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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 .

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Groups Quasi-isometric to Complex Hyperbolic Space
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by Richard Chow PDF
Trans. Amer. Math. Soc. 348 (1996), 1757-1769 Request permission

Abstract:

We show that any finitely generated group quasi-isometric to complex hyperbolic space is a finite extension of a properly discontinuous, cocompact subgroup of the isometry group.
References
  • J. W. Cannon and Daryl Cooper, A characterization of cocompact hyperbolic and finite-volume hyperbolic groups in dimension three, Trans. Amer. Math. Soc. 330 (1992), no. 1, 419–431. MR 1036000, DOI 10.1090/S0002-9947-1992-1036000-0
  • A. Casson and D. Jungreis, Convergence groups and Seifert fibered 3-manifolds, Inventiones Math. 118 (1994), 441–456.
  • D. B. A. Epstein, Complex hyperbolic geometry, Analytical and geometric aspects of hyperbolic space (Coventry/Durham, 1984) London Math. Soc. Lecture Note Ser., vol. 111, Cambridge Univ. Press, Cambridge, 1987, pp. 93–111. MR 903851
  • Herbert Federer, Geometric measure theory, Die Grundlehren der mathematischen Wissenschaften, Band 153, Springer-Verlag New York, Inc., New York, 1969. MR 0257325
  • David Gabai, Convergence groups are Fuchsian groups, Ann. of Math. (2) 136 (1992), no. 3, 447–510. MR 1189862, DOI 10.2307/2946597
  • F. W. Gehring and J. C. Kelly, Quasi-conformal mappings and Lebesgue density, Discontinuous Groups and Riemann Surfaces, Proceedings of the 1973 Conference at the University of Maryland (L. Greenberg, ed.), Ann. of Math. Studies 79, Princeton University Press (1974), 171–179.
  • S. M. Gersten, Bounded cocycles and combings of groups, Internat. J. Algebra Comput. 2 (1992), no. 3, 307–326. MR 1189238, DOI 10.1142/S0218196792000190
  • Étienne Ghys, Les groupes hyperboliques, Astérisque 189-190 (1990), Exp. No. 722, 203–238 (French). Séminaire Bourbaki, Vol. 1989/90. MR 1099877
  • É. Ghys and P. de la Harpe (eds.), Sur les groupes hyperboliques d’après Mikhael Gromov, Progress in Mathematics, vol. 83, Birkhäuser Boston, Inc., Boston, MA, 1990 (French). Papers from the Swiss Seminar on Hyperbolic Groups held in Bern, 1988. MR 1086648, DOI 10.1007/978-1-4684-9167-8
  • W. M. Goldman, Complex Hyperbolic Geometry, preprint.
  • Mikhael Gromov, Groups of polynomial growth and expanding maps, Inst. Hautes Études Sci. Publ. Math. 53 (1981), 53–73. MR 623534
  • Stephen Donkin, Invariants of unipotent radicals, Math. Z. 198 (1988), no. 1, 117–125. MR 938033, DOI 10.1007/BF01183043
  • M. Gromov and P. Pansu, Rigidity of lattices: an introduction, Geometric topology: recent developments (Montecatini Terme, 1990) Lecture Notes in Math., vol. 1504, Springer, Berlin, 1991, pp. 39–137. MR 1168043, DOI 10.1007/BFb0094289
  • A. Koranyi and H. M. Reimann, Foundations for the theory of quasiconformal mappings on the Heisenberg group, Advances in Math. 111 (1995), 1–87.
  • A. Korányi and H. M. Reimann, Quasiconformal mappings on the Heisenberg group, Invent. Math. 80 (1985), no. 2, 309–338. MR 788413, DOI 10.1007/BF01388609
  • Hans Maass, Siegel’s modular forms and Dirichlet series, Lecture Notes in Mathematics, Vol. 216, Springer-Verlag, Berlin-New York, 1971. Dedicated to the last great representative of a passing epoch. Carl Ludwig Siegel on the occasion of his seventy-fifth birthday. MR 0344198
  • G. D. Mostow, Quasi-conformal mappings in $n$-space and the rigidity of hyperbolic space forms, Inst. Hautes Études Sci. Publ. Math. 34 (1968), 53–104. MR 236383
  • Pierre Pansu, Métriques de Carnot-Carathéodory et quasiisométries des espaces symétriques de rang un, Ann. of Math. (2) 129 (1989), no. 1, 1–60 (French, with English summary). MR 979599, DOI 10.2307/1971484
  • E. Rieffel, Groups coarse quasi-isometric to $\mathbf {H}^2 \times \mathbf {R}$, preprint.
  • R. Schwartz, The quasi-isometry classification of rank one lattices, Publ. I.H.E.S. (to appear).
  • Audrey Terras, Harmonic analysis on symmetric spaces and applications. II, Springer-Verlag, Berlin, 1988. MR 955271, DOI 10.1007/978-1-4612-3820-1
  • W. P. Thurston, The geometry and topology of 3-manifolds, Lecture Notes, Princeton Univ., 1978.
  • Pekka Tukia, On quasiconformal groups, J. Analyse Math. 46 (1986), 318–346. MR 861709, DOI 10.1007/BF02796595
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Additional Information
  • Richard Chow
  • Affiliation: Department of Mathematics, National University of Singapore, Singapore 0511
  • Address at time of publication: Department of Mathematics, University of California, Los Angeles, California 90024
  • Email: rchow@math.ucla.edu
  • Received by editor(s): January 30, 1995
  • Received by editor(s) in revised form: May 4, 1995
  • © Copyright 1996 American Mathematical Society
  • Journal: Trans. Amer. Math. Soc. 348 (1996), 1757-1769
  • MSC (1991): Primary 20F32, 30C65
  • DOI: https://doi.org/10.1090/S0002-9947-96-01522-X
  • MathSciNet review: 1329530