Uniform hyperbolicity of the graphs of nonseparating curves via bicorn curves

Rasmussen, Alexander

doi:10.1090/proc/14880

Uniform hyperbolicity of the graphs of nonseparating curves via bicorn curves
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by Alexander J. Rasmussen PDF

Proc. Amer. Math. Soc. 148 (2020), 2345-2357 Request permission

Abstract:

We show that the graphs of nonseparating curves for oriented finite ctype surfaces are uniformly hyperbolic. Our proof follows the proof of uniform hyperbolicity of the graphs of curves for closed surfaces due to Przytycki–Sisto, while introducing new arguments using homology to certify that certain curves are nonseparating. As demonstrated by Aramayona–Valdez, this also proves that the graph of nonseparating curves for any oriented infinite type surface with finite positive genus is hyperbolic.

References

Tarik Aougab, Uniform hyperbolicity of the graphs of curves, Geom. Topol. 17 (2013), no. 5, 2855–2875. MR 3190300, DOI 10.2140/gt.2013.17.2855
Javier Aramayona, Ariadna Fossas, and Hugo Parlier, Arc and curve graphs for infinite-type surfaces, Proc. Amer. Math. Soc. 145 (2017), no. 11, 4995–5006. MR 3692012, DOI 10.1090/proc/13608
Javier Aramayona and Ferrán Valdez, On the geometry of graphs associated to infinite-type surfaces, Math. Z. 289 (2018), no. 1-2, 309–322. MR 3803791, DOI 10.1007/s00209-017-1952-6
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
Juliette Bavard and Anthony Genevois, Big mapping class groups are not acylindrically hyperbolic, Math. Slovaca 68 (2018), no. 1, 71–76. MR 3764317, DOI 10.1515/ms-2017-0081
Jason Behrstock, Bruce Kleiner, Yair Minsky, and Lee Mosher, Geometry and rigidity of mapping class groups, Geom. Topol. 16 (2012), no. 2, 781–888. MR 2928983, DOI 10.2140/gt.2012.16.781
Jason A. Behrstock and Yair N. Minsky, Dimension and rank for mapping class groups, Ann. of Math. (2) 167 (2008), no. 3, 1055–1077. MR 2415393, DOI 10.4007/annals.2008.167.1055
Brian H. Bowditch, Uniform hyperbolicity of the curve graphs, Pacific J. Math. 269 (2014), no. 2, 269–280. MR 3238474, DOI 10.2140/pjm.2014.269.269
Jeffrey F. Brock, Richard D. Canary, and Yair N. Minsky, The classification of Kleinian surface groups, II: The ending lamination conjecture, Ann. of Math. (2) 176 (2012), no. 1, 1–149. MR 2925381, DOI 10.4007/annals.2012.176.1.1
Danny Calegari. Big mapping class groups and dynamics. https://lamington.wordpress.com/ 2009/06/22/big-mapping-class-groups-and-dynamics/, 2009.
Matt Clay, Kasra Rafi, and Saul Schleimer, Uniform hyperbolicity of the curve graph via surgery sequences, Algebr. Geom. Topol. 14 (2014), no. 6, 3325–3344. MR 3302964, DOI 10.2140/agt.2014.14.3325
F. Dahmani, V. Guirardel, and D. Osin, Hyperbolically embedded subgroups and rotating families in groups acting on hyperbolic spaces, Mem. Amer. Math. Soc. 245 (2017), no. 1156, v+152. MR 3589159, DOI 10.1090/memo/1156
Matthew Gentry Durham, Federica Fanoni, and Nicholas G. Vlamis, Graphs of curves on infinite-type surfaces with mapping class group actions, Ann. Inst. Fourier (Grenoble) 68 (2018), no. 6, 2581–2612 (English, with English and French summaries). MR 3897975
Ariadna Fossas and Hugo Parlier, Curve graphs on surfaces of infinite type, Ann. Acad. Sci. Fenn. Math. 40 (2015), no. 2, 793–801. MR 3409704, DOI 10.5186/aasfm.2015.4049
Ursula Hamenstädt, Hyperbolicity of the graph of nonseparating multicurves, Algebr. Geom. Topol. 14 (2014), no. 3, 1759–1778. MR 3212583, DOI 10.2140/agt.2014.14.1759
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
Howard Masur and Saul Schleimer, The geometry of the disk complex, J. Amer. Math. Soc. 26 (2013), no. 1, 1–62. MR 2983005, DOI 10.1090/S0894-0347-2012-00742-5
Howard A. Masur and Yair N. Minsky, Geometry of the complex of curves. I. Hyperbolicity, Invent. Math. 138 (1999), no. 1, 103–149. MR 1714338, DOI 10.1007/s002220050343
Yair Minsky, The classification of Kleinian surface groups. I. Models and bounds, Ann. of Math. (2) 171 (2010), no. 1, 1–107. MR 2630036, DOI 10.4007/annals.2010.171.1
Priyam Patel and Nicholas G. Vlamis, Algebraic and topological properties of big mapping class groups, Algebr. Geom. Topol. 18 (2018), no. 7, 4109–4142. MR 3892241, DOI 10.2140/agt.2018.18.4109
Piotr Przytycki and Alessandro Sisto, A note on acylindrical hyperbolicity of mapping class groups, Hyperbolic geometry and geometric group theory, Adv. Stud. Pure Math., vol. 73, Math. Soc. Japan, Tokyo, 2017, pp. 255–264. MR 3728501, DOI 10.2969/aspm/07310255