Measuring complexity of curves on surfaces

  • Macarena Arenas
  • Max Neumann-CotoEmail author
Original Paper


We consider the relations between different measures of complexity for free homotopy classes of curves on a surface \(\Sigma \), including the minimum number of self-intersections, the minimum length of the words representing them in \(\pi _1(\Sigma )\), and the minimum degree of the coverings of \(\Sigma \) to which they lift as embeddings.


Immersed curves on surfaces Self-intersections Coverings Word length 

Mathematics Subject Classification (2000)

Primary 57M05 57M10 Secondary 20F05 



The authors would like to thank the referee for helpful comments and corrections. The first author is grateful to the Instituto de Matemáticas, UNAM, for an Undergraduate Fellowship during which this work was completed.


  1. 1.
    Aougab, T., Gaster, J., Patel, P., Sapir, J.: Building hyperbolic metrics suited to closed curves and applications to lifting simply. Math. Res. Lett. 24(3), 593–617 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Hass, J., Scott, P.: Shortening curves on surfaces. Topology 33(1), 25–43 (1994)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Hempel, J.: Residual finiteness of surface groups. Proc. Am. Math. Soc. 32(1), 323 (1972)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Neumann-Coto, M.: A characterization of shortest geodesics on surfaces. Algebr. Geom. Topol. 1, 349–368 (2001)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Patel, P.: On a theorem of Peter Scott. Proc. Am. Math. Soc. 142(8), 2891–2906 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Rivin, I.: Geodesics with one self-intersection, and other stories. Adv. Math. 231, 2391–2412 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Scott, P.: Subgroups of surface groups are almost geometric. J. Lond. Math. Soc. (2) 17, 555–565 (1978)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer Nature B.V. 2019

Authors and Affiliations

  1. 1.Facultad de CienciasUniversidad Nacional Autónoma de MéxicoMexico CityMexico
  2. 2.Department of Mathematics and StatisticsMcGill UniversityMontrealCanada
  3. 3.Instituto de MatemáticasUniversidad Nacional Autónoma de MéxicoMexico CityMexico

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