Let be the -dimensional sense-preserving Möbius group or the unitary group . Let be a sequence of subgroups in the group generated by , where . If for each as , then we say that algebraically converges to .
Let us suppose that for each , the group is a Kleinian group. When is 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 be a subgroup of containing a loxodromic element. Denote by the set
where is the set of all loxodromic elements in and
The author proves in the first theorem that given a sequence of subgroups of , if each is discrete, then the algebraic limit group is either a complex Kleinian group, or it is elementary, or is not finite.
In the second theorem it is proved that if the groups are complex Kleinian groups and satisfies the IP-condition, then the algebraic limit group is a complex Kleinian group, where we say that satisfies the IP-condition if the following conditions are satisfied: for any sequence , if for each , and as with being the identity or parabolic, then has uniformly bounded torsion.