Geometry & Topology Monographs 1 (1998), The Epstein Birthday Schrift, paper no. 25, pages 511-549.

Shapes of polyhedra and triangulations of the sphere

William P Thurston

Abstract. The space of shapes of a polyhedron with given total angles less than 2\pi at each of its n vertices has a Kaehler metric, locally isometric to complex hyperbolic space CH^{n-3}. The metric is not complete: collisions between vertices take place a finite distance from a nonsingular point. The metric completion is a complex hyperbolic cone-manifold. In some interesting special cases, the metric completion is an orbifold. The concrete description of these spaces of shapes gives information about the combinatorial classification of triangulations of the sphere with no more than 6 triangles at a vertex.

Keywords. Polyhedra, triangulations, configuration spaces, braid groups, complex hyperbolic orbifolds

AMS subject classification. Primary: 51M20. Secondary: 51F15, 20H15, 57M50.

E-print: arXiv:math.GT/9801088

Submitted: 15 November 1997. (Revised: 27 November 1998.) Published: 4 December 1998.

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William P Thurston
Mathematics Department
University of California at Davis
Davis, CA 95616, USA

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