Hostname: page-component-586b7cd67f-r5fsc Total loading time: 0 Render date: 2024-11-27T22:57:03.006Z Has data issue: false hasContentIssue false
  • English
  • Français

EXHAUSTION OF THE CURVE GRAPH VIA RIGID EXPANSIONS

Part of: Special aspects of infinite or finite groups Low-dimensional topology

Published online by Cambridge University Press:  06 July 2018

JESÚS HERNÁNDEZ HERNÁNDEZ
JESÚS HERNÁNDEZ HERNÁNDEZ*
Affiliation:
Centro de Ciencias Matemáticas, UNAM, Campus Morelia, Morelia, Mich. 58190, Mexico e-mail: [email protected], https://sites.google.com/site/jhdezhdez/
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

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.

MSC classification

Primary: 20F65: Geometric group theory
Secondary: 57M07: Topological methods in group theory
Type
Research Article
Information
Glasgow Mathematical Journal , Volume 61 , Issue 1 , January 2019 , pp. 195 - 230
Copyright
Copyright © Glasgow Mathematical Journal Trust 2018 

References

REFERENCES

1. Aramayona, J. and Leininger, C., Finite rigid sets in curve complexes, J. Topology Anal. 5 (2) (2013), 183–203.Google Scholar
2. Aramayona, J. and Leininger, C., Exhausting curve complexes by finite rigid sets, Pac. J. Math. 282 (2) (2016), 257–283.Google Scholar
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. 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. Farb, B. and Margalit, D., A primer on mapping class groups, vol. 49 (Princeton University Press, Princeton, NJ, 2012).Google Scholar
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. 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. Hernández Hernández, J., Combinatorial rigidity of complexes of curves and multicurves (2016), Ph.D. Thesis, Aix-Marseille Université.Google Scholar
9. Hernández Hernández, J., Edge-preserving maps of curve graphs, Topology Appl. (2018), (in press).Google Scholar
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. 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. Irmak, E., Complexes of nonseparating curves and mapping class groups, Michigan Math. J. 54 (1) (2006), 81–110.Google Scholar
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. Ivanov, N. V., Automorphisms of complexes of curves and of Teichmüller spaces, Internat. Math. Res. Not. 1997 (14) (1997), 651–666.Google Scholar
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. 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. Luo, F., Automorphisms of the complex of curves, Topology 39 (2) (2000), 283–298.Google Scholar
18. Shackleton, K. J., Combinatorial rigidity in curve complexes and mapping class groups, Paci. J. Math. 230 (1) (2007), 217–232.Google Scholar