Hyperideal circle patterns

Details Hyperideal circle patterns Hyperideal circle patterns

2004-07-03

arxiv.org/abs/math/0407043v2

Math. Res. Lett., 12 (2005):1, 85-112.

Mathematics

Geometric Topology

11 pages, 2 figures. Updated versions will be posted on http://picard.ups-tlse.fr/~schlenker Revised version: some corrections

Scientific paper

A ``hyperideal circle pattern'' in $S^2$ is a finite family of oriented circles, similar to the ``usual'' circle patterns but such that the closed disks bounded by the circles do not cover the whole sphere. Hyperideal circle patterns are directly related to hyperideal hyperbolic polyhedra, and also to circle packings. To each hyperideal circle pattern, one can associate an incidence graph and a set of intersection angles. We characterize the possible incidence graphs and intersection angles of hyperideal circle patterns in the sphere, the torus, and in higher genus surfaces. It is a consequence of a more general result, describing the hyperideal circle patterns in the boundaries of geometrically finite hyperbolic 3-manifolds (for the corresponding $\C P^1$-structures). This more general statement is obtained as a consequence of a theorem of Otal \cite{otal,bonahon-otal} on the pleating laminations of the convex cores of geometrically finite hyperbolic manifolds.

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