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 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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• On the Moduli Space of Singular Euclidean Surfaces
• On the Geometry of the Moduli Space of Real Binary Octics
• Complex hyperbolic cone structures on the configuration spaces
• The Geometry of Polygons in R^5 and Quaternions
• Configuration spaces of points on the circle and hyperbolic Dehn fillings, II
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Cited by in corpus (3)

• Shapes of polyhedra, mixed volumes, and hyperbolic geometry
• Small deformations of polygons
• Dual-cyclic polytopes of convex planar polygons with fixed vertex angles