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Local-global principles in circle packings
| Content Provider | Scilit |
|---|---|
| Author | Fuchs, Elena Stange, Katherine E. Zhang, Xin |
| Copyright Year | 2019 |
| Description | Journal: Compositio Mathematica 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 $\operatorname{PSL}_{2}(\mathbb{Z})$ . |
| Ending Page | 1170 |
| Starting Page | 1118 |
| ISSN | 00221295 |
| e-ISSN | 15705846 |
| DOI | 10.1112/s0010437x19007139 |
| Journal | Compositio Mathematica |
| Issue Number | 6 |
| Volume Number | 155 |
| Language | English |
| Publisher | Wiley-Blackwell |
| Publisher Date | 2019-06-01 |
| Access Restriction | Open |
| Subject Keyword | Journal: Compositio Mathematica Mathematical Physics Circle Packings Local Global Principle Local To Global |
| Content Type | Text |
| Resource Type | Article |
| Subject | Algebra and Number Theory |
