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The set of circle packing points in the Teichmüller space of a surface of finite conformal type is dense

Published online by Cambridge University Press:  24 October 2008

Philip L. Bowers  and
Kenneth Stephenson
Philip L. Bowers
Affiliation:
Department of Mathematics, The Florida State University, Tallahassee, FL 32306-3027, U.S.A.
Kenneth Stephenson
Affiliation:
Department of Mathematics, The University of Tennessee, Knoxville, TN 37999-1300, U.S.A.
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Extract

W. Thurston initiated interest in circle packings with his provocative suggestion at the International Symposium in Celebration of the Proof of the Bieberbach Conjecture (Purdue University, 1985) that a result of Andreev[2] had an interpretation in terms of circle packings that could be applied systematically to construct geometric approximations of classical conformal maps. Rodin and Sullivan [11] verified Thurston's conjecture in the setting of hexagonal packings, and more recently Stephenson [12] has announced a proof for more general combinatorics. Inspired by Thurston's work and motivated by the desire to discover and exploit discrete versions of classical results in complex variable theory, Beardon and Stephenson [4, 5] initiated a study of the geometry of circle packings, particularly in the hyperbolic setting. This topic is a recent example among many of the beautiful and sometimes unexpected interplay between Geometry, Topology, and Cornbinatorics that is evident in much of the topological research of the past decade, and that has its roots in the seminal work of the great geometrically-minded mathematicians – Riemann, Klein, Poincaré – of the last century. A somewhat surprising example of this interplay concerns us here; namely, the fact that the combinatorial information encoded in a simplicial triangulation of a topological surface can determine its geometry.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 111 , Issue 3 , May 1992 , pp. 487 - 513
Copyright
Copyright © Cambridge Philosophical Society 1992

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References

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