EXHAUSTION OF THE CURVE GRAPH VIA RIGID EXPANSIONS

HERNÁNDEZ, JESÚS HERNÁNDEZ

Glasgow mathematical journal, Tome 61 (2019) no. 1, pp. 195-230

Résumé

For an orientable surface S of finite topological type with genus g ≥ 3, we construct a finite set of curves whose union of iterated rigid expansions is the curve graph $\mathcal{C}$(S). The set constructed, and the method of rigid expansion, are closely related to Aramayona and Leiniger's finite rigid set in Aramayona and Leininger, J. Topology Anal.5(2) (2013), 183–203 and Aramayona and Leininger, Pac. J. Math.282(2) (2016), 257–283, and in fact a consequence of our proof is that Aramayona and Leininger's set also exhausts the curve graph via rigid expansions.

Détail

DOI : 10.1017/S0017089518000174

Bibliographie

[1] 1. Aramayona, J. and Leininger, C., Finite rigid sets in curve complexes, J. Topology Anal. 5 (2) (2013), 183–203. Google Scholar

[2] 2. Aramayona, J. and Leininger, C., Exhausting curve complexes by finite rigid sets, Pac. J. Math. 282 (2) (2016), 257–283. Google Scholar

[3] 3. Behrstock, J. and Margalit, D., Curve complexes and finite index subgroups of mapping class groups, Geometriae Dedicata 118 (1) (2006), 71–85. Google Scholar

[4] 4. Birman, J. S., Broaddus, N. and Menasco, W. W., Finite rigid sets and homologically non-trivial spheres in the curve complex of a surface, J. Topol. Anal. 7 (1) (2015), 47–71. Google Scholar

[5] 5. Farb, B. and Margalit, D., A primer on mapping class groups, vol. 49 (Princeton University Press, Princeton, NJ, 2012). Google Scholar

[6] 6. Harer, J. L., The virtual cohomological dimension of the mapping class group of an orientable surface, Invent. Math. 84 (1) (1986), 157–176. Google Scholar

[7] 7. Harvey, W. J., Geometric structure of surface mapping class groups, in Homological group theory (Wall, C. T. C. editor), London Math. Soc. Lecture Notes, vol. 36 (Cambridge University Press, Cambridge, NY, 1979), 255–269. Google Scholar

[8] 8. Hernández Hernández, J., Combinatorial rigidity of complexes of curves and multicurves (2016), Ph.D. Thesis, Aix-Marseille Université. Google Scholar

[9] 9. Hernández Hernández, J., Edge-preserving maps of curve graphs, Topology Appl. (2018), (in press). Google Scholar

[10] 10. Humphries, S. P., Generators for the mapping class group, in Topology of low-dimensional manifolds (Fenn, R. editor), Lecture Notes in Math., vol. 722 (Springer-Verlag, Berlin, Heidelberg, 1979), 44–47. Google Scholar

[11] 11. Irmak, E., Superinjective simplicial maps of complexes of curves and injective homomorphisms of subgroups of mapping class groups, Topology 43 (3) (2004), 513–541. Google Scholar

[12] 12. Irmak, E., Complexes of nonseparating curves and mapping class groups, Michigan Math. J. 54 (1) (2006), 81–110. Google Scholar

[13] 13. Irmak, E., Superinjective simplicial maps of complexes of curves and injective homomorphisms of subgroups of mapping class groups, Topology Appl. 153 (8) (2006), 1309–1340. Google Scholar

[14] 14. Ivanov, N. V., Automorphisms of complexes of curves and of Teichmüller spaces, Internat. Math. Res. Not. 1997 (14) (1997), 651–666. Google Scholar

[15] 15. Korkmaz, M., Automorphisms of complexes of curves on punctured spheres and on punctured tori, Topology Appl. 95 (2) (1999), 85–111. Google Scholar

[16] 16. Lickorish, W. B. R., A finite set of generators for the homeotopy group of a 2-manifold, Math. Proc. Cambridge Philos. Soc. 60 (4) (1964), 769–778. Google Scholar

[17] 17. Luo, F., Automorphisms of the complex of curves, Topology 39 (2) (2000), 283–298. Google Scholar

[18] 18. Shackleton, K. J., Combinatorial rigidity in curve complexes and mapping class groups, Paci. J. Math. 230 (1) (2007), 217–232. Google Scholar