From spaces of polygons to spaces of polyhedra following Bavard, Ghys and Thurston

2003/08/20 by Francois Fillastre, Fillastre, Francois
Mathematics · #FOS: Mathematics #Metric Geometry (math.MG) #math.MG

paper · pdf · doi:10.48550/arxiv.math/0308187

arxiv created 2009/09/07 · arxiv updated 2009/12/01

Abstract

After work of W. P. Thurston, C. Bavard and É. Ghys constructed particular hyperbolic polyhedra from spaces of deformations of Euclidean polygons. We present this construction as a straightforward consequence of the theory of mixed-volumes. The gluing of these polyhedra can be isometrically embedded into complex hyperbolic cone-manifolds constructed by Thurston from spaces of deformations of Euclidean polyhedra. It is then possible to deduce the metric structure of the spaces of polygons embedded in complex hyperbolic orbifolds discovered by P. Deligne and G. D. Mostow.

Related

• Shapes of polyhedra and triangulations of the sphere
• A remark on spaces of flat metrics with cone singularities of constant\n sign curvatures
• Intrinsic geometry of convex ideal polyhedra in hyperbolic 3-space
• Constructing hyperbolic polyhedra using Newton's Method
• Sur la rigidité de polyèdres hyperboliques en dimension 3 : cas de volume fini, cas hyperidéal, cas fuchsien