Skip to main content
SpringerLink
Log in

Measuring complexity of curves on surfaces

  • Original Paper
  • Published:
Geometriae Dedicata Aims and scope Submit manuscript

Abstract

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.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8
Fig. 9
Fig. 10
Fig. 11
Fig. 12

Similar content being viewed by others

References

  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)

    Article  MathSciNet  Google Scholar 

  2. Hass, J., Scott, P.: Shortening curves on surfaces. Topology 33(1), 25–43 (1994)

    Article  MathSciNet  Google Scholar 

  3. Hempel, J.: Residual finiteness of surface groups. Proc. Am. Math. Soc. 32(1), 323 (1972)

    Article  MathSciNet  Google Scholar 

  4. Neumann-Coto, M.: A characterization of shortest geodesics on surfaces. Algebr. Geom. Topol. 1, 349–368 (2001)

    Article  MathSciNet  Google Scholar 

  5. Patel, P.: On a theorem of Peter Scott. Proc. Am. Math. Soc. 142(8), 2891–2906 (2014)

    Article  MathSciNet  Google Scholar 

  6. Rivin, I.: Geodesics with one self-intersection, and other stories. Adv. Math. 231, 2391–2412 (2012)

    Article  MathSciNet  Google Scholar 

  7. Scott, P.: Subgroups of surface groups are almost geometric. J. Lond. Math. Soc. (2) 17, 555–565 (1978)

    Article  MathSciNet  Google Scholar 

Download references

Acknowledgements

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.

Author information

Author notes

  1. Macarena Arenas

    Present address: Department of Mathematics and Statistics, McGill University, Montreal, QC, H3A 0B9, Canada

Authors and Affiliations

  1. Facultad de Ciencias, Universidad Nacional Autónoma de México, Ciudad Universitaria, 04510, Mexico City, Mexico

    Macarena Arenas

  2. Instituto de Matemáticas, Universidad Nacional Autónoma de México, Ciudad Universitaria, 04510, Mexico City, Mexico

    Max Neumann-Coto

Authors
  1. Macarena Arenas
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Max Neumann-Coto
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Max Neumann-Coto.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Arenas, M., Neumann-Coto, M. Measuring complexity of curves on surfaces. Geom Dedicata 204, 25–41 (2020). https://doi.org/10.1007/s10711-019-00443-3

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10711-019-00443-3

Keywords

Mathematics Subject Classification (2000)

Search

Navigation