The Gromov boundary of the ray graph
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- by Juliette Bavard and Alden Walker PDF
- Trans. Amer. Math. Soc. 370 (2018), 7647-7678 Request permission
Abstract:
The ray graph is a Gromov-hyperbolic graph on which the mapping class group of the plane minus a Cantor set acts by isometries. We give a description of the Gromov boundary of the ray graph in terms of cliques of long rays on the plane minus a Cantor set. As a consequence, we prove that the Gromov boundary of the ray graph is homeomorphic to a quotient of a subset of the circle.References
- Juliette Bavard, Hyperbolicité du graphe des rayons et quasi-morphismes sur un gros groupe modulaire, Geom. Topol. 20 (2016), no. 1, 491–535 (French, with English and French summaries). MR 3470720, DOI 10.2140/gt.2016.20.491
- François Béguin, Sylvain Crovisier, and Frédéric Le Roux, Construction of curious minimal uniquely ergodic homeomorphisms on manifolds: the Denjoy-Rees technique, Ann. Sci. École Norm. Sup. (4) 40 (2007), no. 2, 251–308 (English, with English and French summaries). MR 2339286, DOI 10.1016/j.ansens.2007.01.001
- M. Bonk and O. Schramm, Embeddings of Gromov hyperbolic spaces, Geom. Funct. Anal. 10 (2000), no. 2, 266–306. MR 1771428, DOI 10.1007/s000390050009
- Martin R. Bridson and André Haefliger, Metric spaces of non-positive curvature, Grundlehren der mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 319, Springer-Verlag, Berlin, 1999. MR 1744486, DOI 10.1007/978-3-662-12494-9
- Danny Calegari, Big mapping class groups and dynamics, https://lamington.wordpress. com/2009/06/22/big-mapping-class-groups-and-dynamics/
- Louis Funar, Christophe Kapoudjian, and Vlad Sergiescu, Asymptotically rigid mapping class groups and Thompson’s groups, Handbook of Teichmüller theory. Volume III, IRMA Lect. Math. Theor. Phys., vol. 17, Eur. Math. Soc., Zürich, 2012, pp. 595–664. MR 2952772, DOI 10.4171/103-1/11
- Sebastian Hensel, Piotr Przytycki, and Richard C. H. Webb, 1-slim triangles and uniform hyperbolicity for arc graphs and curve graphs, J. Eur. Math. Soc. (JEMS) 17 (2015), no. 4, 755–762. MR 3336835, DOI 10.4171/JEMS/517
- Ilya Kapovich and Nadia Benakli, Boundaries of hyperbolic groups, Combinatorial and geometric group theory (New York, 2000/Hoboken, NJ, 2001) Contemp. Math., vol. 296, Amer. Math. Soc., Providence, RI, 2002, pp. 39–93. MR 1921706, DOI 10.1090/conm/296/05068
- Erica Klarreich, The boundary at infinity of the curve complex and the relative Teichmüller Space, preprint
- Frédéric Le Roux, Étude topologique de l’espace des homéomorphismes de Brouwer. I, Topology 40 (2001), no. 5, 1051–1087 (French, with English and French summaries). MR 1860541, DOI 10.1016/S0040-9383(00)00024-0
- Edwin E. Moise, Geometric topology in dimensions $2$ and $3$, Graduate Texts in Mathematics, Vol. 47, Springer-Verlag, New York-Heidelberg, 1977. MR 0488059
- Witsarut Pho-On, Infinite unicorn paths and Gromov boundaries, Groups Geom. Dyn. 11 (2017), no. 1, 353–370. MR 3641844, DOI 10.4171/GGD/399
Additional Information
- Juliette Bavard
- Affiliation: Institut de Mathématiques de Jussieu-Paris Rive Gauche, UPMC
- Address at time of publication: Univ Rennes, CNRS, IRMAR - UMR 6625, F-35000 Rennes, France
- MR Author ID: 1154220
- Email: juliette.bavard@univ-rennes1.fr
- Alden Walker
- Affiliation: Center for Communications Research, La Jolla, California 92121
- MR Author ID: 925092
- Email: akwalke@ccrwest.org
- Received by editor(s): November 2, 2016
- Received by editor(s) in revised form: January 19, 2017
- Published electronically: April 17, 2018
- © Copyright 2018 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 370 (2018), 7647-7678
- MSC (2010): Primary 20F65, 37E30, 57M60, 05C63
- DOI: https://doi.org/10.1090/tran/7204
- MathSciNet review: 3852444