FROM 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 Coxeter polyhedra. This construction relies on a parametrization of polygons in theplane 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, and he constructs complex hyperbolic orbifolds. We will consider sets built in a canonical way from Bavard and Ghys' polyhedra. We will show that, thanks to a theoreme of Aleksandrov, they are real forms of Thurston's orbifolds.

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

Domaines: Géométrie métrique [math.MG]

Francois Fillastre. FROM SPACES OF POLYGONS TO SPACES OF POLYHEDRA FOLLOWING BAVARD, GHYS AND THURSTON. 2004. ⟨hal-00000544v2⟩