Graphs of curves on infinite-type surfaces with mapping class group actions
Annales de l'Institut Fourier, Volume 68 (2018) no. 6, pp. 2581-2612.

We study when the mapping class group of an infinite-type surface S admits an action with unbounded orbits on a connected graph whose vertices are simple closed curves on S. We introduce a topological invariant for infinite-type surfaces that determines in many cases whether there is such an action. This allows us to conclude that, as non-locally compact topological groups, many big mapping class groups have nontrivial coarse geometry in the sense of Rosendal.

Nous étudions le cas où le groupe modulaire d’une surface de type infini admet une action avec orbites non bornées sur un graphe connexe dont les sommets sont des courbes fermées simples de S. Nous définissons un invariant topologique pour surfaces de type infini qui détecte dans de nombreux cas s’il y a une telle action. Nous en déduissons que beaucoup de gros groupes modulaires, en tant que groupes topologiques non localement compacts, ont géométrie grossière non banale au sens de Rosendal.

Published online:
DOI: 10.5802/aif.3217
Classification: 57S05, 37E30, 20F65, 57M07
Keywords: mapping class groups, surface homeomorphisms, curve graphs, infinite-type surfaces
Durham, Matthew Gentry 1; Fanoni, Federica 2; Vlamis, Nicholas G. 3

1 University of California, Riverside 900 University Ave Riverside, CA 92521 (USA)
2 Universität Heidelberg Im Neuenheimer Feld 205 69120 Heidelberg (Germany)
3 CUNY Queens College Department of Mathematics 65-30 Kissena Blvd Flushing, NY 11367 (USA)
License: CC-BY-ND 4.0
Copyrights: The authors retain unrestricted copyrights and publishing rights
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Durham, Matthew Gentry; Fanoni, Federica; Vlamis, Nicholas G. Graphs of curves on infinite-type surfaces with mapping class group actions. Annales de l'Institut Fourier, Volume 68 (2018) no. 6, pp. 2581-2612. doi : 10.5802/aif.3217.

[1] Ahlfors, Lars V.; Sario, Leo Riemann surfaces, Princeton Mathematical Series, 26, Princeton University Press, 1960, xi+382 pages | MR | Zbl

[2] Aramayona, Javier; Fossas, Ariadna; Parlier, Hugo Arc and curve graphs for infinite-type surfaces, Proc. Am. Math. Soc., Volume 145 (2017) no. 11, pp. 4995-5006 | DOI | MR | Zbl

[3] Aramayona, Javier; Valdez, Ferrán On the geometry of graphs associated to infinite-type surfaces (2016) (To appear in Math. Z.)

[4] 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

[5] Behrstock, Jason; Kleiner, Bruce; Minsky, Yair; Mosher, Lee Geometry and rigidity of mapping class groups, Geom. Topol., Volume 16 (2012) no. 2, pp. 781-888 | DOI | MR

[6] Behrstock, Jason; Minsky, Yair Dimension and rank for mapping class groups, Ann. Math., Volume 167 (2008) no. 3, pp. 1055-1077 | DOI | MR

[7] Bestvina, Mladen; Bromberg, Ken; Fujiwara, Koji Constructing group actions on quasi-trees and applications to mapping class groups, Publ. Math., Inst. Hautes Étud. Sci., Volume 122 (2010), pp. 1-64 | Zbl

[8] Bowditch, Brian H. Uniform hyperbolicity of the curve graphs, Pac. J. Math., Volume 269 (2014) no. 2, pp. 269-280 | DOI | MR

[9] Bowditch, Brian H., 2016 (Personal communication)

[10] Bowditch, Brian H. Large-scale rigidity properties of the mapping class groups, Pac. J. Math., Volume 293 (2018) no. 1, pp. 1-73 | Zbl

[11] Bridson, Martin R.; Haefliger, André Metric spaces of non-positive curvature, Grundlehren der Mathematischen Wissenschaften, 319, Springer, 1999, xxii+643 pages | DOI | MR

[12] Calegari, Danny Circular groups, planar groups, and the Euler class, Proceedings of the Casson Fest (Geometry and Topology Monographs), Volume 7 (2004), pp. 431-491 | Zbl

[13] Calegari, Danny Big mapping class groups and dynamics (2009) (Geometry and the imagination,

[14] Calegari, Danny Mapping class groups: the next generation (2009) (Geometry and the imagination,

[15] Cantwell, John; Conlon, Lawrence Endperiodic Automorphisms of Surfaces and Foliations (2010) ( | arXiv

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

[17] Fossas, Ariadna; Parlier, Hugo Curve graphs on surfaces of infinite type, Ann. Acad. Sci. Fenn. Math., Volume 40 (2015) no. 2, pp. 793-801 | Zbl

[18] Funar, Louis Braided Houghton groups as mapping class groups, An. Ştiinţ. Univ. Al. I. Cuza Iaşi. Mat. (N.S.), Volume 53 (2007) no. 2, pp. 229-240 | Zbl

[19] Funar, Louis; Kapoudjian, Christophe On a universal mapping class group of genus zero, Geom. Funct. Anal., Volume 14 (2004) no. 5, pp. 965-1012

[20] Funar, Louis; Kapoudjian, Christophe The braided Ptolemy-Thompson group is finitely presented, Geom. Topol., Volume 12 (2008) no. 1, pp. 475-530

[21] Funar, Louis; Kapoudjian, Christophe An infinite genus mapping class group and stable cohomology, Commun. Math. Phys., Volume 287 (2009) no. 3, pp. 784-804 | DOI | MR | Zbl

[22] Gilman, Robert H. On the definition of word hyperbolic groups, Math. Z., Volume 242 (2002) no. 3, pp. 529-541 | DOI | MR

[23] Harer, John L. Stability of the homology of the mapping class groups of orientable surfaces, Ann. Math., Volume 121 (1985) no. 2, pp. 215-249 | DOI | MR

[24] Harer, John L. The virtual cohomological dimension of the mapping class group of an orientable surface, Invent. Math., Volume 84 (1986) no. 1, pp. 157-176 | DOI | MR

[25] Hensel, Sebastian; Przytycki, Piotr; Webb, Richard C. H. 1-slim triangles and uniform hyperbolicity for arc graphs and curve graphs, J. Eur. Math. Soc., Volume 17 (2015) no. 4, pp. 755-762 | DOI | MR | Zbl

[26] Kechris, Alexander S. Classical descriptive set theory, Graduate Texts in Mathematics, 156, Springer, 1995, xviii+402 pages | DOI | MR

[27] Mann, Kathryn; Rosendal, Christian Large scale geometry of homeomorphism groups (2016) (to appear in Ergodic Theory Dyn. Syst.)

[28] Margalit, Dan; McCammond, Jon Geometric presentations for the pure braid group, J. Knot Theory Ramifications, Volume 18 (2009) no. 1, pp. 1-20 | DOI | MR

[29] Masur, Howard; Minsky, Yair Geometry of the complex of curves. I. Hyperbolicity, Invent. Math., Volume 138 (1999) no. 1, pp. 103-149 | DOI | MR

[30] Masur, Howard; Schleimer, Saul The geometry of the disk complex, J. Am. Math. Soc., Volume 26 (2013) no. 1, pp. 1-62 | DOI | MR

[31] Mazurkiewicz, Stefan; Sierpiński, Wacław Contribution à la topologie des ensembles dénombrables, Fundamenta Mathematicae, Volume 1 (1920) no. 1, pp. 17-27 | Zbl

[32] Mj, Mahan Cannon-Thurston maps for surface groups, Ann. Math., Volume 179 (2014) no. 1, pp. 1-80 | DOI | MR

[33] Putman, Andrew A note on the connectivity of certain complexes associated to surfaces, Enseign. Math., Volume 54 (2008) no. 3–4, pp. 287-301 | Zbl

[34] Rasmussen, Alexander J. Uniform hyperbolicity of the graphs of nonseparating curves via bicorn curves (2017) (

[35] Richards, Ian On the classification of noncompact surfaces, Trans. Am. Math. Soc., Volume 106 (1963), pp. 259-269 | MR

[36] Rosendal, Christian Coarse geometry of topological groups (2016) ( rosendal/PapersWebsite/Coarse-Geometry-Book17.pdf)

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