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One of the most useful tools for studying hyperbolic 3-manifolds is the technique of ideal triangulations, introduced by Thurston to understand the hyperbolic structure of the complement of the figure-eight knot. If a 3-manifold is equipped with an ideal triangulation, one tries to construct a hyperbolic structure on the manifold by defining the structure on each tetrahedron and then by requiring global compatibility. Straight hyperbolic ideal tetrahedra are parameterized by complex numbers with positive imaginary part, and compatibility translates into algebraic equations in the parameters. In most of this work we consider generalized solutions of the compatibility equations, without restrictions on the imaginary part, and we investigate which such solutions define a global structure. We begin by facing, and essentially solving in full generality, the analogous two-dimensional Euclidean problem. We then study explicit examples of cusped 3-manifold, exhibiting a variety of different phenomena. Finally, we introduce a certain notion of geometric solution, we prove existence and uniqueness results for such solutions, and we characterize them in terms of the volume of their (suitably defined) holonomy. The last part of the thesis is devoted to the study of the volume function on the character variety of a hyperbolic 3-manifold. Our main result here is the proof of a rigidity theorem for representations of maximal volume.
Résumé
One of the most useful tools for studying hyperbolic 3-manifolds is the technique of ideal triangulations, introduced by Thurston to understand the hyperbolic structure of the complement of the figure-eight knot. If a 3-manifold is equipped with an ideal triangulation, one tries to construct a hyperbolic structure on the manifold by defining the structure on each tetrahedron and then by requiring global compatibility. Straight hyperbolic ideal tetrahedra are parameterized by complex numbers with positive imaginary part, and compatibility translates into algebraic equations in the parameters. In most of this work we consider generalized solutions of the compatibility equations, without restrictions on the imaginary part, and we investigate which such solutions define a global structure. We begin by facing, and essentially solving in full generality, the analogous two-dimensional Euclidean problem. We then study explicit examples of cusped 3-manifold, exhibiting a variety of different phenomena. Finally, we introduce a certain notion of geometric solution, we prove existence and uniqueness results for such solutions, and we characterize them in terms of the volume of their (suitably defined) holonomy. The last part of the thesis is devoted to the study of the volume function on the character variety of a hyperbolic 3-manifold. Our main result here is the proof of a rigidity theorem for representations of maximal volume.
