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Transactions of the American Mathematical Society

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

 
 

 

Intersections of analytically and geometrically finite subgroups of Kleinian groups


Author: James W. Anderson
Journal: Trans. Amer. Math. Soc. 343 (1994), 87-98
MSC: Primary 30F40; Secondary 57M50
DOI: https://doi.org/10.1090/S0002-9947-1994-1207578-0
MathSciNet review: 1207578
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Abstract: We consider the intersection of pairs of subgroups of a Kleinian group of the second kind K whose limit sets intersect, where one subgroup G is analytically finite and the other J is geometrically finite, possibly infinite cyclic. In the case that J is infinite cyclic generated by M, we show that either some power of M lies in G or there is a doubly cusped parabolic element Q of G with the same fixed point as M. In the case that J is nonelementary, we show that the intersection of the limit sets of G and J is equal to the limit set of the intersection $ G \cap J$ union with a subset of the rank 2 parabolic fixed points of K. Hence, in both cases, the limit set of the intersection is essentially equal to the intersection of the limit sets. The main facts used in the proof are results of Beardon and Pommerenke [4] and Canary [6], which yield that the Poincaré metric on the ordinary set of an analytically finite Kleinian group G is comparable to the Euclidean distance to the limit set of G.


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  • [1] J. W. Anderson, Intersections of topologically tame subgroups of Kleinian groups, preprint. MR 1335369 (96i:30035)
  • [2] -, On the finitely generated intersection property for Kleinian groups, Complex Variables 17 (1991), 111-112. MR 1123809 (92i:30042)
  • [3] A. Beardon and B. Maskit, Limit points of Kleinian groups and finite sided fundamental polyhedra, Acta Math. 132 (1974), 1-12. MR 0333164 (48:11489)
  • [4] A. Beardon and C. Pommerenke, The Poincaré metric of plane domains, J. London Math. Soc. (2) 18 (1978), 475-483. MR 518232 (80a:30020)
  • [5] R. Canary, Covering theorems for hyperbolic 3-manifolds, preprint. MR 1316167 (95m:57024)
  • [6] -, The Poincaré metric and a conformal version of a theorem of Thurston, Duke Math. J. 64 (1991), 349-359. MR 1136380 (92k:57020)
  • [7] J. Hempel, The finitely generated intersection property for Kleinian groups, Knot Theory and Manifolds, (D. Rolfsen, ed.), Lecture Notes in Math., vol. 1144, Springer-Verlag, Berlin and New York, 1985. MR 823280 (88j:20049)
  • [8] I. Kra, Automorphic forms and Kleinian groups, Benjamin, Reading, MA, 1972. MR 0357775 (50:10242)
  • [9] B. Maskit, Intersections of component subgroups of Kleinian groups, Discontinuous Groups and Riemann Surfaces, (L. Greenberg, ed.), Ann. of Math. Stud., no. 79, Princeton Univ. Press, Princeton, NJ, 1974. MR 0355037 (50:7514)
  • [10] -, Kleinian groups, Springer-Verlag, Berlin, 1988. MR 959135 (90a:30132)
  • [11] J. Morgan, On Thurston's uniformization theorem for three-dimensional manifolds, The Smith Conjecture (H. Bass and J. Morgan, eds.), Academic Press, San Diego, 1984. MR 758464
  • [12] T. Soma, Function groups in Kleinian groups, Math. Ann. 292 (1992), 181-190. MR 1141792 (93j:30048)
  • [13] P. Susskind, Kleinian groups with intersecting limit sets, J. Analyse Math. 52 (1989), 26-38. MR 981494 (90f:57045)
  • [14] P. Susskind and G. A. Swarup, Limit sets of geometrically finite hyperbolic groups, Amer. J. Math. 114 (1992), 233-250. MR 1156565 (94d:57066)

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DOI: https://doi.org/10.1090/S0002-9947-1994-1207578-0
Article copyright: © Copyright 1994 American Mathematical Society

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