On Klein's combination theorem. IV

Author:
Bernard Maskit

Journal:
Trans. Amer. Math. Soc. **336** (1993), 265-294

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.

**[A]**William Abikoff,*The residual limit sets of Kleinian groups*, Acta Math.**130**(1973), 127–144. MR**0404613****[M-M]**A. Marden and B. Maskit,*On the isomorphism theorem for Kleinian groups*, Invent. Math.**51**(1979), no. 1, 9–14. MR**524274**, 10.1007/BF01389909**[M1]**Bernard Maskit,*On Klein’s combination theorem*, Trans. Amer. Math. Soc.**120**(1965), 499–509. MR**0192047**, 10.1090/S0002-9947-1965-0192047-1**[M2]**Bernard Maskit,*On Klein’s combination theorem. II*, Trans. Amer. Math. Soc.**131**(1968), 32–39. MR**0223570**, 10.1090/S0002-9947-1968-0223570-1**[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. 297-316.**[M4]**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****[M5]**Bernard Maskit,*Kleinian groups*, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 287, Springer-Verlag, Berlin, 1988. MR**959135**

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DOI:
https://doi.org/10.1090/S0002-9947-1993-1137258-0

Article copyright:
© Copyright 1993
American Mathematical Society