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discrete group; geometric convergence; uniformly bounded torsion
One of the basic questions in the Kleinian group theory is to understand both algebraic and geometric limiting behavior of sequences of discrete subgroups. In this paper we consider the geometric convergence in the setting of the isometric group of the real or complex hyperbolic space. It is known that if $\Gamma $ is a non-elementary finitely generated group and $\rho _{i}\colon \Gamma \rightarrow {\rm SO}(n,1)$ a sequence of discrete and faithful representations, then the geometric limit of $\rho _{i}(\Gamma )$ is a discrete subgroup of ${\rm SO}(n,1)$. We generalize this result by showing that for a sequence of discrete and non-elementary subgroups $\{G_{j}\}$ of ${\rm SO}(n,1)$ or ${\rm PU}(n,1)$, if $\{G_{j}\}$ has uniformly bounded torsion, then its geometric limit is either elementary, or discrete and non-elementary.
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