Intersections of analytically and geometrically finite subgroups of Kleinian groups
Author:
James W. Anderson
Journal:
Trans. Amer. Math. Soc. 343 (1994), 8798
MSC:
Primary 30F40; Secondary 57M50
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 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.
 [1]
James
W. Anderson, Intersections of topologically tame subgroups of
Kleinian groups, J. Anal. Math. 65 (1995),
77–94. MR
1335369 (96i:30035), http://dx.doi.org/10.1007/BF02788766
 [2]
James
W. Anderson, On the finitely generated intersection property for
Kleinian groups, Complex Variables Theory Appl. 17
(1991), no. 12, 111–112. MR 1123809
(92i:30042)
 [3]
Alan
F. Beardon and Bernard
Maskit, Limit points of Kleinian groups and finite sided
fundamental polyhedra, Acta Math. 132 (1974),
1–12. MR
0333164 (48 #11489)
 [4]
A.
F. Beardon and Ch.
Pommerenke, The Poincaré metric of plane domains, J.
London Math. Soc. (2) 18 (1978), no. 3,
475–483. MR
518232 (80a:30020), http://dx.doi.org/10.1112/jlms/s218.3.475
 [5]
Richard
D. Canary, Covering theorems for hyperbolic 3manifolds,
Lowdimensional topology (Knoxville, TN, 1992) Conf. Proc. Lecture Notes
Geom. Topology, III, Int. Press, Cambridge, MA, 1994, pp. 21–30.
MR
1316167 (95m:57024)
 [6]
Richard
D. Canary, The Poincaré metric and a conformal version of a
theorem of Thurston, Duke Math. J. 64 (1991),
no. 2, 349–359. MR 1136380
(92k:57020), http://dx.doi.org/10.1215/S0012709491064173
 [7]
John
Hempel, The finitely generated intersection property for Kleinian
groups, Knot theory and manifolds (Vancouver, B.C., 1983) Lecture
Notes in Math., vol. 1144, Springer, Berlin, 1985,
pp. 18–24. MR 823280
(88j:20049), http://dx.doi.org/10.1007/BFb0075010
 [8]
Irwin
Kra, Automorphic forms and Kleinian groups, W. A. Benjamin,
Inc., Reading, Mass., 1972. Mathematics Lecture Note Series. MR 0357775
(50 #10242)
 [9]
Bernard
Maskit, Intersections of component subgroups of Kleinian
groups, Discontinuous groups and Riemann surfaces (Proc. Conf., Univ.
Maryland, College Park, Md., 1973) Princeton Univ. Press, Princeton,
N.J., 1974, pp. 349–367. Ann. of Math. Studies, No. 79. MR 0355037
(50 #7514)
 [10]
Bernard
Maskit, Kleinian groups, Grundlehren der Mathematischen
Wissenschaften [Fundamental Principles of Mathematical Sciences],
vol. 287, SpringerVerlag, Berlin, 1988. MR 959135
(90a:30132)
 [11]
John
W. Morgan, On Thurston’s uniformization theorem for
threedimensional manifolds, The Smith conjecture (New York, 1979)
Pure Appl. Math., vol. 112, Academic Press, Orlando, FL, 1984,
pp. 37–125. MR
758464, http://dx.doi.org/10.1016/S00798169(08)616372
 [12]
Teruhiko
Soma, Function groups in Kleinian groups, Math. Ann.
292 (1992), no. 1, 181–190. MR 1141792
(93j:30048), http://dx.doi.org/10.1007/BF01444616
 [13]
Perry
Susskind, Kleinian groups with intersecting limit sets, J.
Analyse Math. 52 (1989), 26–38. MR 981494
(90f:57045), http://dx.doi.org/10.1007/BF02820470
 [14]
Perry
Susskind and Gadde
A. Swarup, Limit sets of geometrically finite hyperbolic
groups, Amer. J. Math. 114 (1992), no. 2,
233–250. MR 1156565
(94d:57066), http://dx.doi.org/10.2307/2374703
 [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), 111112. MR 1123809 (92i:30042)
 [3]
 A. Beardon and B. Maskit, Limit points of Kleinian groups and finite sided fundamental polyhedra, Acta Math. 132 (1974), 112. MR 0333164 (48:11489)
 [4]
 A. Beardon and C. Pommerenke, The Poincaré metric of plane domains, J. London Math. Soc. (2) 18 (1978), 475483. MR 518232 (80a:30020)
 [5]
 R. Canary, Covering theorems for hyperbolic 3manifolds, preprint. MR 1316167 (95m:57024)
 [6]
 , The Poincaré metric and a conformal version of a theorem of Thurston, Duke Math. J. 64 (1991), 349359. 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, SpringerVerlag, 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, SpringerVerlag, Berlin, 1988. MR 959135 (90a:30132)
 [11]
 J. Morgan, On Thurston's uniformization theorem for threedimensional 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), 181190. MR 1141792 (93j:30048)
 [13]
 P. Susskind, Kleinian groups with intersecting limit sets, J. Analyse Math. 52 (1989), 2638. MR 981494 (90f:57045)
 [14]
 P. Susskind and G. A. Swarup, Limit sets of geometrically finite hyperbolic groups, Amer. J. Math. 114 (1992), 233250. MR 1156565 (94d:57066)
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Additional Information
DOI:
http://dx.doi.org/10.1090/S00029947199412075780
PII:
S 00029947(1994)12075780
Article copyright:
© Copyright 1994
American Mathematical Society
