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Article: Local-global principles in circle packings

TitleLocal-global principles in circle packings
Authors
Keywordslocal-to-global
Kleinian group
circle method
Apollonian circle packing
Issue Date2019
PublisherFoundation Compositio Mathematica. The Journal's web site is located at https://compositio.nl/compositio.html
Citation
Compositio Mathematica, 2019, v. 155 n. 6, p. 1118-1170 How to Cite?
AbstractWe 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 A⩽PSL2(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 A possesses a spectral gap property, which we prove for any infinite-covolume, geometrically finite, Zariski dense Kleinian group in PSL2(OK) containing a Zariski dense subgroup of PSL2(Z) .
Persistent Identifierhttp://hdl.handle.net/10722/289728
ISSN
2019 Impact Factor: 1.2
2015 SCImago Journal Rankings: 2.965

 

DC FieldValueLanguage
dc.contributor.authorFuchs, E-
dc.contributor.authorStange, KE-
dc.contributor.authorZhang, X-
dc.date.accessioned2020-10-22T08:16:37Z-
dc.date.available2020-10-22T08:16:37Z-
dc.date.issued2019-
dc.identifier.citationCompositio Mathematica, 2019, v. 155 n. 6, p. 1118-1170-
dc.identifier.issn0010-437X-
dc.identifier.urihttp://hdl.handle.net/10722/289728-
dc.description.abstractWe 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 A⩽PSL2(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 A possesses a spectral gap property, which we prove for any infinite-covolume, geometrically finite, Zariski dense Kleinian group in PSL2(OK) containing a Zariski dense subgroup of PSL2(Z) .-
dc.languageeng-
dc.publisherFoundation Compositio Mathematica. The Journal's web site is located at https://compositio.nl/compositio.html-
dc.relation.ispartofCompositio Mathematica-
dc.rightsCompositio Mathematica. Copyright © Foundation Compositio Mathematica.-
dc.subjectlocal-to-global-
dc.subjectKleinian group-
dc.subjectcircle method-
dc.subjectApollonian circle packing-
dc.titleLocal-global principles in circle packings-
dc.typeArticle-
dc.identifier.emailZhang, X: xz27@hku.hk-
dc.identifier.authorityZhang, X=rp02608-
dc.description.naturelink_to_subscribed_fulltext-
dc.identifier.doi10.1112/S0010437X19007139-
dc.identifier.scopuseid_2-s2.0-85080869231-
dc.identifier.hkuros317202-
dc.identifier.volume155-
dc.identifier.issue6-
dc.identifier.spage1118-
dc.identifier.epage1170-
dc.publisher.placeUnited Kingdom-

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