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Algebraic and geometric convergence of Kleinian groups. (English) Zbl 0738.30032
The purpose of this paper is to compare two different notions of convergence of sequences of finitely generated Kleinian groups namely that of algebraic convergence (i.e. convergence of marked elements) and geometric convergence (i.e. convergence of Dirichlet fundamental polyhedra with a fixed centre). The latter concept can be given more intrinsically as the authors point out; if \(G_ n(n\geq 1)\) is a sequence of Kleinian groups then for an infinite subset \(S\) of \(\mathbb{N}\) we let \(G_{\infty}(S)\) be the derived set of \(\bigcup_{j\in S}G_ j\); if this does not depend on \(S\) and is non-elementary then one says that the sequence \(G_ n\) is geometrically convergent to the common \(G_{\infty}(S)=:G_{\infty}\). It is not in general true that if the algebraic limit exists then it is the geometric limit. One of the main results of this paper gives conditions for this to happen. Here the \(G_ n\) should converge algebraically to \(G_{\infty}\) which is assumed to be geometrically finite and of the second kind (i.e. the ordinary set is non-trivial). Then the sequence \(G_ n\) will converge geometrically to \(G_{\infty}\) if and only if the ordinary sets of the \(G_ n\) converge to that of \(G\) in the sense of Carathéodory (i.e. every compact subset of the latter lies in all of the former from some point on). The other main theorem describes what happens when the two limits do not coincide.

30F40 Kleinian groups (aspects of compact Riemann surfaces and uniformization)
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