Geodesic length functions and Teichmüller spaces
Author:
Feng Luo
Journal:
Electron. Res. Announc. Amer. Math. Soc. 2 (1996), 34-41
MSC (1991):
Primary 32G15, 30F60
DOI:
https://doi.org/10.1090/S1079-6762-96-00008-X
MathSciNet review:
1405967
Full-text PDF Free Access
Abstract |
References |
Similar Articles |
Additional Information
Abstract: Given a compact orientable surface $\Sigma$, let $\mathcal {S}(\Sigma )$ be the set of isotopy classes of essential simple closed curves in $\Sigma$. We determine a complete set of relations for a function from $\mathcal {S}(\Sigma )$ to $\mathbf {R}$ to be the geodesic length function of a hyperbolic metric with geodesic boundary on $\Sigma$. As a consequence, the Teichmüller space of hyperbolic metrics with geodesic boundary on $\Sigma$ is reconstructed from an intrinsic combinatorial structure on $\mathcal {S}(\Sigma )$. This also gives a complete description of the image of Thurston’s embedding of the Teichmüller space.
- Francis Bonahon, The geometry of Teichmüller space via geodesic currents, Invent. Math. 92 (1988), no. 1, 139–162. MR 931208, DOI https://doi.org/10.1007/BF01393996
- Max Dehn, Papers on group theory and topology, Springer-Verlag, New York, 1987. Translated from the German and with introductions and an appendix by John Stillwell; With an appendix by Otto Schreier. MR 881797
- R. Fricke and F. Klein, Vorlesungen über die Theorie der Automorphen Funktionen, Teubner, Leipizig, 1897–1912.
- Travaux de Thurston sur les surfaces, Astérisque, vol. 66, Société Mathématique de France, Paris, 1979 (French). Séminaire Orsay; With an English summary. MR 568308
- Linda Keen, Intrinsic moduli on Riemann surfaces, Ann. of Math. (2) 84 (1966), 404–420. MR 203000, DOI https://doi.org/10.2307/1970454
- W. B. R. Lickorish, A representation of orientable combinatorial $3$-manifolds, Ann. of Math. (2) 76 (1962), 531–540. MR 151948, DOI https://doi.org/10.2307/1970373
- F. Luo, Geodesic length functions and Teichmüller spaces, preprint.
- ---, Non-separating simple closed curves in a compact surface, Topology, in press.
- Bernard Maskit, On Klein’s combination theorem. II, Trans. Amer. Math. Soc. 131 (1968), 32–39. MR 223570, DOI https://doi.org/10.1090/S0002-9947-1968-0223570-1
- Yoshihide Okumura, On the global real analytic coordinates for Teichmüller spaces, J. Math. Soc. Japan 42 (1990), no. 1, 91–101. MR 1027542, DOI https://doi.org/10.2969/jmsj/04210091
- ---, Global real analytic length parameters for Teichmüller spaces, Hiroshima Math. J. 26 (1) (1996), 165–179.
- Paul Schmutz, Die Parametrisierung des Teichmüllerraumes durch geodätische Längenfunktionen, Comment. Math. Helv. 68 (1993), no. 2, 278–288 (German). MR 1214232, DOI https://doi.org/10.1007/BF02565819
- T. Sorvali, Parametrization for free Möbius groups, Ann. Acad. Sci. Fenn. 579 (1974), 1–12.
- William P. Thurston, On the geometry and dynamics of diffeomorphisms of surfaces, Bull. Amer. Math. Soc. (N.S.) 19 (1988), no. 2, 417–431. MR 956596, DOI https://doi.org/10.1090/S0273-0979-1988-15685-6
- Scott A. Wolpert, Geodesic length functions and the Nielsen problem, J. Differential Geom. 25 (1987), no. 2, 275–296. MR 880186
- F. Bonahon, The geometry of Teichmüller space via geodesic currents, Invent. Math. 92 (1988), 139–162.
- M. Dehn, Papers on group theory and topology, J. Stillwell (ed.), Springer-Verlag, Berlin and New York, 1987.
- R. Fricke and F. Klein, Vorlesungen über die Theorie der Automorphen Funktionen, Teubner, Leipizig, 1897–1912.
- A. Fathi, F. Laudenbach, and V. Poenaru, Travaux de Thurston sur les surfaces, vol. 66–67, Astérisque, Société Mathématique de France, 1979.
- L. Keen, Intrinsic moduli on Riemann surfaces, Ann. Math. 84 (1966), 405–420.
- R. Lickorish, A representation of orientable combinatorial 3-manifolds, Ann. Math. 72 (1962), 531–540.
- F. Luo, Geodesic length functions and Teichmüller spaces, preprint.
- ---, Non-separating simple closed curves in a compact surface, Topology, in press.
- B. Maskit, On Klein’s combination theorem, Trans. Amer. Math. Soc. 131 (1968), 32–39.
- Y. Okumura, On the global real analytic coordinates for Teichmüller spaces, J. Math. Soc. Japan 42 (1990), 91–101.
- ---, Global real analytic length parameters for Teichmüller spaces, Hiroshima Math. J. 26 (1) (1996), 165–179.
- P. Schmutz, Die Parametrisierung des Teichmüllerraumes durch geodätishe Längenfunktionen, Comment. Math. Helv. 68 (1993), 278–288.
- T. Sorvali, Parametrization for free Möbius groups, Ann. Acad. Sci. Fenn. 579 (1974), 1–12.
- W. Thurston, On the geometry and dynamics of diffeomorphisms of surfaces, Bul. Amer. Math. Soc. 19 (1988), 417–438.
- S. Wolpert, Geodesic length functions and the Nielsen problem, J. Diff. Geom. 25 (1987), 275–296.
Similar Articles
Retrieve articles in Electronic Research Announcements of the American Mathematical Society
with MSC (1991):
32G15,
30F60
Retrieve articles in all journals
with MSC (1991):
32G15,
30F60
Additional Information
Feng Luo
Affiliation:
Department of Mathematics, Rutgers University, New Brunswick, NJ 08903
MR Author ID:
251419
Email:
fluo@math.rutgers.edu
Keywords:
Hyperbolic metrics,
geodesics,
Teichmüller spaces
Received by editor(s):
April 9, 1996
Communicated by:
Walter Neumann
Article copyright:
© Copyright 1996
American Mathematical Society