Hyperideal polyhedra in hyperbolic 3-space

Bao, Xiliang; Bonahon, Francis

Bulletin de la Société Mathématique de France, Tome 130 (2002) no. 3, pp. 457-491.

Un polyèdre hyperidéal est un polyèdre non-compact de l’espace hyperbolique ${ℍ}^3$ de dimension $3$ qui, dans le modèle projectif pour ${ℍ}^3\subset {\mathrm{ℝ\mathbb{P}}}^3$, est simplement l’intersection de ${ℍ}^3$ avec un polyèdre projectif dont les sommets sont tous en dehors de ${ℍ}^3$ et dont toutes les arêtes rencontrent ${ℍ}^3$. Nous classifions ces polyèdres hyperidéaux, à isométrie de ${ℍ}^3$ près, en fonction de leur type combinatoire et de leurs angles diédraux.

A hyperideal polyhedron is a non-compact polyhedron in the hyperbolic $3$-space ${ℍ}^3$ which, in the projective model for ${ℍ}^3\subset {\mathrm{ℝ\mathbb{P}}}^3$, is just the intersection of ${ℍ}^3$ with a projective polyhedron whose vertices are all outside ${ℍ}^3$ and whose edges all meet ${ℍ}^3$. We classify hyperideal polyhedra, up to isometries of ${ℍ}^3$, in terms of their combinatorial type and of their dihedral angles.

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