Géométrie hyperbolique effective et triangulations idéales canoniques en dimension trois

Abstract : We study certain decompositions of M into ideal polyhedra, where M is a cusped hyperbolic 3-manifold. A result of Epstein and Penner states that such a decomposition exists: in particular, the so-called Delaunay decomposition, which is canonical in a geometric sense.

In Chapter 1, we find the Delaunay decomposition for M a punctured-torus bundle over the circle. The method is to ``guess'' the combinatorics of the decomposition, then find positive dihedral angles for its combinatorial polyhedra: by a theorem of Rivin, any critical point of the volume functional in the deformation space of dihedral angles gives the hyperbolic metric. The inequalities involved in showing that such a critical point exists also imply that the decomposition is indeed Delaunay.

In Chapter 2, we extend the method to certain link complements (notably, 2-bridge links). In Chapter 3 we extend it to convex cores of quasifuchsian punctured-torus groups (here the decomposition is infinite, and has some non-polyhedral pieces). As a
corollary, we reprove the Pleating Lamination Theorem for punctured-torus groups. In Chapter 4, we partially extend the method to arborescent link complements: without finding critical points, we characterize hyperbolic arborescent links.

In Chapter 5, extending a proposition of Chapter 3, we show that certain Laurent polynomials, which generalize the Markoff numbers, have only positive coefficients.
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François Guéritaud. Géométrie hyperbolique effective et triangulations idéales canoniques en dimension trois. Mathématiques [math]. Université Paris Sud - Paris XI, 2006. Français. ⟨tel-00119465⟩

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