Skip to main content

The geometry of Markoff forms

  • Conference paper
  • First Online:
Number Theory

Part of the book series: Lecture Notes in Mathematics ((LNM,volume 1240))

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

Access this chapter

Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 34.99
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 46.00
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. H. Cohn, Approach to Markoff's minimal forms through modular functions, Annals of Math. 61 (1955) 1–12.

    Article  MATH  Google Scholar 

  2. _____, Representation of Markoff's binary quadratic forms by geodesics on a perforated torus, Acta Arith. 18 (1971) 125–136.

    MathSciNet  MATH  Google Scholar 

  3. _____, Markoff forms and primitive words, Math. Annalen, 196 (1972) 8–22.

    Article  MathSciNet  MATH  Google Scholar 

  4. _____, Some direct limits of primitive homotopy words and of Markoff geodesics, Discontinuous Groups and Riemann Surfaces, Annals Math. Studies, vol. 79, Princeton, 1974, 81–98.

    Google Scholar 

  5. _____, Minimal geodesics on Fricke's torus-covering, Riemann Surfaces and Related Topics, Annals Math. Studies, vol. 97, Princeton, 1980, 73–85.

    Google Scholar 

  6. A. Haas, Diophantine approximation on hyperbolic Riemann surfaces, to appear in Acta Math. 156:1–2.

    Google Scholar 

  7. J. Lehner & M. Sheingorn, Simple closed geodesics on H+/Γ (3) arise from the Markov spectrum, Bulletin of the A.M.S., 11 (1984), 359–362.

    Article  MathSciNet  MATH  Google Scholar 

  8. A. A. Markoff, Sur les formes binaires indefinies, I, Math. Ann 15 (1879), 281–309; II, 17 (1880), 379–400.

    Article  MathSciNet  Google Scholar 

  9. J. Nielsen, Die isomorphismen der allgemeinen unendlichen gruppe mit zwei erzeugenden, Math. Ann. 78 (1918), 385–397.

    Article  MathSciNet  MATH  Google Scholar 

  10. A. L Schmidt, Minimum of quadratic forms with respect to Fuchsian groups, I, J. Reine Angev. Math. 286/287(1976), 341–368.

    MathSciNet  MATH  Google Scholar 

  11. C. Series, The geometry of Markoff numbers, The Math. Intel. 7 (1985).

    Google Scholar 

  12. M. Sheingorn, Characterization of simple closed geodesics on Fricke surfaces, Duke Math. Jnl. 52 (1985), 535–545.

    Article  MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Editor information

David V. Chudnovsky Gregory V. Chudnovsky Harvey Cohn Melvyn B. Nathanson

Rights and permissions

Reprints and permissions

Copyright information

© 1987 Springer-Verlag

About this paper

Cite this paper

Haas, A. (1987). The geometry of Markoff forms. In: Chudnovsky, D.V., Chudnovsky, G.V., Cohn, H., Nathanson, M.B. (eds) Number Theory. Lecture Notes in Mathematics, vol 1240. Springer, Berlin, Heidelberg. https://doi.org/10.1007/BFb0072978

Download citation

  • DOI: https://doi.org/10.1007/BFb0072978

  • Published:

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-17669-5

  • Online ISBN: 978-3-540-47756-3

  • eBook Packages: Springer Book Archive

Publish with us

Policies and ethics