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I - Hyperbolic Geometry

Published online by Cambridge University Press:  05 January 2012

By
Aline Aigon-Dupuy ,
Peter Buser  and
Klaus-Dieter Semmler
Edited by
Jens Bolte  and
Frank Steiner
Aline Aigon-Dupuy
Affiliation:
Ecole Polytechnique Fédérale de Lausanne
Peter Buser
Affiliation:
Ecole Polytechnique Fédérale de Lausanne
Klaus-Dieter Semmler
Affiliation:
Ecole Polytechnique Fédérale de Lausanne
Jens Bolte
Affiliation:
Royal Holloway, University of London
Frank Steiner
Affiliation:
Universität Ulm, Germany
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Summary

Summary. The aim of these lectures is to provide a self-contained introduction into the geometry of the hyperbolic plane and to develop computational tools for the construction and the study of discrete groups of non-Euclidean motions.

The first lecture provides a minimal background in the classical geometries: spherical, hyperbolic and Euclidean. In Lecture 2 we present the Poincaré model of the hyperbolic plane. This is the most widely used model in the literature, but possibly not always the most suitable one for effective computation, and so we present in Lectures 4 and 5 an independent approach based on what we call the matrix model. Between the two approaches, in Lecture 3, we introduce the concepts of a Fuchsian group, its fundamental domains and the hyperbolic surfaces. In the final lecture we bring everything together and study, as a special subject, the construction of hyperbolic surfaces of genus 2 based on geodesic octagons.

The course has been designed such that the reader may work himself linearly through it. The prerequisites in differential geometry are kept to a minimum and are largely covered, for example by the first chapters of the books by Do Carmo [8] or Lee [15]. The numerous exercises in the text are all of a computational rather than a problem-to-solve nature and are, hopefully, not too hard to do.

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Information
Hyperbolic Geometry and Applications in Quantum Chaos and Cosmology , pp. 1 - 82
Publisher: Cambridge University Press
Print publication year: 2011

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References

1. A., Aigon-Dupuy, P., Buser, M., Cibils, A., Künzle, F., Steiner: Hyperbolic octagons and Teichmüller space in genus 2. J. Math. Phys. 46(3), 033513 (2005).Google Scholar
2. A., Aigon, R., Silhol: Hyperbolic hexagons and algebraic curves in genus 3. J. London Math. Soc. (2), 66, 671–690 (2002).Google Scholar
3. R., Aurich, M., Sieber, F., Steiner: Quantum chaos of the Hadamard-Gutzwiller model. Phys. Rev. Lett., 61, 483–487 (1988).Google Scholar
4. R., Aurich, E. B., Bogomolny, F., Steiner: Periodic orbits on the regular hyperbolic octagon. Physica D, 48, 91–101 (1991).Google Scholar
5. A. F., Beardon: The geometry of discrete groups, volume 91 of Graduate Texts in Mathematics (Springer-Verlag, New York, 1995).Google Scholar
6. P., Buser: Geometry and spectra of compact Riemann surfaces, volume 106 of Progress in Mathematics (Birkhäuser, Boston, 1992).Google Scholar
7. P., Buser, R., Silhol: Some remarks on the uniformizing function in genus 2. Geom. Dedicata 115, 121–133 (2005).Google Scholar
8. M. P., Do Carmo: Riemannian geometry, Mathematics: Theory and Applications (Birkhäuser, Boston, 1992).Google Scholar
9. D. B. A., Epstein, C., Petronio: An exposition of Poincaré's polyhedron theorem. Enseign. Math. (2) 40, 113–170 (1994).Google Scholar
10. H. M., Farkas, I, Kra: Riemann surfaces, volume 71 of Graduate Texts in Mathematics, second edition (Springer-Verlag, New York, 1992).
11. W., Fenchel: Elementary geometry in hyperbolic space, vol. 11 of de Gruyter Studies in Mathematics (Walter de Gruyter & Co., Berlin, 1989).Google Scholar
12. W., Fenchel, J., Nielsen: Discontinuous groups of isometries in the hyperbolic plane, vol. 29 of de Gruyter Studies in Mathematics (Walter de Gruyter & Co., Berlin, 2003).Google Scholar
13. H., Helling: Diskrete Untergruppen von SL(2,ℝ). Invent. Math. 17, 217–229 (1972).Google Scholar
14. H., Helling: Über den Raum der kompakten Riemannschen Flächen vom Geschlecht 2. J. Reine Angew. Math. 268/269, 286–293 (1974).Google Scholar
15. J. M., Lee: Riemannian manifolds. An introduction to curvature, volume 176 of Graduate Texts in Mathematics (Springer-Verlag, New York, 1997).Google Scholar
16. W. S., Massey: A basic course in algebraic topology, volume 127 of Graduate Texts in Mathematics (Springer-Verlag, New York, 1991).Google Scholar
17. H., Poincaré: Sur les fonctions Fuchsiennes. C. R. Acad. Sci. Paris 92, 333–335 (1881). Reprinted in: Oeuvres de Henri Poincaré. Tome II, Éditions Jacques Gabay, Sceaux, 1995).Google Scholar
18. K.-D., Semmler: A fundamental domain for the Teichmüller space of compact Riemann surfaces of genus 2. PhD Thesis, École Polytechnique Fédérale de Lausanne, 1988.
19. M., Seppälä, T., Sorvali: Geometry of Riemann surfaces and Teichmüller spaces, volume 169 of North-Holland Mathematics Studies (North-Holland Publishing Co., Amsterdam, 1992).Google Scholar

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  • Hyperbolic Geometry
  • Edited by Jens Bolte, Royal Holloway, University of London, Frank Steiner, Universität Ulm, Germany
  • Book: Hyperbolic Geometry and Applications in Quantum Chaos and Cosmology
  • Online publication: 05 January 2012
  • Chapter DOI: https://doi.org/10.1017/CBO9781139108782.002
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Save book to Dropbox

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  • Hyperbolic Geometry
  • Edited by Jens Bolte, Royal Holloway, University of London, Frank Steiner, Universität Ulm, Germany
  • Book: Hyperbolic Geometry and Applications in Quantum Chaos and Cosmology
  • Online publication: 05 January 2012
  • Chapter DOI: https://doi.org/10.1017/CBO9781139108782.002
Available formats
×

Save book to Google Drive

To save content items to your account, please confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your account. Find out more about saving content to Google Drive.

  • Hyperbolic Geometry
  • Edited by Jens Bolte, Royal Holloway, University of London, Frank Steiner, Universität Ulm, Germany
  • Book: Hyperbolic Geometry and Applications in Quantum Chaos and Cosmology
  • Online publication: 05 January 2012
  • Chapter DOI: https://doi.org/10.1017/CBO9781139108782.002
Available formats
×