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Three techniques for obtaining algebraic circle packings | | Larsen Louder
; Andrey M. Mishchenko
; Juan Souto
; | | Date: |
4 Apr 2013 | | Abstract: | The main purpose of this article is to demonstrate three techniques for
proving algebraicity statements about circle packings. We give proofs of three
related theorems: (1) that every finite simple planar graph is the contact
graph of a circle packing on the Riemann sphere, equivalently in the complex
plane, all of whose tangency points, centers, and radii are algebraic, (2) that
every flat conformal torus which admits a circle packing whose contact graph
triangulates the torus has algebraic modulus, and (3) that if R is a compact
Riemann surface of genus at least 2, having constant curvature -1, which admits
a circle packing whose contact graph triangulates R, then R is isomorphic to
the quotient of the hyperbolic plane by a subgroup of PSL_2(real algebraic
numbers). The statement (1) is original, while (2) and (3) have been previously
proved in the Ph.D. thesis of McCaughan.
Our first proof technique is to apply Tarski’s Theorem, a result from model
theory, which says that if an elementary statement in the theory of real-closed
fields is true over one real-closed field, then it is true over any real closed
field. This technique works to prove (1) and (2). Our second proof technique is
via an algebraicity result of Thurston on finite co-volume discrete subgroups
of the orientation-preserving-isometry group of hyperbolic 3-space. This
technique works to prove (1). Our first and second techniques had not
previously been applied in this area. Our third and final technique is via a
lemma in real algebraic geometry, and was previously used by McCaughan to prove
(2) and (3). We show that in fact it may be used to prove (1) as well. | | Source: | arXiv, 1304.1488 | | Services: | Forum | Review | PDF | Favorites | |
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