On Klein's combination theorem. IV
Author:
Bernard Maskit
Journal:
Trans. Amer. Math. Soc. 336 (1993), 265294
MSC:
Primary 30F40
MathSciNet review:
1137258
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Abstract: This paper contains an expansion of the combination theorems to cover the following problems. New rank parabolic subgroups are produced, while, as in previous versions, all elliptic and parabolic elements are tracked. A proof is given that the combined group is analytically finite if and only if the original groups are; in the analytically finite case, we also give a formula for the hyperbolic area of the combined group (i.e., the hyperbolic area of the set of discontinuity on the sphere modulo ) in terms of the hyperbolic areas of the original groups. There is also a new variation on the first combination theorem in which the common subgroup has finite index in one of the two groups.
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William
Abikoff, The residual limit sets of Kleinian groups, Acta
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 [MM]
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Marden and B.
Maskit, On the isomorphism theorem for Kleinian groups,
Invent. Math. 51 (1979), no. 1, 9–14. MR 524274
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Maskit, On Klein’s combination
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Bernard
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, On Kleins' combination theorem. III. Advances in the Theory of Riemann Surfaces, Ann. of Math. Stud., no. 66, Princeton Univ. Press, Princeton, N.J., 1971, pp. 297316.
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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,
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Maskit, Kleinian groups, Grundlehren der Mathematischen
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 [A]
 William Abikoff, The residual limit sets of Kleinian groups, Acta Math. 130 (1973), 127144. MR 0404613 (53:8413)
 [MM]
 Albert Marden and Bernard Maskit, On the isomorphism theorem for Kleinian groups, Invent. Math. 51 (1979), 914. MR 524274 (80f:30037)
 [M1]
 Bernard Maskit, On Klein's combination theorem, Trans. Amer. Math. Soc. 120 (1965), 499509. MR 0192047 (33:274)
 [M2]
 , On Klein's combination theorem. II, Trans. Amer. Math. Soc. 131 (1968), 3239. MR 0223570 (36:6618)
 [M3]
 , On Kleins' combination theorem. III. Advances in the Theory of Riemann Surfaces, Ann. of Math. Stud., no. 66, Princeton Univ. Press, Princeton, N.J., 1971, pp. 297316.
 [M4]
 , Intersections of component subgroups of Kleinian groups, Discontinuous Groups and Riemann Surfaces, Ann. of Math. Stud., no. 79, Princeton Univ. Press, Princeton, N.J., 1974, pp. 349367. MR 0355037 (50:7514)
 [M5]
 , Kleinian groups, SpringerVerlag, Berlin, Heidelberg, and New York, 1988. MR 959135 (90a:30132)
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Additional Information
DOI:
http://dx.doi.org/10.1090/S00029947199311372580
PII:
S 00029947(1993)11372580
Article copyright:
© Copyright 1993
American Mathematical Society
