Skip to Main Content

Transactions of the American Mathematical Society

Published by the American Mathematical Society since 1900, Transactions of the American Mathematical Society is devoted to longer research articles in all areas of pure and applied mathematics.

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

The 2020 MCQ for Transactions of the American Mathematical Society is 1.48.

What is MCQ? The Mathematical Citation Quotient (MCQ) measures journal impact by looking at citations over a five-year period. Subscribers to MathSciNet may click through for more detailed information.

 

A characterization of cocompact hyperbolic and finite-volume hyperbolic groups in dimension three
HTML articles powered by AMS MathViewer

by J. W. Cannon and Daryl Cooper
Trans. Amer. Math. Soc. 330 (1992), 419-431
DOI: https://doi.org/10.1090/S0002-9947-1992-1036000-0
PDF | Request permission

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
    P. Buser and H. Karcher, Gromov’s almost flat manifolds, Astérisque 81 (1981).
  • James W. Cannon, The combinatorial structure of cocompact discrete hyperbolic groups, Geom. Dedicata 16 (1984), no. 2, 123–148. MR 758901, DOI 10.1007/BF00146825
  • Tim Bedford, Michael Keane, and Caroline Series (eds.), Ergodic theory, symbolic dynamics, and hyperbolic spaces, Oxford Science Publications, The Clarendon Press, Oxford University Press, New York, 1991. Papers from the Workshop on Hyperbolic Geometry and Ergodic Theory held in Trieste, April 17–28, 1989. MR 1130170
  • William J. Floyd, Group completions and limit sets of Kleinian groups, Invent. Math. 57 (1980), no. 3, 205–218. MR 568933, DOI 10.1007/BF01418926
  • M. Gromov, Hyperbolic manifolds, groups and actions, Riemann surfaces and related topics: Proceedings of the 1978 Stony Brook Conference (State Univ. New York, Stony Brook, N.Y., 1978) Ann. of Math. Stud., vol. 97, Princeton Univ. Press, Princeton, N.J., 1981, pp. 183–213. MR 624814
  • Mikhael Gromov, Infinite groups as geometric objects, Proceedings of the International Congress of Mathematicians, Vol. 1, 2 (Warsaw, 1983) PWN, Warsaw, 1984, pp. 385–392. MR 804694
  • Mikhael Gromov, Groups of polynomial growth and expanding maps, Inst. Hautes Études Sci. Publ. Math. 53 (1981), 53–73. MR 623534
  • Bernard Maskit, Kleinian groups, Grundlehren der mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 287, Springer-Verlag, Berlin, 1988. MR 959135
  • G. D. Mostow, Strong rigidity of locally symmetric spaces, Annals of Mathematics Studies, No. 78, Princeton University Press, Princeton, N.J.; University of Tokyo Press, Tokyo, 1973. MR 0385004
  • Dennis Sullivan, On the ergodic theory at infinity of an arbitrary discrete group of hyperbolic motions, Riemann surfaces and related topics: Proceedings of the 1978 Stony Brook Conference (State Univ. New York, Stony Brook, N.Y., 1978) Ann. of Math. Stud., vol. 97, Princeton Univ. Press, Princeton, N.J., 1981, pp. 465–496. MR 624833
    • W. P. Thurston, The geometry and topology of $3$-manifolds, Lecture Notes, Princeton Univ., 1978.
  • P. Tukia and J. Väisälä, Quasiconformal extension from dimension $n$ to $n+1$, Ann. of Math. (2) 115 (1982), no. 2, 331–348. MR 647809, DOI 10.2307/1971394
Similar Articles
Bibliographic Information
  • © Copyright 1992 American Mathematical Society
  • 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