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On Klein's Combination Theorem. IV

Bernard Maskit
Transactions of the American Mathematical Society
Vol. 336, No. 1 (Mar., 1993), pp. 265-294
DOI: 10.2307/2154347
Stable URL: http://www.jstor.org/stable/2154347
Page Count: 30
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On Klein's Combination Theorem. IV
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Abstract

This paper contains an expansion of the combination theorems to cover the following problems. New rank 1 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 2-sphere modulo G) 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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