Abstract
We prove that, for 3g−3+n>1 and (g,n)≠(1,2), the group of Weil–Petersson isometries of the Teichmüller space T g,n coincides with the extended mapping class group.
Similar content being viewed by others
References
Abikoff, W.: Augmented Teichmuller spaces, Bull. Amer. Math. Soc. 82 (1971), 333–334.
Abikoff, W.: Degenerating families of Riemann surfaces, Ann. of Math. 105 (1977), 29–44.
Ahlfors, L.: Some remarks on Teichmüller's space of Riemann surfaces, Ann. Math. 74 (1961), 171–191.
Ballman, W.: Lectures on Spaces of Nonpositive Curvature, Birkhäuser, Basel, 1995.
Bers, L.: Spaces of Degenerating Riemann Surfaces in Discontinous Groups and Riemann Surfaces, Ann. of Math Stud. 79, Princeton Univ. Press, Princeton, New Jersey, 1974.
Bridson, M. and Haefliger, A.: Metric Spaces of Non-Positive Curvature, Springer, Berlin, 1999.
Chu, T.: The Weil—Petersson metric in moduli space, Chinese J. Math. 4 (1976), 29–51.
Earle, C. and Kra, I.: On isometries between Teichmuller spaces, Duke J. Math 41 (1974), 583–591.
Ivanov, N.: Automorphism of complexes of curves and of Teichmuller spaces, Internat. Math. Res. Notices 14 (1997), 651–666.
Korkmaz, M.: Automorphisms of complexes of curves in punctured spheres and on punctured tori, Topology Appl. 95 (1999), 85–111.
Luo, F.: Automorphisms of the complex of curves, Topology 39 (2000), 283–298.
Masur, H.: The extension of the Weil—Petersson metric to the boundary of Teichmuller space, Duke Math J. 43 (1976), 623–635.
Mumford, D.: A remark on Mahler's compactness theorem, Proc. Amer. Math. Soc. 28 (1971), 289–294.
Nag, S.: Complex Analytic Theory of Teichmuller Space, Wiley, New York, 1988.
Royden, H.: Automorphisms and isometries of Teichmuller spaces. Advances in the theory of Riemann surfaces, Ann. of Math. Stud. 66, Princeton, Univ. Press, Princeton, New Jersey, 1970, pp. 369–383.
Royden, H.: Oral communication.
Schoen, R. and Yau, S.-T.: Compact group actions and the topology of manifolds with nonpositive curvature, Topology 18 (1979), 361–380.
Strebel, K.: Quadratic Differentials, Springer, Berlin, 1984.
Trapani, S.: On the determinant of the bundle of meromorphic quadratic differentials on the Deligne-Mumford compactification of the moduli space of Riemann surfaces, Math. Ann. 293 (1992), 681–705.
Tromba, A.: On a natural algebraic affine connection on the space of almost complex structures and the curvature of Teichmuller space with respect to its Weil—Petersson metric, Manuscripta Math. 56 (1986), 475–497.
Tromba, A.: Teichmuller Theory in Riemannian Geometry, Birkhäuser, Basel, 1992.
Wolpert, S.: Noncompleteness of the Weil—Petersson metric for Teichmuller space, Pacific J. Math. 61 (1975), 573–577.
Wolpert, S.: The finite Weil—Petersson diameter of Riemann space, Pacific J. Math. 70 (1977), 281–288.
Wolpert, S.: Chern forms and the Riemann tensor for the moduli space of curves, Invent. Math. 85 (1986), 119–145.
Wolpert, S.: Geodesic length functions and the Nielsen problem, J.Differential Geom. 25 (1987), 275–295.
Yamada, S.: Weil—Petersson completion of Teichmuller spaces and mapping class group actions, Manuscript.
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Masur, H., Wolf, M. The Weil–Petersson Isometry Group. Geometriae Dedicata 93, 177–190 (2002). https://doi.org/10.1023/A:1020300413472
Issue Date:
DOI: https://doi.org/10.1023/A:1020300413472