There are two distinct but related combination theorems: the amalgamated free product, and the HNN extension. The basic outlines of these, in a purely abstract setting, are given in sections A and D. For Kleinian groups, the purely abstract setting is sufficient to prove that the combined group G is discrete and has the named group theoretic structure, but does not suffice to give a clear understanding of Ω/G or of ℍ3/G; nor does it yield sufficient information to decide whether or not G is geometrically finite. The necessary inequality is given in section B. The combination theorems themselves (these are sometimes known as the Klein-Maskit combination theorems) are given in sections C and E. We state and prove these theorems only for discrete subgroups of M. The minor modifications required for the case that G contains orientation reversing elements are left to the reader.
KeywordsNormal Form Limit Point Kleinian Group Simple Closed Curve Finite Subgroup
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