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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.

##### MSC:
 57M50 General geometric structures on low-dimensional manifolds 57N10 Topology of general $$3$$-manifolds (MSC2010)
##### Keywords:
Hyperbolic structure; cone 3-manifolds; local rigidity
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