Homeomorphic subsurfaces and the omnipresent arcs
Annales Henri Lebesgue, Volume 4 (2021), pp. 1565-1593.

Metadata

Keywords Infinite-type surfaces, subsurfaces, arcs, arc graphs, mapping class groups.

Abstract

In this article, we are concerned with various aspects of arcs on surfaces. In the first part, we deal with topological aspects of arcs and their complements. We use this understanding, in the second part, to construct an interesting action of the mapping class group on a subgraph of the arc graph. This subgraph naturally emerges from a new characterisation of infinite-type surfaces in terms of homeomorphic subsurfaces.


References

[AFP17] 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

[Bav16] 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

[BF02] Bestvina, Mladen; Fujiwara, Koji Bounded cohomology of subgroups of mapping class groups, Geom. Topol., Volume 6 (2002), pp. 69-89 | DOI | MR | Zbl

[BKMM12] 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 | Zbl

[Bow14] Bowditch, Brian H. Uniform hyperbolicity of the curve graphs, Pac. J. Math., Volume 269 (2014) no. 2, pp. 269-280 | DOI | MR | Zbl

[Bow18] Bowditch, Brian H. Large-scale rigidity properties of the mapping class groups, Pac. J. Math., Volume 293 (2018) no. 1, pp. 1-73 | DOI | MR | Zbl

[Bro10] Brouwer, Luitzen E. J. On the structure of perfect sets of points, KNAW, Proceedings, Volume 12 (1910), pp. 785-794

[DFV18] Durham, Matthew Gentry; Fanoni, Federica; Vlamis, Nicholas G. Graphs of curves on infinite-type surfaces with mapping class group actions, Ann. Inst. Fourier, Volume 68 (2018) no. 6, pp. 2581-2612 | DOI | Numdam | MR | Zbl

[FHV19] Fanoni, Federica; Hensel, Sebastian; Vlamis, Nicholas G. Big mapping class groups acting on homology (2019) (https://arxiv.org/abs/1905.12509, to appear in Indiana University Mathematics Journal)

[FP15] Fossas, Ariadna; Parlier, Hugo Curve graphs on surfaces of infinite type, Ann. Acad. Sci. Fenn., Math., Volume 40 (2015) no. 2, pp. 793-801 | DOI | MR | Zbl

[Har85] 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 | Zbl

[Har86] 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 | Zbl

[HPW15] 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

[Kec95] Kechris, Alexander S. Classical descriptive set theory, Graduate Texts in Mathematics, 156, Springer, 1995 | DOI | MR | Zbl

[LPRT19] Lanier, Justin; Patel, Priyam; Randecker, Anja; Tao, Jing Surfaces of infinite type, 2019 (Available at http://aimpl.org/genusinfinity)

[MR19] Mann, Kathryn; Rafi, Kasra Large scale geometry of big mapping class groups (2019) (https://arxiv.org/abs/1912.10914)

[MS20] Mazurkiewicz, Stefan; Sierpiński, Wacław Contribution à la topologie des ensembles dénombrables, Fundam. Math., Volume 1 (1920) no. 1, pp. 17-27 | DOI | Zbl

[MS13] Masur, Howard; Schleimer, Saul The geometry of the disk complex, J. Am. Math. Soc., Volume 26 (2013) no. 1, pp. 1-62 | DOI | MR | Zbl

[Ric63] Richards, Ian On the classification of noncompact surfaces, Trans. Am. Math. Soc., Volume 106 (1963), pp. 259-269 | DOI | MR | Zbl

[SG75] Schoenfeld, Alan H.; Gruenhage, Gary An alternate characterization of the Cantor set, Proc. Am. Math. Soc., Volume 53 (1975) no. 1, pp. 235-236 | DOI | MR | Zbl