Sur la rigidit\'e de poly\`edres hyperboliques en dimension 3 : cas de volume fini, cas hyperid\'eal, cas fuchsien

Abstract

A hyperbolic semi-ideal polyedron is a polyedron whose vertices lie inside the hyperbolic space $\mathbf{H}^{3}$ or at infinity. A hyperideal polyedron is, in the projective model, the intersection of $\mathbf{H}^{3}$ with a projective polyhedron whose vertices all lie outside of $\mathbf{H}^{3}$, and whose edges all meet $\mathbf{H}^{3}$. We classify semi-ideal polyhedra in terms of their dual metric, using the results of Rivin and Hodgson in \cite{comp} et \cite{idea}. This result is used to obtain the classification of hyperideal polyhedra in terms of their combinatorial type and their dihedral angles. These two results are generalized to the case of fuchsian polyhedra.