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Shapes of polyhedra and triangulations of the sphere. (English) Zbl 0931.57010
Rivin, Igor (ed.) et al., The Epstein Birthday Schrift dedicated to David Epstein on the occasion of his 60th birthday. Warwick: University of Warwick, Institute of Mathematics, Geom. Topol. Monogr. 1, 511-549 (1998).
The author develops a global theory to describe all triangulations of the sphere $$S^2$$ such that each vertex has six or fewer triangles at any vertex. This is closely related to negative curvature and involves studying the space of shapes of a polyhedron with given total angles less than $$2\pi$$ at each of its $$n$$ vertices. The author shows that this space has a Kähler metric, locally isometric to the complex hyperbolic space $$\mathbb C\mathbb H^{n-3}$$. The metric is not complete. The metric completion appears to be a complex hyperbolic cone-manifold, and in some interesting cases it is an orbifold. The concrete description of these spaces of shapes allows the author to obtain information about the combinatorial classification of triangulations of the sphere with no more than six triangles at a vertex.
For the entire collection see [Zbl 0901.00063].

##### MSC:
 57M50 General geometric structures on low-dimensional manifolds 20H15 Other geometric groups, including crystallographic groups
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