Laminations and 2-filling rays on infinite type surfaces
Annales de l'Institut Fourier, Volume 73 (2023) no. 6, pp. 2305-2369.

The loop graph of an infinite type surface is an infinite diameter hyperbolic graph first studied in detail by Juliette Bavard. An important open problem in the study of infinite type surfaces is to describe the boundary of the loop graph as a space of geodesic laminations. We approach this problem by constructing the first examples of 2-filling rays on infinite type surfaces. Such rays have strong filling properties while failing to correspond to points on the boundary of the loop graph. As such they may be thought of as “fake boundary points.” We give multiple constructions using both a hands-on combinatorial approach and an approach using train tracks and automorphisms of flat surfaces. In addition, our approaches are sufficiently robust to describe all 2-filling rays with certain other basic properties as well as to produce uncountably many distinct mapping class group orbits.

Le graphe des lacets d’une surface de type infini est un graphe hyperbolique de diamètre infini étudié en détail pour la première fois par Juliette Bavard. Un problème ouvert important dans l’étude des surfaces de type infini est de décrire le bord du graphe des lacets comme un espace de laminations géodésiques. Nous abordons ce problème en construisant les premiers exemples de rayons de 2-remplissage sur des surfaces de type infini. De tels rayons ont des propriétés fortes de remplissage tout en ne correspondant pas à des points au bord du graphe des lacets. En tant que tels, ils peuvent être considérés comme des « faux points au bord » . Nous donnons plusieurs constructions en utilisant à la fois une approche combinatoire pratique et une autre approche en utilisant les voies ferrées et les automorphismes des surfaces planes. En outre, nos approches sont suffisamment robustes pour décrire tous les rayons de 2-remplissage avec certaines autres propriétés de base ainsi que pour produire un ensemble infini non dénombrable de l’orbite de groupe modulaire.

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DOI: 10.5802/aif.3581
Classification: 57M60, 57N05
Keywords: Big mapping class groups, ray graph, train track.
Mot clés : Gros groupe modulaire, graphe des rayons, voie ferrée.
Chen, Lvzhou 1; Rasmussen, Alexander J. 2

1 Department of Mathematics Purdue University West Lafayette, IN (USA)
2 Department of Mathematics University of Utah Salt Lake City, UT (USA)
License: CC-BY-ND 4.0
Copyrights: The authors retain unrestricted copyrights and publishing rights
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Chen, Lvzhou; Rasmussen, Alexander J. Laminations and $2$-filling rays on infinite type surfaces. Annales de l'Institut Fourier, Volume 73 (2023) no. 6, pp. 2305-2369. doi : 10.5802/aif.3581. https://aif.centre-mersenne.org/articles/10.5802/aif.3581/

[1] Bavard, Juliette Hyperbolicité du graphe des rayons et quasi-morphismes sur un gros groupe modulaire, Geom. Topol., Volume 20 (2016) no. 1, pp. 491-535 | DOI | MR | Zbl

[2] Bavard, Juliette An infinite clique of high-filling rays in the plane minus a Cantor set (2021) (https://arxiv.org/abs/2106.01962)

[3] Bavard, Juliette; Walker, Alden The Gromov boundary of the ray graph, Trans. Amer. Math. Soc., Volume 370 (2018) no. 11, pp. 7647-7678 | DOI | MR

[4] Bavard, Juliette; Walker, Alden Two simultaneous actions of big mapping class groups (2018) (https://arxiv.org/abs/1806.10272)

[5] Bonahon, Francis Closed Curves on Surfaces, Available at https://dornsife.usc.edu/francis-bonahon/preprints/, 2000

[6] Calegari, Danny Big mapping class groups and dynamics, 2009 (https://lamington.wordpress.com/2009/06/22/big-mapping-class-groups-and-dynamics/)

[7] Calegari, Danny; Chen, Lvzhou Big mapping class groups and rigidity of the simple circle, Ergodic Theory Dynam. Systems, Volume 41 (2021) no. 7, pp. 1961-1987 | DOI | MR | Zbl

[8] de Carvalho, André; Hall, Toby Unimodal generalized pseudo-Anosov maps, Geom. Topol., Volume 8 (2004), pp. 1127-1188 | DOI | MR | Zbl

[9] Farb, Benson; Margalit, Dan A primer on mapping class groups, Princeton Mathematical Series, 49, Princeton University Press, Princeton, NJ, 2012, xiv+472 pages | MR | Zbl

[10] Fathi, Albert; Laudenbach, François; Poénaru, Valentin Thurston’s work on surfaces, Mathematical Notes, 48, Princeton University Press, Princeton, NJ, 2012, xvi+254 pages | MR | Zbl

[11] Klarreich, Erica The boundary at infinity of the curve complex and the relative Teichmüller space (1999) (https://arxiv.org/abs/1803.10339)

[12] Moore, R. L. Concerning upper semi-continuous collections of continua, Trans. Amer. Math. Soc., Volume 27 (1925) no. 4, pp. 416-428 | DOI | MR

[13] Penner, R. C.; Harer, J. L. Combinatorics of train tracks, Annals of Mathematics Studies, 125, Princeton University Press, Princeton, NJ, 1992, xii+216 pages | DOI | MR | Zbl

[14] Schaffer-Cohen, Anschel Graphs of curves and arcs quasi-isometric to big mapping class groups (2020) (https://arxiv.org/abs/2006.14760)

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