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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: [{\it T. Jørgensen} and {\it P. Klein}, Quart. J. Math. 33, 325--332 (1982; Zbl 0499.30033); {\it G. J. Martin}, Acta Math. 163, No. 3--4, 253--289 (1989; Zbl 0698.20037); {\it 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.
30F40Kleinian groups
20H10Fuchsian groups and their generalizations (group theory)
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