Large-scale geometry of the saddle connection graph
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- by Valentina Disarlo, Huiping Pan, Anja Randecker and Robert Tang PDF
- Trans. Amer. Math. Soc. 374 (2021), 8101-8129
Abstract:
We prove that the saddle connection graph associated to any half-translation surface is $4$–hyperbolic and uniformly quasi-isometric to the regular countably infinite-valent tree. Consequently, the saddle connection graph is not quasi-isometrically rigid. We also characterise its Gromov boundary as the set of straight foliations with no saddle connections. In our arguments, we give a generalisation of the unicorn paths in the arc graph which may be of independent interest.References
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Additional Information
- Valentina Disarlo
- Affiliation: Mathematisches Institut, Universität Heidelberg, Im Neuenheimer Feld 205, 69120 Heidelberg, Germany
- MR Author ID: 983316
- Email: vdisarlo@mathi.uni-heidelberg.de
- Huiping Pan
- Affiliation: Department of Mathematics, Jinan University, 601 Huangpu Road, Tianhe, Guangzhou, Guangdong 510632, People’s Republic of China
- MR Author ID: 1173565
- ORCID: 0000-0001-9892-8074
- Email: panhp@jnu.edu.cn
- Anja Randecker
- Affiliation: Mathematisches Institut, Universität Heidelberg, Im Neuenheimer Feld 205, 69120 Heidelberg, Germany
- MR Author ID: 1284825
- ORCID: 0000-0001-5991-4542
- Email: randecker@mathi.uni-heidelberg.de
- Robert Tang
- Affiliation: Department of Pure Mathematics, Xi’an Jiaotong–Liverpool University, 111 Ren’ai Road, Suzhou Industrial Park, Suzhou, Jiangsu 215123, People’s Republic of China
- MR Author ID: 1000078
- ORCID: 0000-0001-7021-4117
- Email: robert.tang@xjtlu.edu.cn
- Received by editor(s): January 20, 2021
- Received by editor(s) in revised form: March 15, 2021
- Published electronically: August 18, 2021
- Additional Notes: The first author acknowledges support from the Olympia Morata Habilitation Programme of Universität Heidelberg and from the European Research Council under ERC-Consolidator grant 614733 (GEOMETRICSTRUCTURES). The second author acknowledges the support of National Natural Science Foundation of China NSFC 11901241. The third author acknowledges the support of NYU-ECNU Institute of Mathematical Sciences at NYU Shanghai. The fourth author acknowledges support from a JSPS KAKENHI Grant-in-Aid for Early-Career Scientists (No. 19K14541)
- © Copyright 2021 by the authors
- Journal: Trans. Amer. Math. Soc. 374 (2021), 8101-8129
- MSC (2020): Primary 57K20; Secondary 20F65, 53C10
- DOI: https://doi.org/10.1090/tran/8448
- MathSciNet review: 4328693