Algebraic convergence of finitely generated Kleinian groups in all dimensions
Let \(\{G_{i,r}\}\), \(G_{i,r}=\langle g_{11},\dots, g_{1r}\rangle\), be a sequence of \(r\)-generator subgroups of the \(n\)-dimensional orientation preserving Möbius group such that for each \(k\in \{1,\dots, r\}\) the sequence \(\{g_{ik}\}\) converges to a Möbius transformation \(g_k\). Then we say that \(\{G_{i,r}\}\) converges algebraically to the group \(G= \langle g_1,\dots, g_r\rangle\). We are interested in the case that all \(G_{i,r}\) are Kleinian groups, that is, all \(G_{i,r}\) are discrete and non-elementary. The question arises now under what conditions also the algebraic limes \(G\) is a Kleinian group. If \(\{G_{i,r}\}\) has uniformly bounded torsion, that is, if there exists an integer \(N\geq 1\) such that \(\text{ord}(g)\leq N\) or \(\text{ord}(g)=\infty\) for all \(g\in G_{i,r}\) for some \(i\), then \(G\) is indeed a Kleinian group. Here the author extends this result by replacing the hypothesis of uniformly bounded torsion by a weaker and more technical hypothesis which is called Condition A.
scientific article; zbMATH DE number 5669171