Algebraic convergence theorems of complex Kleinian groups. (English) Zbl 1262.30041

Let \(\mathbb{G}\) be the \(n\)-dimensional sense-preserving Möbius group \(M(\mathbb{\overline R}^n)\) or the unitary group \(U(1,n;\mathbb{C})\). Let \(\{G_{r,i}\}\) be a sequence of subgroups in the group \(\mathbb{G}\) generated by \(g_{1,i},g_{2,i},\dots,g_{r,i}\), where \(r=1,2,\ldots\). If for each \(t\) \((1\leq t\leq r),\) \(g_{t,i}\rightarrow g_t \in \mathbb{G}\) as \(i\rightarrow \infty\), then we say that \(\{G_{r,i}\}\) algebraically converges to \(G_r=\left<g_1,g_2,\dots,g_r\right>\).
Let us suppose that for each \(i\), the group \(G_{r,i}\) is a Kleinian group. When is \(G_r\) a Kleinian group? This problem has been studied by several authors: [T. Jørgensen and P. Klein, Quart. J. Math. 33, 325–332 (1982; Zbl 0499.30033); G. J. Martin, Acta Math. 163, No. 3–4, 253–289 (1989; Zbl 0698.20037); X. Wang, Isr. J. Math. 162, 221–233 (2007; Zbl 1161.20047)].
In the paper under review two algebraic convergence theorems are obtained. Let \(G\) be a subgroup of \(U(1,n;\mathbb{C})\) containing a loxodromic element. Denote by \(W(G)\) the set
\[ \bigcap_{f\in h(G)}G_{\text{fix}(f)}, \] where \(h(G)\) is the set of all loxodromic elements in \(G\) and
\[ G_{\text{fix}(f)}=\big\{g\in G : \text{fix}(f)\subset \text{fix}(g)\big\}. \]
The author proves in the first theorem that given a sequence \(\{G_{r,i}\}\) of subgroups of \(U(1,n;\mathbb{C})\), if each \(G_{r,i}\) is discrete, then the algebraic limit group \(G_r\) is either a complex Kleinian group, or it is elementary, or \(W(G_r)\) is not finite.
In the second theorem it is proved that if the groups \(G_{r,i}\) are complex Kleinian groups and \(\{G_{r,i}\}\) satisfies the IP-condition, then the algebraic limit group \(G_r\) is a complex Kleinian group, where we say that \(\{G_{r,i}\}\) satisfies the IP-condition if the following conditions are satisfied: for any sequence \(\{f_{i_k}\}\) \((f_{i_k} \in G_{r,i_k})\), if \(\text{card}\big(\text{fix}(f_{i_k})\big)=\infty\) for each \(k\), and \(f_{i_k}\rightarrow f\) as \(k\rightarrow \infty\) with \(f\) being the identity or parabolic, then \(\{f_{i_k}\}\) has uniformly bounded torsion.


30F40 Kleinian groups (aspects of compact Riemann surfaces and uniformization)
20H10 Fuchsian groups and their generalizations (group-theoretic aspects)
Full Text: DOI


[1] DOI: 10.1515/9783110808056
[2] DOI: 10.1006/aima.2000.1970 · Zbl 0980.20040
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