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Thurston’s spinning construction and solutions to the hyperbolic gluing equations for closed hyperbolic 3-manifolds. (English) Zbl 1272.57004

The paper under review uses Thurston’s spinning to construct the hyperbolic structure of a hyperbolic closed 3-manifold starting from an essential triangulation. A semisimplicial triangulation is called essential if no edge is a nulhomotopic loop. Notice that after subdivision every triangulation is essential.
Starting from a triangulation and a solution of the gluing equation, each simplex is realized as a hyperbolic simplex. W. P. Thurston’s spinning construction was introduced in [Ann. Math. (2) 124, 203–246 (1986; Zbl 0668.57015)], and it pushes the vertices of a lift of a simplex in hyperbolic space to an ideal simplex. Thus in the compact manifold the ends of this ideal simplex spin in the manifold. The authors prove that, if one starts from the right solution or from the simplices in the hyperbolic manifold, it corresponds to a noncomplete hyperbolic structure whose completion gives the initial hyperbolic manifold.
Thuston’s spinning construction has a well defined volume, which equals the volume of the initial holonomy of the solution to the gluing equations. Thus it is always bounded above (in absolute value) by the volume of the hyperbolic manifold, and it equals the volume of the hyperbolic manifold iff this construction gives the hyperbolic structure, using volume rigidity.
Since for Seifert manifolds the volume is always zero, and the set of solutions of the gluing equations has finitely many components, this construction also gives an algorithm to decide whether an irreducible atoroidal manifold is hyperbolic or small Seifert.

MSC:

57M25 Knots and links in the \(3\)-sphere (MSC2010)
57N10 Topology of general \(3\)-manifolds (MSC2010)

Citations:

Zbl 0668.57015
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References:

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