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

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

2003-08-20

arxiv.org/abs/math/0308187v4

Mathematics

Metric Geometry

Scientific paper

After work of W. P. Thurston, C. Bavard and \'E. 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.

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