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Regenerating hyperbolic cone structures from Nil. (English) Zbl 1032.57015
Let \(M\) be a 3-dimensional \(Nil\)-orbifold, whose singular set is a circle transverse to the Seifert fibration. The first result here is that \(M\) is the limit, after rescaling, of a sequence of hyperbolic cone-manifolds \((M_\alpha)_{\alpha\in (\pi-\epsilon, \pi)}\), with the same singular set as \(M\) and with cone angle \(\alpha\). When \(\alpha\rightarrow \pi\), the limit of \((M_\alpha)_{\alpha\in (\pi-\epsilon, \pi)}\) without scaling is a Euclidean 2-dimensional orbifold; the scaling which is needed is by a factor \((\pi-\alpha)^{-1/3}\) in the horizontal directions, and \((\pi-\alpha)^{-2/3}\) in the vertical direction.
This statement is actually a consequence of a careful study of the space of geometric structures near \(M\), parametrized by their Dehn-filling coefficients. Near \(M\), the space of Dehn-filling coefficients has 3 open components, one of which corresponds to hyperbolic structures and the other to spherical structures, with a curve of Euclidean structures in between.
As another consequence, some spherical cone-manifolds structures are not locally rigid: there are other cone-manifolds which are arbitrarily close, with the same underlying topology and the same cone angles. This corresponds to a ”folding” in the description above, with some Dehn filling coefficients corresponding to two different spherical structures.
The proof rests on a local study of the representations of the underlying space of \(M\) near its \(Nil\)-orbifold structure.

57M50 General geometric structures on low-dimensional manifolds
57N10 Topology of general \(3\)-manifolds (MSC2010)
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