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 $ 4$-dimensional Kleinian group


Authors: B. H. Bowditch and G. Mess
Journal: Trans. Amer. Math. Soc. 344 (1994), 391-405
MSC: Primary 57S30; Secondary 20H10, 30F40, 57M50
DOI: https://doi.org/10.1090/S0002-9947-1994-1240944-6
MathSciNet review: 1240944
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: We give an example of a 4-dimensional Kleinian group which is finitely generated but not finitely presented, and is a subgroup of a cocompact Kleinian group.


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

  • [Ah] L. V. Ahlfors, Finitely generated Kleinian groups, Amer. J. Math. 86 (1964), 413-429. MR 0167618 (29:4890)
  • [ApT] B. N. Apanasov and A. V. Tetenov, Nontrivial cobordisms with geometrically finite hyperbolic structures, J. Differential Geom. 28 (1988), 407-422. MR 965222 (89k:57033)
  • [Bea] A. Beardon, The geometry of discrete groups, Graduate Texts in Math. 91, Springer-Verlag, 1983. MR 698777 (85d:22026)
  • [BesC] M. Bestvina and D. Cooper, A wild Cantor set as the limit set of a conformal group action on $ {S^3}$, Proc. Amer. Math. Soc. 99 (1987), 623-626. MR 877028 (88b:57015)
  • [Bo] F. Bonahon, Bouts des variétés hyperboliques de dimension 3, Ann. of Math. 124 (1986), 71-158. MR 847953 (88c:57013)
  • [D] M. Davis, A hyperbolic 4-manifold, Proc. Amer. Math. Soc. 93 (1985), 325-328. MR 770546 (86h:57016)
  • [FMc] M. Feighn and D. McCullough, Finiteness theorems for 3-manifolds with boundary, Amer. J. Math. 109 (1987), 1155-1169; Correction, ibid. 112 (1990), 41-45.
  • [FMe] M. Feighn and G. Mess, Conjugacy classes of finite subgroups of Kleinian groups, Amer. J. Math. 113 (1991), 179-188. MR 1087807 (92a:57042)
  • [GLT] M. Gromov, M. B. Lawson, and W. P. Thurston, Hyperbolic 4-manifolds, and conformally flat 3-manifolds, Publ. Math. I.H.E.S. 68 (1988), 27-45. MR 1001446 (90k:57021)
  • [Ka1] M. Kapovich, Flat conformal structures on 3-manifolds. I, Siberian Math. J. 30 (1989), no. 5, 60-73. MR 1025290 (91b:57017)
  • [Ka2] -, On absence of Sullivan's cusp finiteness theorem in higher dimensions, preprint I.H.E.S. (1990).
  • [KaP] M. Kapovich and L. Potyagailo, On absence of Ahlfors' finiteness theorem for Kleinian groups in dimension 3, Topology Appl. 40 (1991), 83-91. MR 1114093 (92j:57023)
  • [Kui] N. H. Kuiper, Hyperbolic 4-manifolds and tessellations, Publ. Math. I.H.E.S. 68 (1988), 47-76. MR 1001447 (90k:57022)
  • [KulS] R. Kulkarni and P. Shalen, On Ahlfors' finiteness theorem, Adv. Math. 76 (1989), 155-169. MR 1013665 (91e:57067)
  • [L] D. Long, Immersions and embeddings of totally geodesic surfaces, Bull London Math. Soc. 19 (1987), 481-484. MR 898729 (89g:57014)
  • [M] S. Matsumoto, Foundations of flat conformal structures, Aspects of Low Dimensional Manifolds, Adv. Stud. in Pure Math., vol. 20, Academic Press, 1992, pp. 167-261. MR 1208312 (93m:57014)
  • [N] V. V. Nikulin, On the classification of arithmetic groups generated by reflection in Lobachevskiĭ spaces, Math. USSR Izv. 18 (1982), 99-123. MR 607579 (83b:57025)
  • [P1] L. Potyagailo, The problem of finiteness for Kleinian groups in 3-space, Proc. Internat. Conf. "Knots-90" (A. Kawauchi, ed., Osaka, 1992), De Gruyter (to appear). MR 1177449 (93h:57025)
  • [P2] -, Finitely generated Kleinian groups in 3-space and 3-manifolds of infinite topological type, Trans. Amer. Math. Soc. (to appear).
  • [Sc1] P. Scott, Compact submanifolds of 3-manifolds, J. London Math. Soc. (2) 7 (1973), 246-250. MR 0326737 (48:5080)
  • [Sc2] -, Subgroups of surface groups are almost geometric, J. London Math. Soc. (2) 17 (1978), 555-565; Correction, ibid. 32 (1985), 217-220. MR 0494062 (58:12996)
  • [Se] A. Selberg, On discontinuous groups in higher dimensional symmetric spaces, Contributions to Function Theory, Tata Inst. of Fund. Res., Bombay, 1960, pp. 147-164. MR 0130324 (24:A188)
  • [Su1] D. Sullivan, A finiteness theorem for cusps, Acta Math. 147 (1981), 289-299. MR 639042 (83f:30043)
  • [Su2] -, Travaux de Thurston sur les groupes quasifuchsiens et les variétés hyperboliques de dimension 3 fibrées sur $ {S^1}$, Lecture Notes in Math., vol. 842, Springer Verlag, 1980.
  • [T] W. P. Thurston, The geometry and topology of three-manifolds, Notes, Princeton Univ. Math., Department 1979.
  • [V] A. Yu. Vesnin, Three-dimensional hyperbolic manifolds of the Loebell type, Siberian Math. J. 28 (1987), no. 3, 731-733. MR 924975 (89f:57022)

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC: 57S30, 20H10, 30F40, 57M50

Retrieve articles in all journals with MSC: 57S30, 20H10, 30F40, 57M50


Additional Information

DOI: https://doi.org/10.1090/S0002-9947-1994-1240944-6
Article copyright: © Copyright 1994 American Mathematical Society

American Mathematical Society