From spaces of polygons to spaces of polyhedra following Bavard, Ghys and Thurston
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 mixedvolumes. The gluing of these polyhedra can be isometrically embedded into complex hyperbolic conemanifolds 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.
 Publication:

arXiv Mathematics eprints
 Pub Date:
 August 2003
 DOI:
 10.48550/arXiv.math/0308187
 arXiv:
 arXiv:math/0308187
 Bibcode:
 2003math......8187F
 Keywords:

 Mathematics  Metric Geometry