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Local-global principles in circle packings

Part of: Other groups of matrices Diophantine equations Discrete geometry Riemann surfaces Lie groups

Published online by Cambridge University Press:  20 May 2019

Elena Fuchs ,
Katherine E. Stange  and
Xin Zhang
Elena Fuchs
Affiliation:
Department of Mathematics, UC Davis, One Shields Avenue, Davis, CA 95616, USA email efuchs@math.ucdavis.edu
Katherine E. Stange
Affiliation:
Department of Mathematics, University of Colorado, Campus Box 395, Boulder, Colorado 80309-0395, USA email kstange@math.colorado.edu
Xin Zhang
Affiliation:
Department of Mathematics, University of Illinois, 1409 West Green Street, Urbana, IL 61801, USA email xz87@illinois.edu
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Abstract

We generalize work by Bourgain and Kontorovich [On the local-global conjecture for integral Apollonian gaskets, Invent. Math. 196 (2014), 589–650] and Zhang [On the local-global principle for integral Apollonian 3-circle packings, J. Reine Angew. Math. 737, (2018), 71–110], proving an almost local-to-global property for the curvatures of certain circle packings, to a large class of Kleinian groups. Specifically, we associate in a natural way an infinite family of integral packings of circles to any Kleinian group ${\mathcal{A}}\leqslant \text{PSL}_{2}(K)$ satisfying certain conditions, where $K$ is an imaginary quadratic field, and show that the curvatures of the circles in any such packing satisfy an almost local-to-global principle. A key ingredient in the proof is that ${\mathcal{A}}$ possesses a spectral gap property, which we prove for any infinite-covolume, geometrically finite, Zariski dense Kleinian group in $\operatorname{PSL}_{2}({\mathcal{O}}_{K})$ containing a Zariski dense subgroup of $\operatorname{PSL}_{2}(\mathbb{Z})$.

Keywords

local-to-globalKleinian groupcircle methodApollonian circle packing

MSC classification

Primary: 52C26: Circle packings and discrete conformal geometry 30F40: Kleinian groups 11D85: Representation problems
Secondary: 20H10: Fuchsian groups and their generalizations 22E40: Discrete subgroups of Lie groups

Information

Type
Research Article
Information
Compositio Mathematica , Volume 155 , Issue 6 , June 2019 , pp. 1118 - 1170
Copyright
© The Authors 2019 

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Footnotes

Fuchs has been supported by NSF DMS-1501970, the Sloan Foundation, and the BSF. Stange has been supported by NSF EAGER DMS-1643552 and NSF CAREER CNS-1652238.

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