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)

 
 

 

A characterization of cocompact hyperbolic and finite-volume hyperbolic groups in dimension three


Authors: J. W. Cannon and Daryl Cooper
Journal: Trans. Amer. Math. Soc. 330 (1992), 419-431
MSC: Primary 22E40; Secondary 30F40, 53C70, 57M15
DOI: https://doi.org/10.1090/S0002-9947-1992-1036000-0
MathSciNet review: 1036000
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: We show that a cocompact hyperbolic group in dimension $ 3$ is characterized by certain properties of its word metric which depend only on the group structure and not on any action on hyperbolic space. We prove a similar theorem for finite-volume hyperbolic groups in dimension $ 3$.


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

  • [BK] P. Buser and H. Karcher, Gromov's almost flat manifolds, Astérisque 81 (1981).
  • [C1] J. W. Cannon, The combinatorial structure of cocompact discrete hyperbolic groups, Geom. Dedicata 16 (1984), 123-148. MR 758901 (86j:20032)
  • [C2] -, The theory of negatively curved spaces and groups, Ergodic Theory, Symbolic Dynamics and Hyperbolic Spaces (Tim Bedford, Michael Keane and Caroline Series, eds.), Oxford Univ. Press, Oxford, New York and Tokyo, 1991, pp. 315-369. MR 1130170 (93e:58002)
  • [F] W. J. Floyd, Group completions and limit sets of Kleinian groups, Invent. Math. 57 (1980), 205-218. MR 568933 (81e:57002)
  • [G1] M. Gromov, Hyperbolic manifolds, Groups and Actions, Proc. Stony Brook Conf., Ann. of Math. Studies No. 97, Princeton Univ. Press, Princeton, N.J., 1981, pp. 183-215. MR 624814 (82m:53035)
  • [G2] -, Infinite groups as geometric objects, Proc. Internat. Congr. Math., Elsevier, New York, 1983, pp. 385-392. MR 804694 (87c:57033)
  • [G3] -, Groups of polynomial growth and expanding maps, Inst. Hautes Études Sci. Publ. Math. 53 (1981), 53-73. MR 623534 (83b:53041)
  • [Ma] Bernard Maskit, Kleinian groups, Springer-Verlag, Berlin, Heidelberg and New York, 1988. MR 959135 (90a:30132)
  • [M] G. D. Mostow, Strong rigidity of locally symmetric spaces, Ann. of Math. Studies No. 78, Princeton Univ. Press, N.J., 1972. MR 0385004 (52:5874)
  • [S] D. Sullivan, On the ergodic theory at infinity of an arbitrary discrete group of hyperbolic motions, Proc. Stony Brook Conf., Ann. of Math. Studies No. 97, Princeton Univ. Press, Princeton, N.J., 1981, pp. 465-496. MR 624833 (83f:58052)
  • [T] W. P. Thurston, The geometry and topology of $ 3$-manifolds, Lecture Notes, Princeton Univ., 1978.
  • [TV] P. Tukia and J. Väisälä, Quasiconformal extension from dimension $ n$ to $ n + 1$, Ann. of Math. (2) 115 (1982), 331-348. MR 647809 (84i:30030)

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC: 22E40, 30F40, 53C70, 57M15

Retrieve articles in all journals with MSC: 22E40, 30F40, 53C70, 57M15


Additional Information

DOI: https://doi.org/10.1090/S0002-9947-1992-1036000-0
Article copyright: © Copyright 1992 American Mathematical Society

American Mathematical Society