3, 525–576 (2011; Zbl 1234.53014)], and H. Weiss [ibid. 71, No. 3, 437–506 (2005; Zbl 1098.53038)].
When the cone angles are less than \(\pi\), the topology of the singular locus does not change under deformation. This paper however shows that when the angles are larger than \(\pi\), then there exist deformations such that the vertices of the singular locus of valency \(\geq 4\) can split along new singular edges. To describe such splittings one has to look at the link of a singular vertex (which is a sphere with cone points, one for each singular edge adjacent to the vertex). A splitting of the vertex is described in terms of curves in the link, so that the corresponding edges are split by this curve. The authors give sufficient conditions in terms of the geometry of this curve so that the splitting exists. Moreover they give precise parameters for the deformation space with this given topology, namely the cone angles of the old singular edges and the length of the new edges.
The result uses a cohomology theorem of Weiss, and the paper mainly deals with representation spaces. In particular, it relies on the results of W. M. Goldman [in: Dani, S. G. (ed.) et al., Algebraic groups and arithmetic. Proceedings of the international conference, Mumbai, India, December 17–22, 2001. New Delhi: Narosa Publishing House/Published for the Tata Institute of Fundamental Research. 375–407 (2004; Zbl 1089.53060)] on the complex symplectic structure of the representations of a surface in \(SL(2,\mathbb C)\) as well as the geometric interpretation of certain infinitesimal deformations.

MSC:
57M50 General geometric structures on low-dimensional manifolds
58D27 Moduli problems for differential geometric structures
53C35 Differential geometry of symmetric spaces
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