Mathematical Research Letters

Volume 12 (2005)

Number 1

Hyperideal circle patterns

Pages: 85 – 102



J. M. Schlenker (Laboritorie Emile Picard)


A "hyperideal circle pattern” in $S^2$ is a finite family of oriented circles, similar to an "usual” circle pattern but such that the closed disks bounded by the circles do not cover the whole sphere. Hyperideal circle patters 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 graphics and intersection angles of hyperideal circle patterns in the sphere, the torus, and in higher genus surfaces. It it a consequence of a more general result, describing the hyperideal circle patternsin the boundaries of geometrically finite hyperbolic 3-manifolds (for the corresponding $\mathbb{C}P^1$-structures). this more general statement is obtained as a consequence of a theorem of Otal [Ota94, BO01] on the pleating laminations of the convex cores of geometrically finite hyperbolic manifolds.

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