Remote Access Transactions of the American Mathematical Society
Green Open Access

Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



Hyperbolic structures for surfaces of infinite type

Author: Ara Basmajian
Journal: Trans. Amer. Math. Soc. 336 (1993), 421-444
MSC: Primary 30F35; Secondary 30F40, 57M50
MathSciNet review: 1087051
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: Our main objective is to understand the geometry of hyperbolic structures on surfaces of infinite type. In particular, we investigate the properties of surfaces called flute spaces which are constructed from infinite sequences of "pairs of pants," each glued to the next along a common boundary geodesic. Necessary and sufficient conditions are supplied for a flute space to be constructed using only "tight pants," along with sufficient conditions on when the hyperbolic structure is complete. An infinite version of the Klein-Maskit combination theorem is derived.

Finally, using the above constructions a number of applications to the deformation theory of infinite type hyperbolic surfaces are examined.

References [Enhancements On Off] (What's this?)

  • [A] W. Abikoff, The real analytic theory of Teichmüller space, Lecture Notes in Math., vol. 820, Springer-Verlag, New York, 1980. MR 590044 (82a:32028)
  • [A-S] L. Ahlfors and L. Sario, Riemann surfaces, Princeton Univ. Press, Princeton, N.J., 1960. MR 0114911 (22:5729)
  • [B-l] A. Basmajian, Hyperbolic invariants for infinitely generated Fuchsian groups, Ph.D. thesis, Stony Brook University, 1987.
  • [B-2] -, Constructing pairs of pants, Ann. Acad. Sci. Fenn. Ser. A I Math. 15 (1990), 65-74. MR 1050782 (91g:57041)
  • [Be] A. Beardon, The geometry of discrete groups, Springer-Verlag, New York, 1983. MR 698777 (85d:22026)
  • [G] F. Gardiner, Teichmüller theory and quadratic differentials, Wiley, New York, 1987. MR 903027 (88m:32044)
  • [K] L. Keen, On infinitely generated Fuchsian groups, J. Indian Math. Soc 35 (1971), 67-85. MR 0306348 (46:5474)
  • [M] B. Maskit, Kleinian groups, Springer-Verlag, New York, 1987. MR 959135 (90a:30132)
  • [N-S] M. Nakai and L. Sario, Classification theory of Riemann surfaces, Springer-Verlag, New York, 1970. MR 0264064 (41:8660)
  • [P-1] N. Purzitsky, A cutting and pasting of non-compact polygons with applications to Fuchsian groups, Acta Math. 143 (1979), 233-250. MR 549777 (80k:30045)
  • [P-2] -, Fricke polygons for infinitely generated fuchsian groups. I, preprint.
  • [T] W. Thurston, The geometry and topology of $ 3$-manifolds, Lecture Notes, 1977.

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC: 30F35, 30F40, 57M50

Retrieve articles in all journals with MSC: 30F35, 30F40, 57M50

Additional Information

Keywords: Fuchsian group, hyperbolic surface, Riemann surface, pair of pants
Article copyright: © Copyright 1993 American Mathematical Society

American Mathematical Society