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Connectedness of curve complex of surface

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Abstract

For a closed orientable surface S g of genus not smaller than 2, C(S g ) is the curve complex on S g whose vertices consist of the isotopy classes of nontrivial circles on S g . It has been showed that any two vertices in C(S g ) can be connected by an edge path, and C(S g ) has an infinite diameter. We show that for 0 ⩽ i ⩽ 3g−5, two i-simplices can be connected by an (i +1)-path in C(S g ), and the diameter of C(S g ) under such a distance is infinite.

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References

  1. Harer J. Stability of the homology of the mapping class group of an orientable surface. Ann Math, 1985, 121: 215–249

    Article  MATH  MathSciNet  Google Scholar 

  2. Harer J. The virtual cohomological dimension of the mapping class group of an orientable surface. Invent Math, 1986, 84: 157–176

    Article  MATH  MathSciNet  Google Scholar 

  3. Harvey W. Boundary structure of the modular group. In: Riemann Surfaces and Related Topics, vol. 97. Princeton, NJ: Princeton University Press, 1981, 245–251

    Google Scholar 

  4. Hempel J. 3-Manifold. Princeton, NJ: Princeton University Press, 1975

    Google Scholar 

  5. Hempel J. 3-Manifolds as viewed from the curve complex. Topology, 2001, 40: 631–657

    Article  MATH  MathSciNet  Google Scholar 

  6. Ivanov N V. On the virtual cohomology dimension of the Teichimüller modular group. In: Lecture Notes in Mathematics, vol. 1060. Berlin: Springer, 1984, 306–318

    Google Scholar 

  7. Ivanov N V. Mapping class groups. In: Daverman R J, Sher R B, eds. Handbook of Geometric Topology. Amsterdam: North-Holland, 2002, 523–633

    Google Scholar 

  8. Ivanov N V. Automorphisms of complexes of curves and of Teichimüller spaces. Internat Math Res Notices, 1997, 14: 651–666

    Article  Google Scholar 

  9. Ivanov N V. Complexes of curves and Teichimüller spaces. Math Notes, 1991, 49: 479–484

    Article  MATH  MathSciNet  Google Scholar 

  10. Masur H A, Minsky Y N. Geometry of the complex of curves I: Hyperbolicity. Invent Math, 1999, 138: 103–149

    Article  MATH  MathSciNet  Google Scholar 

  11. Minsky Y. Combinbatorial and geometrical aspects of hyperbolic 3-manifolds. Lond Math Soc Lec Notes, 2003, 299: 3–40

    MathSciNet  Google Scholar 

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Authors and Affiliations

  1. School of Mathematical Sciences, Dalian University of Technology, Dalian, 116024, China

    DongQi Sun, FengChun Lei & FengLing Li

  2. College of Science, Harbin Engineering University, Harbin, 150001, China

    DongQi Sun

Authors
  1. DongQi Sun
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  2. FengChun Lei
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  3. FengLing Li
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Correspondence to FengChun Lei.

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Sun, D., Lei, F. & Li, F. Connectedness of curve complex of surface. Sci. China Math. 57, 847–854 (2014). https://doi.org/10.1007/s11425-013-4768-9

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  • DOI: https://doi.org/10.1007/s11425-013-4768-9

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