Abstract
For an analytically infinite Riemann surface R, we consider the action of the quasiconformal mapping class group MCG(R) on the Teichmüller space T(R), which preserves the fibers of the projection α: T(R) → AT(R) onto the asymptotic Teichmüller space AT(R). We prove that if MCG(R) has a common fixed point α(p) ∈ AT(R), then it acts discontinuously on the fiber T p over α(p), which is a separable subspace of T(R). In particular, this implies that MCG(R) is a countable group. This is a generalization of a fact that MCG(R) acts discontinuously on T o = T(R) for an analytically finite Riemann surface R.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
P. Buser, Geometry and Spectra of Compact Riemann Surface; Birkhäuser, 1992.
A. Douady and C. Earle, Conformally natural extension of homeomorphisms of the circle, Acta Math. 157 (1986), 23–48.
C. Earle and F. Gardiner, Geometric isomorphisms between infinite dimensional Teichmüller spaces, Trans. Amer. Math. Soc. 348 (1996), 1163–1190.
C. Earle, F. Gardiner and N. Lakic, Asymptotic Teichmüller space, Part I: the complex structure, In the Tradition of Ahlfors and Bers, Contemp. Math. 256, 2000, pp. 17–38.
C. Earle, F. Gardiner and N. Lakic, Asymptotic Teichmüller space, Part II: the metric structure, In the Tradition of Ahlfors and Bers III, Contemp. Math. 355, 2004, pp. 187–219.
C. Earle, V. Markovic and D. Šarić, Barycentric extension and the Bers embedding for asymptotic Teichmüller space, Complex Manifolds and Hyperbolic Geometry, Contemp. Math. 311, 2002, pp. 87–105.
E. Fujikawa, The action of geometric automorphisms of asymptotic Teichmüller spaces, Michigan Math. J. 54 (2006), 269–282.
E. Fujikawa, H. Shiga and M. Taniguchi, On the action of the mapping class group for Riemann surfaces of infinite type, J. Math. Soc. Japan 56 (2004), 1069–1086.
F. Gardiner and N. Lakic, Quasiconformal Teichmüller Theory, Amer. Math. Soc. 2000.
O. Lehto, Univalent Functions and Teichmüller Spaces, Springer, 1986.
K. Matsuzaki, The infinite direct product of Dehn twists acting on infinite dimensional Teichmüller spaces, Kodai Math. J. 26 (2003), 279–287.
K. Matsuzaki, A countable Teichmüller modular group, Trans. Amer. Math. Soc. 357 (2005), 3119–3131.
K. Matsuzaki, A quasiconformal mapping class group acting trivially on the asymptotic Teichmüller space, Proc. Amer. Math. Soc. 135 (2007), 2573–2579.
K. Matsuzaki, A classification of the modular transformations of infinite dimensional Teichmüller spaces, in In the tradition of Ahlfors and Bers: Proceedings of the Ahlfors-Bers Colloquium 2005, Contemp. Math. 432, 2007, to appear.
T. Nakanishi and J. Velling, On inner radii of Teichmüller spaces, Prospects in Complex Geometry, Lecture Notes in Math., vol. 1468, Springer, 1991, pp. 115–126.
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Matsuzaki, K. Quasiconformal mapping class groups having common fixed points on the asymptotic Teichmüller spaces. J Anal Math 102, 1–28 (2007). https://doi.org/10.1007/s11854-007-0015-z
Received:
Revised:
Issue Date:
DOI: https://doi.org/10.1007/s11854-007-0015-z