Large-scale geometry of big mapping class groups

Mann, Kathryn¹ ; Rafi, Kasra²

¹ Department of Mathematics, Cornell University, Ithaca, NY, United States
² Department of Mathematics, University of Toronto, Toronto, ON, Canada

Geometry & topology, Tome 27 (2023) no. 6, pp. 2237-2296.

Résumé

We study the large-scale geometry of mapping class groups of surfaces of infinite type, using the framework of Rosendal for coarse geometry of non-locally-compact groups. We give a complete classification of those surfaces whose mapping class groups have local coarse boundedness (the analog of local compactness). When the end space of the surface is countable or tame, we also give a classification of those surfaces where there exists a coarsely bounded generating set (the analog of finite or compact generation, giving the group a well-defined quasi-isometry type) and those surfaces with mapping class groups of bounded diameter (the analog of compactness).

We also show several relationships between the topology of a surface and the geometry of its mapping class groups. For instance, we show that nondisplaceable subsurfaces are responsible for nontrivial geometry and can be used to produce unbounded length functions on mapping class groups using a version of subsurface projection; while self-similarity of the space of ends of a surface is responsible for boundedness of the mapping class group.

Détail

DOI : 10.2140/gt.2023.27.2237

Keywords: big mapping class group, CB generating set, coarsely bounded sets

Bibliographie

[1] J Aramayona, A Fossas, H Parlier, Arc and curve graphs for infinite-type surfaces, Proc. Amer. Math. Soc. 145 (2017) 4995 | DOI

[2] J Aramayona, P Patel, N G Vlamis, The first integral cohomology of pure mapping class groups, Int. Math. Res. Not. 2020 (2020) 8973 | DOI

[3] J Bavard, Hyperbolicité du graphe des rayons et quasi-morphismes sur un gros groupe modulaire, Geom. Topol. 20 (2016) 491 | DOI

[4] J Bavard, S Dowdall, K Rafi, Isomorphisms between big mapping class groups, Int. Math. Res. Not. 2020 (2020) 3084 | DOI

[5] R Camerlo, S Gao, The completeness of the isomorphism relation for countable Boolean algebras, Trans. Amer. Math. Soc. 353 (2001) 491 | DOI

[6] R Chamanara, Affine automorphism groups of surfaces of infinite type, from: "In the tradition of Ahlfors and Bers, III" (editors W Abikoff, A Haas), Contemp. Math. 355, Amer. Math. Soc. (2004) 123 | DOI

[7] M G Durham, F Fanoni, N G Vlamis, Graphs of curves on infinite-type surfaces with mapping class group actions, Ann. Inst. Fourier (Grenoble) 68 (2018) 2581 | DOI

[8] F Fanoni, S Hensel, N G Vlamis, Big mapping class groups acting on homology, Indiana Univ. Math. J. 70 (2021) 2261 | DOI

[9] W P Hooper, Grid graphs and lattice surfaces, Int. Math. Res. Not. 2013 (2013) 2657 | DOI

[10] J Ketonen, The structure of countable Boolean algebras, Ann. of Math. 108 (1978) 41 | DOI

[11] H A Masur, Y N Minsky, Geometry of the complex of curves, I : Hyperbolicity, Invent. Math. 138 (1999) 103 | DOI

[12] H A Masur, Y N Minsky, Geometry of the complex of curves, II : Hierarchical structure, Geom. Funct. Anal. 10 (2000) 902 | DOI

[13] S Mazurkiewicz, W Sierpiński, Contribution à la topologie des ensembles dénombrables, Fund. Math. 1 (1920) 17 | DOI

[14] P Patel, N G Vlamis, Algebraic and topological properties of big mapping class groups, Algebr. Geom. Topol. 18 (2018) 4109 | DOI

[15] A Randecker, Wild translation surfaces and infinite genus, Algebr. Geom. Topol. 18 (2018) 2661 | DOI

[16] I Richards, On the classification of noncompact surfaces, Trans. Amer. Math. Soc. 106 (1963) 259 | DOI

[17] C Rosendal, Global and local boundedness of Polish groups, Indiana Univ. Math. J. 62 (2013) 1621 | DOI

[18] C Rosendal, Coarse geometry of topological groups, 223, Cambridge Univ. Press (2021) | DOI

[19] S Schleimer, Notes on the complex of curves, lecture notes (2020)