The Pure and Applied Mathematics (한국수학교육학회지시리즈B:순수및응용수학)

Korean Society of Mathematical Education (한국수학교육학회)
권호 표지

DISCRETE PRESENTATIONS OF THE HOLONOMY GROUP OF A ONE-HOLED TORUS

  • Kim, Jpmg-Chan (Department of Mathematics Education, Korea University)
  • Received : 2010.01.28
  • Accepted : 2010.11.22
  • Published : 2010.11.30
Download PDF
⟨ Previous Next ⟩

Abstract

A one-holed torus ${\Sigma}$(l, 1) is a building block of oriented surfaces. In this paper we formulate the matrix presentations of the holonomy group of a one-holed torus ${\Sigma}$(1, 1) by the gluing method. And we present an algorithm for deciding the discreteness of the holonomy group of ${\Sigma}$(1, 1).

Keywords

References

  1. A. Beardon: The Geometry of Discrete Groups. Graduate Texts in Mathematics, 91, Springer-Verlag, 1983.
  2. W.M. Goldman: Geometric structures on manifolds and varieties of representations. Geometry of group representations (Boulder, CO, 1987), 169-198, Contemp. Math., 74.
  3. W.M. Goldman: The modular group action on real SL(2)-characters of a one-holed torus. Geometry and Topology 7 (2003), 443-486. https://doi.org/10.2140/gt.2003.7.443
  4. D. Johnson & J.J. Millson: Deformation spaces associated to compact hyperbolic manifolds. Discrete groups in geometry and analysis (New Haven, Conn., 1984), 48-106, Progr. Math., 67.
  5. L. Keen: Canonical polygons for finitely generated Fuchsian groups. Acta Math. 115 (1965), 1-16.
  6. H.C. Kim: Discrete Conditions for the Holonomy Group of a Pair of Pants. J. Korean Math. Soc. 44 (2007), no. 3, 615-626. https://doi.org/10.4134/JKMS.2007.44.3.615
  7. N. Kuiper: On convex locally projective spaces. Convegno Internazionale di Geometria Differenziale, Italia, 1953, 200-213.
  8. K. Matsuzaki & M. Taniguchi: Hyperbolic manifolds and Kleinian groups, Oxford Science Publications, Oxford University Press, New York, 1998.
  9. D. Sullivan & W. Thurston: Manifolds with canonical coordinates: Some examples. Enseign. Math. 29 (1983), 15-25.
  10. W. Thurston: Three-dimensional geometry and topology. Vol. 1. Princeton Mathematical Series, 35. Princeton University Press, 1997.
  11. S. Wolpert: On the Weil-Petersson geometry of the moduli space of curves. Amer. J. Math. 107 (1985), no. 4, 969-997. https://doi.org/10.2307/2374363