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On sequences of finitely generated discrete groups. (English) Zbl 1234.20060
Bonk, Mario (ed.) et al., In the tradition of Ahlfors-Bers, V. Proceedings of the 4th triennial Ahlfors-Bers colloquium, Newark, NJ, USA, May 8–11, 2008. Providence, RI: American Mathematical Society (AMS) (ISBN 978-0-8218-4732-9/pbk). Contemporary Mathematics 510, 165-184 (2010).
Summary: We consider sequences of discrete subgroups ${{\Gamma }}_{i}={\rho }_{i}\left({\Gamma }\right)$ of a rank 1 Lie group $G$, with ${\Gamma }$ finitely generated. We show that, for algebraically convergent sequences $\left({{\Gamma }}_{i}\right)$, unless ${{\Gamma }}_{i}$’s are (eventually) elementary or contain normal finite subgroups of arbitrarily high order, their algebraic limit is a discrete nonelementary subgroup of $G$. In the case of divergent sequences $\left({{\Gamma }}_{i}\right)$ we show that the resulting action ${\Gamma }⤻T$ on a real tree satisfies certain semistability condition, which generalizes the notion of stability introduced by Rips. We then verify that the group ${\Gamma }$ splits as an amalgam or HNN extension of finitely generated groups, so that the edge group has an amenable image in $\text{Isom}\left(T\right)$.
##### MSC:
 20H15 Other geometric groups, including crystallographic groups 22E40 Discrete subgroups of Lie groups 20E08 Groups acting on trees 20E06 Free products and generalizations (group theory)