SPACES OF POLYGONS TO SPACES OF POLYHEDRA FOLLOWING BAVARD, GHYS AND THURSTON

Francois Fillastre (1, 2)

1 LEP - Laboratoire Émile Picard
2 Institut de mathématique - Institut de mathématique

In their article on polygons and hyperbolic polyhedra (\cite{BavardGyhs}), C. Bavard and E. Ghys built in a simple way examples of Im Hof orthoschems (particular cases of Coxeter polyhedra).This construction relies on a parametrization of polygons in the plane by hyperbolic polyhedra.In his article on shapes of polyhedra and triangulations of the sphere (\cite{Thurart1}), W.P. Thurston studies the sets of euclidean metrics with cone singularities on the sphere. He parametrizes these sets by complex hyperbolic cone-manifolds, and studies when they are orbifolds.We will consider hyperbolic cone-manifolds built in a canonical way from Bavard and Ghys' polyhedra.We will show that they are real forms of Thurston's cone-manifolds.

Mots clés: cone-variété; orbifold; singularité conique; métrique sur la sphère; cocycle; polygone; polyèdre hyperbolique

Domaines: Géométrie métrique [math.MG]

Francois Fillastre. SPACES OF POLYGONS TO SPACES OF POLYHEDRA FOLLOWING BAVARD, GHYS AND THURSTON. 2003. ⟨hal-00000544v1⟩